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Acoustic Impedance Testing for Aeroacoustic Applications

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PAGE 1

ACOUSTIC IMPEDANCE TESTING FO R AEROACOUSTIC APPLICATIONS By TODD SCHULTZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Todd Schultz

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iii ACKNOWLEDGMENTS Financial support for the research pr oject was provided by a NASA-Langley Research Center Grant (Grant # NAG-1-2261). I thank Mr. Michael Jones and Mr. Tony Parrot at the NASA-Langley Rese arch Center for their guidance and support. I also thank the University of Florida, Department of Mechanical and Aerospace Engineering, the NASA Graduate Student Research Progr am Fellowship, and the National Defense Science and Engineering Graduate Fellowship administered by the American Society for Engineering Education for their financial support. My advisors, Mark Sheplak, Lou Cattaf esta, Toshi Nishida, and Tony Schmitz, deserve special thanks. I tha nk all of the students in the Interdisciplinary Microsystems Group, particularly Steve Horowitz, Fei Li u, Ryan Holman, Tai-An Chen and David Martin, for their assistance and friendship. I thank Paul Hubner for his help with data acquisition systems and LabVIEW programming. I would also like to thank the staff at the University of Florida, including Pa m Simon, Becky Hoover, Jan Machnik, Mark Riedy and Teresa Mathia, for helping with the administrative aspects of this project. I also want to thank Ken Reed at TMR Engineering for machining the equipment needed to make this project possible.

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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.............................................................................................................x ABSTRACT...................................................................................................................xvii i CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Research Goals.....................................................................................................10 1.2 Research Contributions.........................................................................................12 1.3 Dissertation Organization.....................................................................................12 2 ACOUSTIC WAVEGUIDE THEORY......................................................................13 2.1 Waveguide Acoustics...........................................................................................13 2.1.1 Solution to the Wave Equation............................................................14 2.1.2 Wave Modes........................................................................................16 2.1.3 Phase Speed.........................................................................................18 2.1.4 Wave Mode Attenuation.............................................................................22 2.1.5 Reflection Coefficient a nd Acoustic Impedance........................................26 2.2 Two-Microphone Method.....................................................................................28 2.2.1 Derivation of the TMM..............................................................................29 2.2.2 Dissipation and Dispersion for Plane Waves.............................................31 3 UNCERTAINTY ANALYSIS FOR THE TWO-MICROPHONE METHOD..........36 3.1 Multivariate Form of the TMM Data Reduction Equations.................................37 3.2 TMM Uncertainty Analysis..................................................................................38 3.2.1 Multivariate Uncertainty Analysis.............................................................39 3.2.2 Monte Carlo Method..................................................................................41 3.2.3 Frequency Response Function Estimate.....................................................41 3.2.4 Microphone Locations................................................................................45 3.2.5 Temperature................................................................................................45 3.2.6 Normalized Acoustic Impedance Uncertainty............................................48

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v 3.3 Numerical Simulations.........................................................................................48 3.3.1 Sound-Hard Sample....................................................................................50 3.3.2 Ideal Impedance Model..............................................................................53 3.4 Experimental Methodology..................................................................................64 3.4.1 Waveguides................................................................................................64 3.4.2 Equipment Description...............................................................................66 3.4.3 Signal Processing........................................................................................66 3.4.4 Procedure....................................................................................................66 4 MODAL DECOMPOSITION METHOD..................................................................68 4.1 Data Reduction Algorithm....................................................................................70 4.1.1 Complex Modal Amplitudes......................................................................71 4.2.2 Reflection Coefficient Matrix.....................................................................73 4.2.3 Acoustic Impedance...................................................................................74 4.2.4 Acoustic Power...........................................................................................75 4.2 Experimental Methodology..................................................................................77 4.2.1 Waveguide..................................................................................................77 4.2.2 Equipment Description...............................................................................78 4.2.3 Signal Processing........................................................................................80 4.2.4 Numerical Study of Uncertainties..............................................................80 5 EXPERIMENTAL RESULTS FOR AC OUSTIC IMPEDANCE SPECIMENS......82 5.1 Ceramic Tubular Honeycomb with 65% Porosity................................................83 5.1.1 TMM Results..............................................................................................84 5.1.2 High Frequency TMM Results...................................................................87 5.1.3 MDM Results and Comparison..................................................................89 5.2 Ceramic Tubular Honeycomb with 73% Porosity................................................97 5.2.1 TMM Results..............................................................................................98 5.2.2 High Frequency TMM Results.................................................................100 5.2.3 MDM Results and Comparison................................................................101 5.3 Rigid Termination...............................................................................................109 5.3.1 TMM Results............................................................................................109 5.3.2 High Frequency TMM Results.................................................................111 5.3.3 MDM Results and Comparison................................................................113 5.4 SDOF Liner........................................................................................................120 5.4.1 TMM Results............................................................................................121 5.4.2 MDM Results and Comparison................................................................123 5.5 Mode Scattering Specimen.................................................................................130 5.5.1 MDM Results...........................................................................................130 6 CONCLUSIONS AND FUTURE WORK...............................................................138 6.1 TMM Uncertainty Analysis................................................................................138 6.2 Modal Decomposition Method...........................................................................140

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vi APPENDIX A VISCOTHERMAL LOSSES....................................................................................143 A.1 Nondimensionalization and Lineariza tion of the Navier-Stokes Equations......144 A.1.1 Continuity................................................................................................147 A.1.2 x -direction Momentum Equation...........................................................147 A.1.3 y-direction Momentum Equation...........................................................148 A.1.4 Thermal Energy Equation........................................................................150 A.1.5 Equation of State for an Ideal Gas...........................................................151 A.1.6 Summary of the Nondimensi onal, Linearized Equations........................152 A.2 Boundary Layer Solution...................................................................................153 A.2.1 Wall Shear Stress.....................................................................................157 A.2.2 Wall Heat Flux.........................................................................................158 A.3 Mainstream Flow...............................................................................................158 A.3.1 Axial Momentum.....................................................................................159 A.3.2 Energy Equation......................................................................................161 A.3.3 Summary of the Mainstream Flow Equations.........................................166 A.3.4 Mainstream Flow Wave Equation...........................................................167 A.3.5 Dissipation and Dispersion Relations......................................................168 B RANDOM UNCERTAINTY ESTIMATES FOR THE FREQUENCY RESPONSE FUNCTION..............................................................................................................173 B.1 Introduction........................................................................................................174 B.2 Uncertainty Analysis..........................................................................................175 B.2.1 Classical Uncertainty Analysis................................................................176 B.2.2 Multivariate Uncertainty Analysis...........................................................177 B.2.2.1 Fundamentals.................................................................................178 B.2.2.2 Multivariate uncertainty propagation............................................181 B.2.2.3 Application: Converting uncertain ty from real and imaginary parts to magnitude and phase.........................................................................182 B.3 Frequency Response Function Estimates...........................................................185 B.3.1 Output Noise Only System Model...........................................................186 B.3.2 Uncorrelated Input/Output Noise System Model....................................191 B.4 Application: Measurement of the FRF Between Two Microphones in a Waveguide...........................................................................................................200 B.5 Conclusions........................................................................................................206 C FREQUENCY RESPONSE FUNCTION BIAS UNCERTAINTY ESTIMATES..208 C.1 Bias Uncertainty.................................................................................................208 C.2 Conclusions........................................................................................................214 D SOUND POWER FOR WAVES PR OPAGATING IN A WAVEGUIDE..............215

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vii E MODAL DECOMPOSITION METHOD NUMERICAL ERROR STUDY..........219 E.1 Signal-to-Noise Ratio.........................................................................................222 E.2 Microphone Phase Mismatch.............................................................................222 E.3 Microphone Locations........................................................................................226 E.4 Speed of Sound...................................................................................................228 E.5 Frequency...........................................................................................................229 E.6 Conclusions........................................................................................................231 F AUXILIARY GRAPHS...........................................................................................233 F.1 CT65...................................................................................................................233 F.2 CT73...................................................................................................................235 F.3 Rigid Termination..............................................................................................237 F.4 SDOF Liner........................................................................................................239 G COMPUTER CODES..............................................................................................241 G.1 TMM Program Files..........................................................................................241 G.1.1 TMM Program Readme File....................................................................241 G.1.2 Pulse to MATLAB Conversion Program................................................242 G.1.3 TMM Main Program...............................................................................244 G.1.4 TMM Subroutine Program......................................................................253 G.1.5 TMM Subroutine for the Analytical Uncertainty in R............................255 G.1.6 TMM Subroutine for the Analytical Uncertainty in Z............................257 G.1.7 TMM Subroutine for the Monte Carlo Uncertainty Estimates................258 G.2 Uncertainty Subroutines....................................................................................259 G.2.1 Frequency Response Function Uncertainty.............................................259 G.2.2 Averaged FRF Uncertainty......................................................................261 G.2.3 Effective Number of degrees of Freedom...............................................262 G.2.4 Numeric Computation of Bivariate Confidence Regions........................263 G.2.5 Analytical Propagation of Uncertain ty from Rectangular Form to Polar Form...............................................................................................................265 G.3 Multivariate Statistics Subroutines....................................................................266 G.3.1 Computation of Bivariate PDF................................................................266 G.3.2 Numerical Computation of Constant PDF Contours...............................267 G.3.3 Multivariate Normal Ra ndom Number Generator...................................268 G.4 Modal Decomposition Programs.......................................................................268 G.4.1 Pulse to MATLAB Conversion Program................................................268 G.4.2 MDM Main Program...............................................................................273 G.4.3 MDM Subroutine to Compute the Decomposition..................................281 G.4.4 MDM Plotting Subroutine.......................................................................284 G.4.5 MDM Mode Scattering Coefficients Plotting Subroutine.......................290 LIST OF REFERENCES.................................................................................................294

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viii BIOGRAPHICAL SKETCH...........................................................................................300

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ix LIST OF TABLES Table page 2-1 Cut-on frequencies in kHz for an 8.5 mm by 8.5 mm waveguide...........................17 2-2 Cut-on frequencies in kHz fo r a 25.4 mm by 25.4 mm waveguide.........................18 2-3 Minimum frequencies to keep effect s of dispersion and dissipation <5%...............35 3-1 Elemental bias and precision error sources for the TMM........................................43 3-2 Nominal values for input para meters of numeric simulations.................................50 4-1 Cut-on frequencies in kHz for the higher-order modes............................................78 4-2 Microphone measurement locations ( a = 25.4 mm)................................................79 A-1 Minimum frequency required for series expansion for the two waveguides for air at 20 C........................................................................................................................170 C-1 Simulation results for the second-order system.....................................................212 E-1 Power in Pa2 for all signals from all simulation sources........................................220

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x LIST OF FIGURES Figure page 1-1 Illustration of the approach, takeoff, a nd cutback flight segments and measurement points......................................................................................................................... .2 1-2 Typical noise sources on an aircraft...........................................................................4 1-3 Component noise levels during approach, cutback, and take-off for a Boeing 767300 with GEAE CF6-80C2 engines...........................................................................4 1-4 Comparative overall noise leve ls of various engine types.........................................5 1-5 Engine cutaway showing the acoustic liner locations................................................7 1-6 An example of a SDOF liner showing the atypical honeycomb and the perforate face sheet....................................................................................................................7 1-7 An example of a 2DOF liner......................................................................................8 2-1 Illustration of the wave guide coordinate system......................................................14 2-2 Illustration of the fi rst four mode shapes.................................................................18 2-3 Illustration of the wave front and the in cidence angle to the waveguide wall and to the termination..........................................................................................................20 2-4 Phase speed versus frequenc y for the first four modes............................................21 2-5 Angle of incidence to the sidewall ve rsus frequency for the first four modes.........21 2-6 Angle of incidence to the termination ve rsus frequency for the first four modes....22 2-7 Attenuation of higher-order modes in th e large waveguide over a distance of 25.4 mm............................................................................................................................2 4 2-8 Attenuation of higher-order modes in th e small waveguide over a distance of 8.5 mm............................................................................................................................2 4 2-9 Attenuation of the first hi gher-order mode ((1,0) or (0,1) ) in the large waveguide at the microphone locations used for the TMM experiments......................................26

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xi 2-10 Reflection and transmission of a wave off an impedance boundary........................28 2-11 Experimental setup for the TMM.............................................................................29 3-1 Flow chart for the Monte Carlo methods.................................................................51 3-2 Absolute uncertainty of 00,00R due to the uncertainties in l, s, and T for 00,000.999 R at f =5 kHz..........................................................................................52 3-3 Absolute uncertainty 00,00R due to the SNR for 00,000.999 R at f =5 kHz..............53 3-4 Estimated value for the (a) reflection co efficient and (b) total uncertainty for the sound-hard boundary................................................................................................54 3-5 Ideal impedance model and estimated valu es for (a) reflection coefficient and (b) normalized specific acoustic impedance..................................................................56 3-6 Absolute uncertainty of (a) 00,00R and (b) s pac due to the uncertainties in l s and T for the ideal impedance model at f =5 kHz...............................................................57 3-7 Absolute uncertainty in (a) 00,00R (b) s pac due to the SNR for the ideal impedance model at f =5 kHz......................................................................................................58 3-8 Total uncertainty in (a) 00,00R and (b) s pac as a function of frequency for the ideal impedance model......................................................................................................60 3-9 The confidence region contours for the resistance and reactance for the ideal impedance model at 5 kHz.......................................................................................62 3-10 Confidence region of the ideal impedance model at 5 kHz with 1% relative input uncertainty and 40 dB SN R......................................................................................63 4-1 Schematic of the experimental setup for the MDM.................................................78 4-2 Schematic of the four restrictor plates......................................................................79 5-1 Photograph of the CT65 material.............................................................................83 5-2 Reflection coefficient for CT65 for the TMM.........................................................86 5-3 Normalized specific acoustic impe dance estimates for CT65 via TMM.................86 5-4 Reflection coefficient for CT65 for the high frequency TMM................................88 5-5 Normalized specific acoustic impedan ce estimates for CT65 via the high frequency TMM........................................................................................................................89

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xii 5-6 Incident pressure field for the MDM for CT65........................................................90 5-7 Reflected pressure fiel d for the MDM for CT65......................................................90 5-8 Absorption coefficient for CT65..............................................................................91 5-9 Comparison of the reflection coefficient estimates for CT65 via all three methods.92 5-10 Mode scattering coefficients for CT65 fr om the (0,0) mode to the other propagating modes.......................................................................................................................93 5-11 Mode scattering coefficients for CT65 fr om the (1,0) mode to the other propagating modes.......................................................................................................................93 5-12 Mode scattering coefficients for CT65 fr om the (0,1) mode to the other propagating modes.......................................................................................................................94 5-13 Mode scattering coefficients for CT65 fr om the (1,1) mode to the other propagating modes.......................................................................................................................94 5-14 Comparison of the acoustic impedance ratio estimates for CT65 via all three methods....................................................................................................................96 5-15 Comparison of the normalized specific acoustic impedance estimates for CT65 via all three methods......................................................................................................97 5-16 Photograph of the CT73 material.............................................................................98 5-17 Reflection coefficient for CT73 for the TMM.........................................................99 5-18 Normalized specific acoustic impeda nce estimates for CT73 via the TMM...........99 5-19 Reflection coefficient for CT73 for the high frequency TMM..............................100 5-20 Normalized specific acoustic impedan ce estimates for CT73 via the high frequency TMM......................................................................................................................101 5-21 Incident pressure field for the MDM for CT73......................................................102 5-22 Reflected pressure fiel d for the MDM for CT73....................................................103 5-23 Absorption coefficient for CT73............................................................................103 5-24 Comparison of the reflection coefficient estimates for CT73 via all three methods.104 5-25 Mode scattering coefficients for CT73 fr om the (0,0) mode to the other propagating modes.....................................................................................................................105

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xiii 5-26 Mode scattering coefficients for CT73 fr om the (1,0) mode to the other propagating modes.....................................................................................................................105 5-27 Mode scattering coefficients for CT73 fr om the (0,1) mode to the other propagating modes.....................................................................................................................106 5-28 Mode scattering coefficients for CT73 fr om the (1,1) mode to the other propagating modes.....................................................................................................................106 5-29 Comparison of the acoustic impedance ratio estimates for CT73 via all three methods..................................................................................................................108 5-30 Comparison of the normalized specific acoustic impedance estimates for CT73 via all three methods....................................................................................................108 5-31 Photograph of the rigid termin ation for the large waveguide................................109 5-32 Reflection coefficient for the rigid termination for the TMM...............................110 5-33 Standing wave ratio for the rigid termination measured by the TMM...................111 5-34 Reflection coefficient fo r the rigid termination for the high frequency TMM......112 5-35 SWR for the rigid termination calcul ated from the high frequency TMM............112 5-36 Triangle restrictor plate..........................................................................................114 5-37 Incident pressure field for the MDM for the rigid termination..............................114 5-38 Reflected pressure field for th e MDM for the rigid termination............................115 5-39 Power absorption coefficient for th e rigid termination for the MDM....................115 5-40 Comparison of the reflection coefficient estimates for the rigid termination via all three methods.........................................................................................................117 5-41 SWR for the rigid termination calculated from the MDM.....................................117 5-42 Mode scattering coefficients for rigid te rmination from the (0,0) mode to the other propagating modes.................................................................................................118 5-43 Mode scattering coefficients for rigid te rmination from the (1,0) mode to the other propagating modes.................................................................................................118 5-44 Mode scattering coefficients for rigid te rmination from the (0,1) mode to the other propagating modes.................................................................................................119 5-45 Mode scattering coefficients for rigid te rmination from the (1,1) mode to the other propagating modes.................................................................................................119

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xiv 5-46 SDOF liner showing the irregular ho neycomb and perforated face sheet.............120 5-47 Reflection coefficient for the SDOF specimen for the TMM................................122 5-48 Normalized specific acoustic impeda nce estimates for the SDOF specimen via TMM......................................................................................................................122 5-49 Incident pressure field for the MDM for the SDOF specimen...............................124 5-50 Reflected pressure field for the MDM for the SDOF specimen............................124 5-51 Power absorption coefficient for the SDOF specimen for the MDM....................126 5-52 Comparison of the reflection coefficien t estimates for the SDOF specimen via the TMM and MDM.....................................................................................................126 5-53 Mode scattering coefficients for SDOF specimen from the (0,0) mode to the other propagating modes.................................................................................................127 5-54 Mode scattering coefficients for SDOF specimen from the (1,0) mode to the other propagating modes.................................................................................................127 5-55 Mode scattering coefficients for SDOF specimen from the (0,1) mode to the other propagating modes.................................................................................................128 5-56 Mode scattering coefficients for SDOF specimen from the (1,1) mode to the other propagating modes.................................................................................................128 5-57 Comparison of the acoustic impedance ratio estimates for the SDOF specimen via the TMM and MDM...............................................................................................129 5-58 Comparison of the normalized specific acoustic impedance estimates for the SDOF specimen via the TMM and MDM.........................................................................129 5-59 Photograph of the mode scattering specimen.........................................................130 5-60 Incident pressure field for the MD M for the mode scattering specimen................132 5-61 Reflected pressure field for the MD M for the mode scattering specimen.............132 5-62 Power absorption coefficient for the m ode scattering specimen for the MDM.....133 5-63 Comparison of the reflection coefficient estimates for the mode scattering specimen via the MDM..........................................................................................................134 5-64 Mode scattering coefficien ts for the mode scattering specimen from the (0,0) mode to the other propagating modes..............................................................................135

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xv 5-65 Mode scattering coefficien ts for the mode scattering specimen from the (1,0) mode to the other propagating modes..............................................................................135 5-66 Mode scattering coefficien ts for the mode scattering specimen from the (0,1) mode to the other propagating modes..............................................................................136 5-67 Mode scattering coefficien ts for the mode scattering specimen from the (1,1) mode to the other propagating modes..............................................................................136 5-68 Comparison of the acoustic impedance ratio estimates for the mode scattering specimen via the MDM..........................................................................................137 5-69 Comparison of the normalized specific acoustic impedance estimates for the mode scattering specimen via the MDM.........................................................................137 A-1 Oscillating flow over a flat plate............................................................................145 A-2 Control volume showing the external fo rces and flows crossing the boundaries..161 A-3 Control volume showing the external heat fluxes and flows crossing the boundaries..............................................................................................................166 B-1 A plot of the raw data and estimates for a randomly generated complex variable..183 B-2 A plot of the raw data and estimates in polar form for a randomly generated complex variable....................................................................................................185 B-3 System model with output noise only....................................................................187 B-4 System model with uncorrelated input/output noise..............................................191 B-5 Bode plot of the true FRF and the experimental estimate......................................199 B-6 Magnitude and phase plot of the uncertainty estimates.........................................199 B-7 The experimentally measured FRF between the two microphones.......................205 B-8 Comparison for the uncertainty estimat ed by the multivariate method and by the direct statistics........................................................................................................205 C-1 FRF for the simulation with random noise.............................................................213 E-1 The rms normalized error for the modal coefficients versus noise power added to the signals...............................................................................................................223 E-2 The rms normalized error for the reflecti on coefficient matrix versus noise power.224 E-3 The rms normalized error versus the numb er of averages for a noise power of 0.01 Pa2...........................................................................................................................224

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xvi E-4 The rms normalized error for the refl ection coefficient versus the number of averages for a noise power of 0.01 Pa2..................................................................225 E-5 The rms normalized error for the modal coe fficients versus a phase error applied to microphone 4 in group 2 for each source...............................................................225 E-6 The rms normalized error for the reflecti on coefficient matrix versus a phase error applied to microphone 4 in group 2 for all sources................................................226 E-7 The rms normalized error for the modal coefficients versus a microphone location error applied to microphone 1 in group 1 for each source.....................................227 E-8 The rms normalized error for the reflec tion coefficient matrix versus a microphone location error applied to microphone 1 in group 1 for all sources.........................227 E-9 The rms normalized error for the modal coe fficients versus a te mperature error. 228 E-10 The rms normalized error for the reflecti on coefficient matrix versus a temperature error........................................................................................................................22 9 E-11 The rms normalized error for the modal coefficients versus frequency................230 E-12 The rms normalized error for the reflecti on coefficient matrix versus frequency.230 F-1 Ordinary coherence function for the TMM measurement of CT65.......................233 F-2 The measured FRF fo r CT65 for the TMM...........................................................234 F-3 Ordinary coherence function for the high frequency TMM measurement of CT65.234 F-4 The measured FRF for CT65 for the high frequency TMM..................................235 F-5 Ordinary coherence function for the TMM measurement of CT73.......................235 F-6 The measured FRF fo r CT73 for the TMM...........................................................236 F-7 Ordinary coherence function for the high frequency TMM measurement of CT73.236 F-8 The measured FRF for CT73 for the high frequency TMM..................................237 F-9 Ordinary coherence function be tween the two microphones for the TMM measurement of the rigid termination....................................................................237 F-10 The measured FRF for the rigid termination for the TMM....................................238 F-11 Ordinary coherence function for the hi gh frequency TMM measurement of the rigid termination.............................................................................................................238 F-12 The measured FRF for the rigid termination for the high frequency TMM...........239

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xvii F-13 Ordinary coherence function for the TMM measurement of the SDOF specimen.239 F-14 The measured FRF for the SDOF specimen for the TMM....................................240

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xviii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ACOUSTIC IMPEDANCE TESTING FOR AEROACOUSTIC APPLICATIONS By Todd Schultz August 2006 Chair: Mark Sheplak Cochair: Louis N. Cattafesta Major Department: Mechanic al and Aerospace Engineering Accurate acoustic propagation models are required to characterize and subsequently reduce aircraft engine noise. These models ultimately rely on acoustic impedance measurements of candidate materi als used in sound-ab sorbing liners. The standard two-microphone method (TMM) is wide ly used to estimate acoustic impedance but is limited in frequency range and does not provide uncertainty estimates, which are essential for data quality assessment and model validation. This dissertation presents a systematic framework to estimate uncertainty and extend the frequency range of acoustic impedance testing. Uncertainty estimation for acoustic impeda nce data using the TMM is made via two methods. The first employs a standard analytical technique based on linear perturbations and provides usef ul scaling information. Th e second uses a Monte Carlo technique that permits the pr opagation of arbitrar ily large uncertainties. Both methods are applied to the TMM for simulated data representative of sound-hard and sound-soft

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xix acoustic materials. The results indicate that the analytical techni que can lead to false conclusions about the magnitude and importance of specific error sources. Furthermore, the uncertainty in acoustic impedance is st rongly dependent on the frequency and the uncertainty in the microphone locations. Next, an increased frequency range of ac oustic impedance testing is investigated via two methods. The first method reduces th e size of the test sp ecimen (from 25.4 mm square to 8.5 mm square) and uses the standard TMM. This method has issues concerning specimen nonuniformity because the small specimens may not be representative of the materi al. The second method increase s the duct cross section and, hence, the required complexi ty of the sound field propaga tion model. A comparison among all three methods is conducted for each of the three specimens: two different ceramic tubular specimens and a single degreeof-freedom liner. The results show good agreement between the TMM and the moda l decomposition method for the larger specimens, but the methods disagree for the sm aller specimen size. The results for the two ceramic tubular materials show a repe ating resonant pattern with a monotonic decrease in the resonant peak s of the acoustic resistance with increasing frequency. Also, significant mode scattering is evident in most of the specimens tested.

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1 CHAPTER 1 INTRODUCTION Modern society has increasingly dema nded a safer, more pleasant living environment. Studies have also shown th at exposure to nois e pollution has adverse health effects such as hearing impairment, reduced speech perception, sleep deprivation, increased stress levels, and general annoyance (Berglund et al. 1999). These results have led to increased noise restric tions on industrial factories, au tomobiles, and aircraft. The common element of these sources is that th ey all produce a complex noise spectrum and broad bandwidth. Yet increased noise restrictions have been readily met because of an increased research effort in acoustics, involving sound generation, propagation, and suppression (Golub et al. 2005). The aerospace industry has been a major fo cus for increased noise regulations due to community noise concerns around commerc ial airports (Motsi nger and Kraft 1991; Berglund, Lindvall and Schwela 1999). Take off, landing and cutback are the flight segments of greatest relevan ce to community noise concerns because of the relative proximity of the aircraft to the community. These flight segments and reference measurement points are shown in Figure 1-1. During these flight segments, the configuration of the aircraft is altered from the clean cruise configuration via the deployment of high-lift devices and landing gear. Also, during take-off the engines operate at full power, further increasing the noi se levels. In order to reduce the fly-over noise, the power to the engines is reduced after take-off, during climb or cutback when the aircraft is still relativel y close to the ground near populat ed areas. The overall noise

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2 level from the aircraft has contributions fr om many sources as shown in Figure 1-2. These contributions can be separated into two broad categories: airframe noise and engine noise. Examples of airframe noise include noise generated from flaps, slats, landing gear, and vertical and horizontal tails. Engine noise consists of jet noise from the exhaust, combustion nois e, turbomachinery noise, and the noise due to the integration of the engines with the airframe. The effectiv e perceived noise levels (EPNL) (Smith 1989) of the component noise sources for the three flight segments are shown in Figure 1-3 for a Boeing 767-300 with GEAE CF6-80C2 engine s. The figure shows that the major contributors to the total aircra ft noise are jet and fan noise for takeoff, jet noise for cutback, and inlet, fan and airframe noise for approach. Thus, to reduce aircraft noise for takeoff and cutback, engine noise should be reduced, whereas airframe noise must be considered for the approach flight segment. 120m Figure 1-1: Illustration of the approach, ta keoff, and cutback flight segments and measurement points (adapted from Smith 1989). Early commercial aircraft used turbojet engines, and the resulting noise was dominated by the jet noise component. With the advent of ultra high bypass turbofan engines, the dominant noise sources for mode rn commercial jet aircraft are now engine

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3 noise (during takeoff) and airframe noise (during approach). Because it was the dominant noise source in earlier jet aircraft, jet noise has been studied for many decades. Lighthill’s analogy can be used to understand the scaling issues in the evolution of the jet engine noise from the first turbojet engines to modern high-bypass-ratio turbofan engines. Lighthill’s analogy relates the mean square va lue of the radiated density perturbations ( 2 ) from a subsonic turbulent jet to the velocity and diameter of the jet as (Dowling and Ffowcs-Williams 1983) 2 228 0 2~, D M r (1.1) where 0 is the mean or atmospheric density, M is the exit Mach number of the jet, D is the diameter of the jet and r is the distance from the jet. Equation (1.1) is only valid for subsonic flows and shows that the magn itude of the sound from a jet is more dependent on the velocity of the jet than the si ze of the jet. In particular, the mean square density perturbations are proporti onal to the eighth power of the Mach number but only to the second power of the diameter. A comp arative chart of the perceived noise levels from various engine types is gi ven in Figure 1-4. The first ge neration of jet aircraft relied on propulsion from a single, high velocity je t from the aft of the engine (Rolls-Royce 1996). This generated a tremendous amount of no ise, as seen from Lighthill’s analogy. Subsequently, noise suppression devices for these engines have been developed to reduce the noise generated from the jets. Th e devices included s uppressor nozzles that promote rapid mixing of the exhaust jet wi th the ambient fluid (Owens 1979). The overall effect is to quickly reduce the velocity of the jet and thus re duce the noise levels. The next development was the low-bypass-ratio turbofan engine (Rolls-Royce 1996).

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4 Engines Main landing gear Slats Flap/side-edge vortices Vertical tail Horizontal tail Tip vortices Nose landing gear Figure 1-2: Typical noise s ources on an aircraft (a dapted from Crighton 1991). 60 65 70 75 80 85 90 95 100 Total Aircraft Total Airframe Jet Combustor Aft fan Inlet EPNL [dB] Approach Cutback Sideline Figure 1-3: Component noise levels during approach, cu tback, and take-off for a Boeing 767-300 with GEAE CF6-80C2 engines (adapted from Golub et al. 2005). For this engine, the majority of the propulsi on force is generated by the bypass flow with an increased area, and at th e exit the bypass air is mixed with the jet, significantly lowering the exit Mach number. The diameter of the engine and jet was enlarged, but noise levels were significantly reduced because of the lower-velocity jet from the fan and the mixing of the two jets. Modern turbofan engines use high-bypass-ratio inlets, with a bypass ratio of approximately th ree or greater (Ro lls-Royce 1996). The diameter of the fans on these engines can thus be 2.5 m or larg er. The diameter of the exiting air flow is increased, the mixing is increased, and the je t velocity is decreased, thus lowering the

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5 propagated noise. The growth of the diameter from these large high-bypass turbofans is now restricted by problems related to the large weight and large frontal area, such as drag (Rolls-Royce 1996). 120 110 100 90 EPNL [dB] Turbojets without noise suppressors Turbojets with noise suppressors Low bypass ratio turbofans with noise suppresors High bypass ratio turbofans with noise suppressors Overall trend years Figure 1-4: Comparative overall noi se levels of various engine types (adapted from RollsRoyce 1996). Since the use of high-bypass-ra tio turbofans has reduced th e perceived noise levels from the jet by approximately 20 dB, othe r noise sources have become important contributors to the overall noise level of the aircra ft (Smith 1989). Two such sources are engine noise (other than je t noise) and airframe noise. To reduce the engine noise, designs have focused on acoustic treatments to the interior of the e ngine nacelles to alter the propagation of the sound and reduce the radi ation of noise from the engine into the far-field (Motsinger and Kraft 1991). These na celle liners are placed at various locations throughout the engine to suppress noise from a particular region, as shown in Figure 1-5. The liners minimize the radiation of sound by altering the acoustic impedance boundary condition along the walls of the nacelle. The acoustic impedance, which is the complex

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6 ratio of the acoustic pressure to the acousti c volume velocity, is a property of the liner configuration and materials. As shown in Figure 1-6, typical si ngle degree-of-freedom (SDOF) liners are a composite structure of a layer of honeycomb support sandwiched between a solid backing sheet a nd a perforated face sheet. Th ese liners act as Helmholtz resonators and are used to attenuate the noise spectrum (Motsinger and Kraft 1991; RollsRoyce 1996). The bandwidth over which a SDOF liner is effective is about one octave, centered around its resonant frequency (Motsi nger and Kraft 1991). If the liner has two layers of honeycomb separated by a second pe rforate face sheet, the liner is called a two degree-of-freedom (2DOF) liner as shown in Figure 1-7. The 2DOF liner has two resonant frequencies and a larg er bandwidth, about two octaves, of effectiveness relative to the SDOF liner, but weigh more than SDOF liner (Motsinger and Kraft 1991; Bielak et al. 1999). Another type of lin er uses a bulk absorber, whic h is designed to attenuate sound over a broad bandwidth. These liners ar e less effective at reducing the propagation of engine at a given frequenc y as compared to the SDOF or 2DOF liners, and usually are not able to provide struct ural support (Motsinger and Kraft 1991; Bielak, Premo and Hersh 1999). Typical materials used for bulk absorbers include woven wire mesh, ceramic tubular materials, and acoustic foams and fibers such as polyurethane, melamine, fiberglass, etc. When designing an engine nacelle for noise suppression, semi-empirical analytical models can be used to find the optimal acoustic impedance for acoustic treatment (Motsinger and Kraft 1991). Potential liner cand idates must be experimentally tested to determine their acoustic impedance. The experimentally measured values can then be used in new models to predict the noise le vels from the engine for that particular

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7 configuration. Scale model and full-size engi ne testing can be done to verify the noise level predictions and to certify the engine. Nacelle Fan Inlet Acoustic Liners Exhaust Stators Wing Mount Struts Figure 1-5: Engine cutaway showing the acoustic liner lo cations (adapted from Groeneweg et al. 1991; Rolls-Royce 1996). Figure 1-6: An example of a SDOF liner showing the atypical honeycomb and the perforate face sheet (courtesy of Pratt and Whitney Aircraft). One of the limiting factors for the computational models is the experimental database for the acoustic properties of any mate rial used for noise control (Kraft et al. 1999; Kraft et al. 2003). Curre nt applications re quire extending the frequency range of Honeycomb layer Perforate face sheet Solid back sheet

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8 acoustic impedance testing out to 20 kHz to accommodate 1/5th-scale aeroacoustic testing (Kraft, Yu, Kwan, Echternach, Syed and Ch ien 1999; Kraft, Yu, Kwan, Beer, Seybert and Tathavadekar 2003). Existing methods for measuring normal-incident acoustic impedance have their limitations. Of th ese, the most noticeable limitation is the frequency range within which the methods are valid. Therefore, existing sound propagation models must extrapolate the acous tic impedance to the frequency range of interest for applications. This introduces a potentially large source of error in the models. Better results could be realized if the actual acoustic impedance of th e materials could be measured in the frequency range of interest. Figure 1-7: An example of a 2DOF liner (adapted from Rolls-Royce 1996). The Two-Microphone Method (TMM) (Seybe rt and Ross 1977; Chung and Blaser 1980; ASTM-E1050-98 1998; ISO-10534-2:1998 1998) and the Multi-Point Method (MPM) (Jones and Parrott 1989; Jones and Stie de 1997) are two techniques to determine the normal-incidence acoustic impedance of materials. For the TMM, a compression driver is mounted at one end of a rigidwalled waveguide and the test specimen is mounted at the other end (Fi gure 2-11). Two microphones ar e flush-mounted in the duct Honeycomb layers Perforate face sheets Solidbacksheet

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9 wall at two locations along the tube near the specimen to measure the incident and reflected waves with respect to the sample. The data are used to estimate the complex reflection coefficient and the corresponding aco ustic impedance of the test specimen. More detailed information on the TMM is presented later in Chapter 2. The test procedure for the MPM is similar to th e TMM except the numb er of microphones is increased and the computations rely on a l east-squares approach. However, the MPM still assumes that only plane waves exist in the waveguide. Since the TMM is supported by ISO and ASTM standards, it is the method used in this dissertation. Both methods have produced results for materials up to a frequency of approximately 12 kHz. The TMM or the MP M can, in theory, be extended to higher frequencies. However, in order to do, the sp ecimen size of the materi al would need to be reduced to maintain the plane wave assump tion, since the upper frequency limit of the method is inversely proportional to the sp ecimen size or waveguide dimensions (ASTME1050-98 1998; ISO-10534-2:1998 1998). The size is limited in order to prevent the propagation of higher-order modes and thus ma intain the plane wave assumption. For square cross-section of lengtha, the maximum frequency for plane waves, planewavef, is (Blackstock 2000) 02planewavec f a (1.2) where 0c is the isentropic speed of sound inside the waveguide. A specimen of 25.4 mm by 25.4 mm is limited to a frequency range up to approximately 6.7 kHz in ambient air using the TMM, but a specimen of 8.5 mm by 8.5 mm has a frequency range up to 20 kHz. Unfortunately, a small specimen size resu lts in installation a nd fabrication issues and in local material variations that ca n cause changes in the measured acoustic

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10 impedance. The installation and fabrication issues arise from having to cut a finite specimen, often resulting in damage to its ed ges. Furthermore, the smaller the specimen size, the larger the percentage of the total ar ea composed of the damaged edges. For the local material variations, testing a larg e number of specimens can quantify these statistical variations. However, this a pproach is time consuming and costly. Another method to increase the frequenc y range is to permit the propagation of higher-order modes (bom 1989; Kraft et al. 2003). This allows large specimens but increases the complexity of the measurement setup and data reduction routine. For a 25.4 mm-square duct, the bandwidth is increased to 13.5 kHz if the first four modes propagate or to 20 kHz if the first nine modes propagate. The advantage to this modal decomposition method (MDM) is that the highe r-order modes can also be modeled as plane waves at oblique angles of inciden ce; thus this method can yield information regarding the effects of angle of incidence. The oblique-i ncidence information can used to verify the local reactivity assumption, which states that the acoustic impedance is independent of the angle of incidence (Dowling and Ffowcs-Williams 1983). Before the frequency range can be extende d, the accuracy of the existing methods must be understood. Accurate uncertainty es timates give insight into how errors scale versus frequency and will aid in the design of new measurement techniques and improved liners. Without understanding uncertain ty, there is no way to ensure that such measurements will meet the needs of the aeroacoustic application. 1.1 Research Goals The focus of this dissertation is to increase the frequenc y range of acoustic impedance measurement technology to the range of interest in aeroacoustic applications and to supply experimental uncertainty estimate s with the data. These data will help to

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11 evaluate potential liner candidates and improve the accuracy of models of the sound field. Also, design procedures and codes that pred ict the acoustic impedance of typical liners can be validated using the acoustic impe dance data with corresponding uncertainty estimates. The implementation of improved experimental techniques and corresponding uncertainty analyses with existing design a nd computational tools will assist in the reduction of time and cost required to meet community noise restrictions. To meet the goal of extending the fr equency range of acoustic impedance measuring technology, two different approaches are used. The first seeks to reduce the size of the cross-section of the waveguide and test specimen to increase the cut-on frequency for the first higher-ord er mode to 20 kHz, allowing the TMM to be used. This will limit the specimen size to 8.5 mm by 8.5 mm and thus the above mentioned specimen size issues may affect the results. Th e second approach is to keep the specimen at 25.4 mm by 25.4 mm and allow for the propagation of higher order modes. A direct Modal Decomposition Method (M DM) is used that allows for and computes the amplitudes of the incident and reflected wave s for the higher-order modes. This allows the frequency range to increas e. This method will provide a comparison for the data measured with the small specimen to elucidat e any issues associated with the specimen size. Also, acoustic impedance data at angles of incidence other than perpendicular to the specimen surface are measured, because higherorder modes can also be thought of as plane waves traveling at an angle with respect to the axis of the duct. Before either path is pursued, two techniques are first developed to estimate the uncertainty for the complex reflection coefficien t. One method is an analytical approach that provides scaling information, and the other is a Monte Carlo method that is not

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12 restricted to small perturbations. The two methods are compared to each other to help determine their strengths and weaknesses. 1.2 Research Contributions The contributions of this dissertation to th e aeroacoustic community are as follows. Development of an analytical and a numerical method for the propagation of experimental uncertainty in data reduction routines with complex variables. Application of the uncertain ty analysis methods to the Two-Microphone Method. Application of a Modal Decomposition Method for measuring normal-incident acoustic impedance in the presence of higher-order modes in the waveguide. Comparison of experimental data with uncertainty estimates for acoustic impedance from the TMM and the MDM. The specimens compared are a rigid termination and two ceramic tubular materials. 1.3 Dissertation Organization This dissertation is organized into six chapters. This chapter introduced and discussed the motivation for the research presen t in this dissertation. The next chapter reviews the theory of acoustic waveguides. The derivation of the TMM is presented there as well. Chapter 3 presents the de rivation and applicati on of the uncertainty methods for the TMM and includes a discussion of the issues present when increasing the bandwidth of the TMM up to 20 kHz. Chapter 4 introduces the MDM, a method that accounts for the propagation of higher-order mo des through the waveguide. This chapter presents the derivation of this method and a discussion of the requirements for the data acquisition hardware to ensure reasonable accuracy. Chapter 5 presents detailed experimental results for differe nt acoustic impedance specimens. The final chapter offers concluding remarks and future directions.

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13 CHAPTER 2 ACOUSTIC WAVEGUIDE THEORY This chapter introduces the basic analyti cal analysis for rectangular duct acoustic waveguides. First, the acousti c wave equation is presented, and its solution is given. Next, a discussion of the soluti on properties is presented. Then this chapter concludes with a derivation of the TMM. 2.1 Waveguide Acoustics A waveguide is a device that is used to contain and di rect the propagation of a wave. A simple example of an acoustic wave guide is a plastic t ube with a sound source at one end. For simple geometries of the in ternal cross section, the exact sound field in the waveguide can be solved from the linear lossless acoustic wave equation, as long as the wave equation assumptions are not vi olated (Pierce 1994; Blackstock 2000). The lossy wave equation can also be solved for some simple cases but an ad hoc method will be introduced in a later section in this ch apter to account for atte nuation. The linear lossless acoustic wave equation assumes that an acoustic wave is isentropic, the pressure perturbations are small compared to the medium’s bulk modulus (2 00c ), and that there is no mean flow of the medium. Under these conditions, the wave equation for pressure fluctuations is 2 2 22 01 0, p p ct (2.1)

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14 where p is the acoustic pressure, 0c is the isentropic propa gation speed given by 0 g ascRT, is the ratio of specific heats, g as R is the ideal gas constant, Tis the absolute temperature, t is time and 2 is the Laplacian operator. First, let the coordinate system for the waveguide be defined as a Cartesian coordinate system with the z-axis aligned with the axis of the tube and the origin located at a co rner of the tube as shown in Figure 2-1. The d-axis as s hown is a useful auxiliary coor dinate axis. Note that the ddirection is the direction along the axis of the tube and is the propagation direction for reflected waves. Figure 2-1: Illustration of the waveguide coordinate system. 2.1.1 Solution to the Wave Equation Solutions to the wave equation are presen t in many sources such as (Rayleigh 1945), (Dowling and Ffowcs-Williams 1983), (Pie rce 1994), (Morse and Ingard 1986), (Kinsler et al. 2000) and (Bl ackstock 2000). For the solution presented in this section, the acoustic pressure signal is assume d to be time-harmonic which states Re,jtpPe (2.2) where P is the complex acoustic pressure amplitude and is the angular frequency. After substituting Equation (2.2) into the acoustic wave equation, it reduces to 220, PkP (2.3)

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15 which is known as Helmholtz’s Equation (P ierce 1994; Blackstock 2000). The constant k is the wavenumber, which is defined as 0. k c (2.4) The general solution to the Helmholtz equation, assuming propagation in the d-direction, is a summation of normal modes given as ,zzjkdjkd mnmnmn mnPxyAeBe (2.5) where 1 j mnA and mn B are the complex modal amplitudes of the incident and reflected wave, respectively, m and n are the mode numbers, zk is the propagation constant, and x y is the transverse factor. The tr ansverse factor is a product of two eigenfunctions determined by the boundary conditions. For the waveguides used in this dissertation, the tube walls ar e assumed to be sound-hard or rigid and therefore do not vibrate or transmit sound. Practically, the sound-hard boundary conditio n can be realized for a gaseous medium by utilizing tube walls ma de of a thick, rigid material, such as steel or aluminum. The only boundary condition provided by a sound-hard boundary for an inviscid flow is that the particle velocity no rmal to the surface is zero at the walls. From the conservation of momentum (i.e. Euler’s Eq uation), this is represented as the normal gradient component of the acoustic pressure be ing equal to zero. Hence, the transverse factor for a rectangular duct with rigid walls is ,coscosmnmn x yxy ab (2.6) where a and b are the side lengths of the wave guide shown in Figure 2-1. The remaining constants, the complex modal amplitudes, mnA and mn B are determined by

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16 two boundary conditions. The first boundary co ndition is a given acoustic impedance at 0 d The second boundary condition would be a known pressure or velocity source at the other end of the waveguide at wgdL where wgL is the length of the waveguide. Applying the boundary conditions, Equation (2.5) can be solved for each mnA and mn B The zjkd mn A e terms represent waves traveling from the source to the other end of the tube. The z jkd mnBeterms represent the reflected wave s returning to the source after bouncing off the sample. The dispersion relation for the rectangul ar waveguide comes for the separation constant from applying a sepa ration of variables solution to Equation (2.3) and is 2 22 0 zmn k cab (2.7) and, for a normal mode to propagate, zk must contain a real-valued component. If zk has an imaginary component, there will be two solutions to Equation (2.7) that will be complex conjugates. For a waveguide, onl y the solution that causes the amplitude to exponentially decay is physically valid from c onservation of energy. This term will force the acoustic pressure amplitude to zero as th e axial distance increases in the direction of propagation, and the wave is deemed an evanescent wave. 2.1.2 Wave Modes The indices m and n represent the mode numbers and are denoted by(,) mn Physically, the indices m and n represent the number of half-wavelengths in the x direction and y -direction, respectively. The frequency at which a mode makes the transition from evanescent to propagating is known as the cut-on frequency. Below the

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17 cut-on frequency, the mode is evanescent. Above the cut-on frequency, the mode is propagating and present along the entire length of the wavegui de. The cut-on frequencies are calculated from the dispersi on relation in Equation (2.7) when0zk 22 0. 2co mnc mn f ab (2.8) The experiments for this dissertation will use two different waveguides. Both of them have a square cross-section. The lengt h of the sides of the first waveguide is 8.5 mm. For air at 298 K and 101.3 kPa, 0343 cms and the cut-on frequencies for the different modes are given in Table 2-1. No tice that the cut-on frequency for the first higher-order mode is approximately 20 kHz. This implies that only plane waves are present below this frequency and that the TMM can be used. Table 2-1: Cut-on frequencies in kH z for an 8.5 mm by 8.5 mm waveguide. m n 0 1 2 3 0 0 20.2 40.4 60.5 1 20.2 28.5 45.1 63.8 2 40.4 45.1 57.1 72.8 3 60.5 63.8 72.8 85.6 The second waveguide that will be used for this experiment has a square crosssection measuring 25.4 mm on each side. For the same conditions as above, the cut-on frequencies for the different modes are given in Table 2-2. Note that the plane wave mode, mode (0,0), is present for all frequencies. Also note that only the (0,0), (1,0), (0,1) and (1,1) modes are present at frequencies less than 13.5 kHz. The (1,0) and (0,1) modes have one half wavelength in the x -direction and y -direction, respectively. The (1,1) mode has a half wavelength in both the x -direction and the y -direction. The mode shapes are

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18 given in Figure 2-2 as observed from the sound source. The node lines indicate where the acoustic pressure is zero. Also note that the wavenumbers are a f unction of the mode. Table 2-2: Cut-on frequencies in kHz for a 25.4 mm by 25.4 mm waveguide. m n 0 1 2 3 0 0 6.75 13.5 20.3 1 6.75 9.55 15.1 21.4 2 13.5 15.1 19.1 24.4 3 20.3 21.4 24.4 28.7 xx x x y y y y (0,0) mode (1,0) mode (0,1) mode (1,1) mode Node Lines Figure 2-2: Illustration of the first four mode shapes. 2.1.3 Phase Speed The speed at which a wave front travels down the axis of the waveguide is known as the phase speed. The phase speed, phc is defined for a rectangular waveguide as (Blackstock 2000)

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19 ph zc k (2.9) and can be found for each mode: 2 22 0,ph mnc mn cab (2.10) which can be rewritten as 0 21ph mn co mnc c f f (2.11) The concept of the phase speed allows for hi gher-order modes to be considered as plane waves traveling at an angle inside the wa veguide, as shown in Figure 2-3. From Equation (2.11), as the frequency is increased the phase speed approaches the isentropic speed of sound but the phase speed at the cut-on frequencies for each mode tends to infinity. The incidence angle with respect to the waveguide wall normal as seen in Figure 2-3, mn is found from the geometric relationship between the phase speed for that mode and the speed of sound. Thus the incidence angle is found from 2 11 0sinsin1co mn mn ph mncf cf (2.12) Another useful angle that is developed from the concept of phase speed is the angle that the wave makes with the normal to a flat termination at the end of the waveguide (0d ), denoted bymn From the geometry given in Figur e 2-3 and from specular reflection, mn is complementary to mn by

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20 2 11 090coscos1co mn mnmn ph mncf cf o. (2.13) d mn Wave front mn Figure 2-3: Illustration of the wave front a nd the incidence angle to the waveguide wall, mn and to the termination, mn The expressions given above for the ph ase speed, Equation (2.11) and the two angles of incidence, Equations (2.12) a nd (2.13), are shown only to depend on the waveguide geometry, the bandwidth of in terest, and the mode number. For the waveguide with the 8.5 mm by 8.5 mm crosssection and a bandwidth of 20 kHz, the phase speed is simply the isentropic speed of sound and the wave is normally incident to the termination. Continuing with the example of a waveguide with a square cross-section of 25.4 mm by 25.4 mm, at room temperature and pressure the phase speed and the two angles of incidence are graphed in Figure 2-4, Figure 2-5 and Figure 26, respectively. A bandwidth of 13.5 kHz is used for the graphs, thus the included modes are (0,0), (1,0), (0,1) and (1,1). Figure 2-4 shows the pha se speed, Figure 2-5 shows the angle of incidence the mode makes to the sidewall of the waveguide, and Figure 2-6 shows the angle of incidence the mode makes to the termination. The plane wave mode is present for all frequencies and is normally incident to the termination for all frequencies as well.

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21 The properties of the other modes vary as a function of frequency. The phase speed approaches infinity asymptotically at the cu t-on frequencies, where th e angle of incidence to the sidewall approaches zero and the angle of incidence to the termination approaches 90 degrees. This shows that the MDM offers the potential to test the impedance of specimens with oblique incident-waves. 0 2000 4000 6000 8000 10000 12000 300 400 500 600 700 800 900 1000cph mn [m/s]f [Hz] (0,0) (0,1) (1,0) (1,1) Figure 2-4: Phase speed versus freq uency for the first four modes. 0 2000 4000 6000 8000 10000 12000 0 10 20 30 40 50 60 70 80 90mn [deg]f [Hz] (0,0) (0,1) (1,0) (1,1) Figure 2-5: Angle of incidence to the sidewall versus frequency for the first four modes.

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22 0 2000 4000 6000 8000 10000 12000 0 10 20 30 40 50 60 70 80 90mn [deg]f [Hz] (0,0) (0,1) (1,0) (1,1) Figure 2-6: Angle of incidence to the termin ation versus frequency for the first four modes. 2.1.4 Wave Mode Attenuation The energy in the evanescent wave modes exponentially decays as the wave propagates down the waveguide. The TMM a ssumes that only the plane wave mode is present at the microphone locations and that all other modes have decayed and can be neglected. To ensure that the evanescent wa ves have decayed sufficiently, the amplitude of a wave should be measured at two differe nt axial locations in the waveguide. This analysis of the decay of the amplitude of the evanescent waves assumes only an incident or right-running wave. This allows Equation (2.5) to be simplified for a single mode to coscos.zjkd mnmnmn PAxye ab (2.14) To measure the loss in amplitude of the evanes cent wave, the ratio of Equation (2.14) is taken for two locations separated by a distance ed to give

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23 coscos coscosze z e zjkdd mn jkd mne jkd mn mnmn Axye Pdd ab e mn Pd Axye ab (2.15) Recall that for an evanescent wave, the wave number is imaginary and thus the amplitude of the evanescent wave will exponentially decay The loss in amplitude can be defined on a decibel scale by 1020log.zekde (2.16) The decay of the evanescent waves for the two waveguides introduced in Section 2.1.2 can be plotted. The distance traveled by the wave is assumed to be 25.4edmm for the large waveguide which is equal to the length of one of the sides of the cross-section. Figure 2-7 shows the attenuation of the highe r-order modes in the large waveguide for modes (0,1), (1,0), up to (3,3) up to their cu t-on frequency. The mode can be determined by comparing the cut-on frequency in the figure to those listed in Table 2-2. For the small waveguide, the distance traveled is assumed to be 8.5edmm which is again the length of one of the sides of the cross section. Figure 2-8 shows the attenuation of the higher-order modes in the small waveguide for modes (0,1), (1,0), up to (3,3) up to their cut-on frequency. The mode can be determin ed by comparing the cut-on frequency in the figure to those listed in Table 2-1. The amp litude of the first evanescent wave ((0,1) and (1,0)) is reduced by -3.8 dB with the freque ncy lowered from the cut-on frequency by only 16 Hz for the large waveguide and onl y 25 Hz for the small waveguide. The attenuation of other higher-order modes is larger. Figure 2-9 shows the attenuation of the fi rst higher-order mode in the large waveguide for two different distances. The tw o distances chosen are the distances from

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24 0 5 10 15 20 25 30 -120 -100 -80 -60 -40 -20 0 Freq [kHz] [dB] Increasing wave mode Figure 2-7: Attenuation of highe r-order modes in the large waveguide over a distance of 25.4 mm. 0 10 20 30 40 50 60 70 80 90 -120 -100 -80 -60 -40 -20 0 Freq [kHz] [dB] Increasing wave mode Figure 2-8: Attenuation of highe r-order modes in the small waveguide over a distance of 8.5 mm. the specimen test surface to the two microphones used in the TMM for the large waveguide. The attenuation shown in this fi gure represents the wo rst case in terms of contamination of the microphone signals with unmodeled deterministic signals that will

PAGE 44

25 bias the estimates from the TMM. The fi gure shows that the a ttenuation approaches 34dB asymptotically for the clos er microphone location and 57dB for the farther microphone location as the frequency approaches zero, but that the attenuation tends to zero near the cut-on frequencies. The 20dB point for the closer microphone location is at approximately 5.47 kHz. Above this frequency, the signal measured by this microphone could be affected by the non-neglig ible amplitude of the higher-order modes propagating from the specimen to the microphone The absolute amplitude of the first higher-order mode may still be negligible when compared to the absolute amplitude of the plane wave mode, because the overall le ngth of the waveguide provides sufficient attenuation such that only plane waves are incident on the specimen and such that the specimen may not strongly scatter incident energy from the plane wave mode into the higher-order modes upon reflection. The data shown in the figures in this s ection demonstrates that the attenuation of the higher-order modes is not an instantane ous effect. At the cut-on frequency, the higher-order modes have an infinite speed and are felt throughout the entire duct. As the frequency decreases away from cut-on, th e amplitude of the evanescent mode is decreased, but only by a finite amount. If the initial amplitude of the evanescent mode is sufficiently high, then the attenuation may not be strong enough to reduce the amplitude below the noise floor of the measurement mi crophones. This may introduce a significant bias error source into the estimates for th e TMM or any other method that assumes no amplitude in the evanescent modes. Nume rical simulation of the sound field at the microphone locations, including an amplitude component for the higher-order modes, would be required to characterize the impact of the unmodeled evanescent modes. The

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26 relative amplitudes between the microphone loca tions are known, as shown in Equation (2.15), and the absolute amplitude at one lo cation could be inferred from experimental data. The simulated signals could then be pr ocessed as experimental data to gauge the amount of bias error that is introduced into the estimates for the reflection coefficient and acoustic impedance. 0 1 2 3 4 5 6 7 -60 -50 -40 -30 -20 -10 0 Freq [kHz] [dB] de=32.1 mm de=52.7 mm Figure 2-9: Attenuation of the first higher-order mode ((1,0) or (0,1)) in the large waveguide at the microphone locations used for the TMM experiments. 2.1.5 Reflection Coefficient and Acoustic Impedance For the remainder of this chapter, only pl ane waves are assumed to propagate, thus restricting the bandwidth for a given waveguide. For this case, the reflection coefficient is the ratio of the acoustic pressure amplitudes of the reflected wave to the incident wave and is a single complex quantity. The plan e wave reflection coefficient is defined as 00 00,00 00 B R A (2.17)

PAGE 46

27 where 00,00R is the plane wave reflection coefficient, and 00A and 00 B are the complex modal amplitudes for the incident and refl ected wave, respectively. The reflection coefficient indicates the degree to which a material reflects sound. However, the reflection coefficient can also be used to calculate the normalized specific acoustic impedance, s pac of a material. The normalized spec ific acoustic impedance is defined by the ratio of the acoustic impedance of the ma terial to that of the medium used during the test. For most cases, the medium is air. The acoustic impedance is defined as the complex ratio of the acoustic pressure to the acoustic volume velocity. The specific acoustic impedance is the complex ratio of th e acoustic pressure to the acoustic particle velocity. The characteristic impedance is the specific acoustic impedance of that particular medium. For the purpose of finding the acoustic impeda nce ratio, consider an incident wave reflecting off the termination of the wave guide as shown in Figure 2-10. The two boundary conditions are applied to the interface (Blackstock 2000): 1. The pressure must be continuous across the interface. 2. The normal component of the particle ve locity must be co ntinuous across the interface. The first boundary condition leads to the following expression 00,0000,001RT (2.18) where 00,00T is the plane wave transmission coe fficient defined as the ratio of the amplitude of the transmitted pressure wave to the amplitude of the incident pressure wave. The second boundary condition l eads to the following expression 00,0000,00 011 coscositrRT ZZ (2.19)

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28 where 0 Z and 1 Z are the specific acoustic impedan ces for medium 0 and medium 1, respectively. The terms 0cosiZ and 1costrZ represent the acoustic impedance for medium 0 and medium 1, respectively, but under the plane wave assumption the incidence angle and the transmission angle are 0o with respect to the specimen surface normal and the acoustic impedance becomes iden tical to the specific acoustic impedance. Thus, Equation (2.19) simplifies to 00,0000,00 011 RT Z Z (2.20) Equations (2.18) and (2.20) can be combined, and then th e resulting expression can be solved for the plane wave specific acoustic impedance ratio, given by 00,00 1 000,001 1spacR Z ZR (2.21) From Equation (2.21), the task of finding the normalized specific acoustic impedance reduces to finding the reflection coefficient of the incident and reflection plane waves. d ZoZ1 piprptrtr mnr Figure 2-10: Reflection and transmission of a wave off an impedance boundary. 2.2 Two-Microphone Method The TMM (Seybert and Ross 1977; Chung and Blaser 1980; ASTM-E1050-98 1998; ISO-10534-2:1998 1998) is a standardized technique for determining the normal incident acoustic impedance. A schematic of the test setup for the TMM is given in

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29 Figure 2-11. The notation used here fo llows the ASTM E1050-98 standard (ASTME1050-98 1998). The advantage of the TMM is the simplicity offered by assuming the sound field is only comprised of plane waves. Therefore, only two unknown coefficients are determined and only two microphones are used. The data reduction equation for the TMM is derived in this section, starting from the basic assumptions and the general solution of the wave equation given in Equa tion (2.5). Afterwards, the effects of dispersion and dissipation ar e addressed briefly. Mic Power Supply Spectrum Analyzer & Signal Generator Power Amplifier Waveguide Compression Driver Specimen Rigid Back Plate Mic 1Mic 2 d Reference Mic s l Figure 2-11: Experimental setup for the TMM. 2.2.1 Derivation of the TMM The TMM assumes that the sound field inside the waveguide is composed solely of plane waves. This simplifies the solution to the wave equation from Equation (2.5) to 0000zzjkdjkdPAeBe. (2.22) This equation can then be recast by using th e definition of the reflection coefficient given in Equation (2.17) to 0000,00zzjkdjkdPAeRe. (2.23)

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30 Now, the two primary unknowns are 00,00R and00A. The two unknowns are solved for by taking measurements of the complex pressure amplitude at two different locations along the waveguide. Let l denote the distance between the test specimen and the closest microphone, 2P, and s denote the distance between the two microphones. The system of equations is 1 0000,00zzjklsjklsPAeRe, (2.24) 2 0000,00zzjkljklPAeRe. (2.25) The complex pressure amplitude of the incident wave is eliminated from the system of equations by taking the ratio of 2P to 1P to get 00,00 2 12 1 00,00,zz zzjkljkl jklsjklseRe P H P eRe (2.26) where 12H is the frequency response function between microphone 1 and microphone 2. Then this new expression is solved for the reflection coefficient and simplified as 2 12 00,00 12ˆ ˆjks jkls jksHe Re eH (2.27) where 121211ˆˆ ˆ EHGG is the estimate of the freque ncy response function between the two microphones, E is the expect ation operator, 12ˆ G is the estimated cross spectrum and 11ˆ G is the estimated autospectrum (Benda t and Piersol 2000). The frequency response function is switched from the exact 12H to the estimate 12ˆ H in Equation (2.27) because 12ˆ H is an unbiased estimate of 12H and reduces to 12H in the case of no measurement noise.

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31 The form of the data reduction equation in Equation (2.27) is the same as the form presented in the ASTM E1050-98 sta ndard (ASTM-E1050-98 1998). The only difference between this form and the form presented in the ISO 10534-2:1998 standard (ISO-10534-2:1998 1998) is the definition of the reference length, l. The ISO standard defines l to be the distance from the surface of the specimen to the microphone farther away (ISO-10534-2:1998 1998). The remainder of this document will use the definition of l used in the derivation in this section that is consistent with the ASTM E1050-98 standard, which is the distance from th e surface of the specimen to the closest microphone (ASTM-E1050-98 1998). After the reflection coefficient is found from Equation (2.27), the normalized specific acous tic impedance is computed from Equation (2.21). 2.2.2 Dissipation and Dispersion for Plane Waves A dispersion relation is an expression that shows how the wave speed depends on frequency. An example of a dispersion re lation was given in E quation (2.11) for the phase speed of the higher-order modes in the waveguide. Dissipation is the removal of energy from the propagating wave. The main mechanisms for the dissipation of wave propagation in ducts are viscous losses a nd thermal conduction in the boundary layer (Ingard and Singhal 1974; Black stock 2000). At high frequenc ies, molecular relaxation can also be another source of attenuation, but this is neglected in this analysis. The boundary layer is a thin region near the boundary where the effects of viscosity and heat transfer are important. The no-slip boundary condition and viscosity produce a transfer of momentum from the flow to the wall and re tards the flow in the boundary layer region. The no-slip boundary condition states that th e velocity of the fl ow must match the velocity of the solid boundaries, which for the cases presented in this dissertation are not

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32 moving. The viscous boundary layer thickness for an oscillatory flow over a stationary plate is 6.5 (2.28) where is the kinematic viscosity, and is defined as the distance from the boundary to the point in the flow where the velocity onl y differs by 1% from the free stream value (White 1991). As the frequency increas es, the viscous boundary layer thickness decreases and the region where the no-slip boundary condition influences the flow is reduced. The thermal boundary layer is the re gion where heat is transferred from the flow to the boundary. The thermal boundary layer thickness is related to the viscous boundary thickness and the Prandtl number by (White 1991) 2~ Prt (2.29) where Pr is the Prandtl number and is the thermal diffusivity. Both the transfer of momentum and the transfer of thermal energy from the flow to the wall work to reduce the amplitude of the pressure wave. To account for dispersion and dissipation in viscothermal flows, the wavenumber is allowed to be complex and is given by kj c (2.30) where c is the speed of sound inside the wa veguide adjusted for dispersion and is the dissipation coefficient. The speed of s ound corrected for viscothermal effects is (Blackstock 2000)

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33 021 11 Prcc S (2.31) where S is the Stokes number given by 2L S (2.32) and 4perimeterLAl is the hydraulic diameter of the waveguide, perimeterl is the wetted perimeter of the cross section and A is the cross-sectional area. The dissipation coefficient for viscothermal effects fo r plane waves is (Ingard and Singhal 1974; Blackstock 2000) 021 1 Pr Sc (2.33) Both Equations (2.31) and (2.33) contain th e Stokes number, which is a nondimensional number that relates a character istic length, in this case the hydraulic diameter, to the viscous boundary layer thickness for oscillati ng flows. In the limit of thin acoustic boundary layers (at high freque ncy), the ratio of the viscous boundary layer thickness to the hydraulic diameter goes to zero, and the St okes number approaches infinity. Thus as 0cc and 0 the lossless wavenumber is rec overed. Physically, as the boundary layer becomes smaller, the effects of viscosity and heat transfer become less important and flow should approach the lossless case as shown. Also, Equa tions (2.30) through (2.33) show that the wavenumber corrected fo r dispersion and dissipa tion is a function of the angular frequency, the thermodynamic stat e, and the geometry of the waveguide. This is in contrast to the wavenumber give n in Equation (2.4) for linear lossless acoustic

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34 motion, which was just a function of the angul ar frequency and the thermodynamic state. The derivations of both Equations (2.31) and (2.33) are given in Appendix A. To consider the relative importance of the effects of dispersion and dissipation, the propagation constants, kd, are compared for the lossless case and for the case with dispersion and dissipation. The propagati on constant for the lossless case is 0.losslesskdd c (2.34) The propagation constant for the ca se dispersion and dissipation is 00 0.thermoviscouscc kdjddj ccc (2.35) Simplifying the ratio of thermoviscouskd to losslesskd yields 2 111. Prthermoviscous losslesskd j kdS (2.36) In order to neglect the effects of dispersion and dissipation, the rati o in Equation (2.36) must be close to unity. This requires that th e last term in the equation is much less than unity and as seen in Equation (2.36), this occu rs at high frequencies. For air at standard temperature and pressure with 1.4 6215.710ms, Pr0.708 (Incropera and DeWitt 2002), Table 2-3 shows the minimum fre quency necessary to keep the last term in Equation (2.36) under a value of 0.05 for th e two waveguides given in this chapter and for the two ceramic tubular specimens, CT73 and CT65, described in Chapter 5. Notice that dispersion and dissipation are important fo r the two ceramic tubular materials in the frequency range of interest for acoustic impedance testing.

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35 Table 2-3: Minimum frequencie s to keep effects of dispersion and dissipation <5%. Waveguide cross-section Frequency [kHz] 25.4 mm x 25.4 mm 0.0067 8.5 mm x 8.5 mm 0.060 CT73 (hydraulic diameter = 1.10 mm) 4.35 CT65 (hydraulic diameter = 0.443 mm) 22.2

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36 CHAPTER 3 UNCERTAINTY ANALYSIS FOR THE TWO-MICROPHONE METHOD Previous studies on the uncertainty of th e TMM have discussed in detail specific error sources due to uncerta inties in spectral estimates (Seybert and Soenarko 1981; Bodn and bom 1986; bom and Bodn 1988) and the microphone spacing and locations (Bodn and bom 1986; bom and Bodn 1988; Katz 2000) and have provided recommendations to minimize the respective error components. However, these efforts did not provide a method to propagate th e estimated uncertainties to the overall uncertainty in the acoustic impedance and refl ection factor. The purpose of this chapter is to provide a systematic framework to accomp lish this task. In particular, a frequencydependent 95% confidence interval is estimated using both multivariate uncertainty analysis and Monte Carlo methods. The multivariate uncertainty analysis is an analytical method that assumes small uncertainties which cause only linear variations in the output quantities, but differs from classical uncertainty methods by allowing multip le, possibly correlated, components to be tracked. As long as the data reduction equation can be cast into a multivariate equation and the derivatives can be found, the mu ltivariate uncertainty method provides a convenient way to propagate the experimental uncertainty. The multivariate technique is required because the measured data and the final output of the TMM are complex variables that are treated as bi variate variables. The input covariance matrix and Jacobian are computed and propagated through the da ta reduction equation (as shown in Appendix B). The multivariate method thus provides anal ytical expressions that are used to extract

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37 important scaling information, while the Mont e Carlo simulations are used to account for the nonlinear perturbations of the input uncertainties observed in practice. The remainder of this chapter is organized as follows. First, the TMM data reduction equations are presented in a multivariate form. Next, a general procedure to estimate the complex uncertainty using the mu ltivariate method is outlined, and a brief discussion of the major error sources and thei r respective frequency scaling follows. The results of numerical simulations to illustrate the relative advantages and disadvantages of the TMM and the multivariate method follow. Specifically, two impedance cases are presented, a sound-hard boundary that is repr esentative of a high-impedance sample, and an “ideal” impedance sample that is represen tative of an optimum impedance for a ducted turbofan. Monte Carlo simulations are comp ared with the results of the multivariate method. 3.1 Multivariate Form of the TMM Data Reduction Equations From Equations (2.27) and (2.21), 00,00R and 00 are complex quantities that are functions of another complex variable ˆH, the multivariate uncertainty analysis method is used to propagate the uncerta inty (Ridler and Salter 2002; Hall 2003; Hall 2004). To employ the multivariate method, the data reduction equations for the plane wave reflection coefficient and the normalized acous tic impedance given in Equations (2.27) and (2.21), respectively, must be separated into the real and imaginary parts denoted by the subscripts R and I respectively. For 00,00R,

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38 22 22 00,00 22 22ˆˆˆ 2cos2cos2cos2 ˆˆˆˆ 12cos2sin ˆˆˆ 2sin2sin2sin2 ˆˆˆˆ 12cos2sinRRI RIRI R I RRI RIRIHklsklHHkls HHHksHks R R R HklsklHHkls HHHksHks (3.1) In this form, the two variates of the refl ection are functions of five input variates, ˆRH, ˆIH, l, s, and k, where ˆRH and ˆIH are the real and imaginary parts of 12ˆ H, respectively. The FRF is also treated as tw o variates instead of a single quantity. The corresponding form for the normalized specific acoustic impedance is 22 2 2 2 21 1 2 1RI RI spac I RIRR R R R R R (3.2) 3.2 TMM Uncertainty Analysis Previous studies of the error sources for the TMM have focused on determining general scaling of the error and an experiment al design that minimizes such errors with the use of a Gaussian input si gnal. Seybert and Soenarko found that the bias error in the FRF due to spectral leakage can be minimized by using a small value for the bin width of the spectral analysis (Seybe rt and Soenarko 1981). Spectra l leakage can be eliminated using a periodic input signal. They also found that locating the microphones too close to the specimen introduced bias and random e rrors that are a function of the measured coherence. To increase the coherence, the microphones should be placed close together relative to the wavelength, but the coherence will always be low when one of the microphone locations coincides with a node in the standing wave pattern. One of the

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39 most important findings was that when the value of s approaches an integer number of half wavelengths, the error increases dramatically. Bodn and bom expanded on these results and found that the bias error of the FRF was impacted by the overall length of the waveguide, the value of the specific acoustic impedance of the specimen, and the location of the microphone s relative to the specimen (Bodn and bom 1986). The random error was a function of the coherence and was influenced by the value of the reflec tion coefficient, outsi de noise sources, and the value of ks. They suggest satisfying 0.10.8 ks to keep the overall error low. In combination with their second study ( bom and Bodn 1988), they concluded that errors in the microphone locations dominated over (1) spatial averag ing effects, (2) any offset the acoustic center has from its assume d location at the geom etric center, and (3) any effects from the finite impedance of the microphones themselves. 3.2.1 Multivariate Uncertainty Analysis The results from the previous studies pr ovide the necessary guidance to quantify and minimize component error sources that, t ogether with the multivariate uncertainty and the Monte Carlo methods, can be used to provide 95% confidence intervals. The multivariate method propagates the uncertain ty estimates through any data reduction equation (Ridler and Salter 2002; Willink a nd Hall 2002; Hall 2003; Hall 2004; Schultz et al. 2005) using T yx sJsJ, (3.3) where ys is the sample covariance matrix of the output variable, xs is the sample covariance matrix of the input variates, J is the Jacobian matrix for the data reduction equation, and the superscript T indicates the transpose. With the sample covariance

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40 matrix of the variable, the 95% confidence region is found fr om the probability statement (Johnson and Wichern 2002) ,1,Prob1 1effeff ypp effp F p 1yysyy, (3.4) where y is a vector representing the multivariate variable, y is the sample mean vector, ys is the sample covariance matrix of the mean, ,1,effppF is the statistic of the F distribution with p variates (two for a complex variable), and 1eff p degrees of freedom for a probability 1 and eff is the effective number of degrees of freedom from the measurements (Willink and Hall 2002). If the entire confidence region is not desired, the confidence level estimates of the uncertainty for each variate can be computed from the equation ncfnUku (3.5) where nu is the estimate of the sample standard deviation for the thn output variate (i.e., the square root of the diagonal elements of ys), and cfk is the coverage factor given by ,1,1effeff cfpp effp kF p (3.6) The Jacobian matrix for the reflection coefficient is 00,00ˆˆ ˆˆRRRRR RI R IIIII RI R RRRR lsT HH R RRRR lsT HH J, (3.7) where, in this model, the wavenumber is tr eated solely as a func tion of temperature and thus, the uncertainty in the wavenumber is solely due to the uncertainty in the

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41 temperature measurement. The Jacobian matrix for the normalized specific acoustic impedance is spacRI RI R R R R J. (3.8) 3.2.2 Monte Carlo Method A Monte Carlo method is also used to co mpute the uncertaintie s of the reflection coefficient and the acoustic impedance ratio. The Monte Carlo method involves assuming distributions for all of the input uncertainties and then randomly perturbing each input variable with a perturbation draw n from its uncertainty distribution (Coleman and Steele 1999). The assumed distributions will be multivariate distributions if the input variates are correlated. No w, the perturbed input variat es are used to compute the outputs, in this case 00,00R and s pac This is repeated until the statistical distribution of the output variable has converged and then the output distributi on is used to estimate the 95% confidence regions. A summary of the uncer tainty sources is given in Table 3-1. 3.2.3 Frequency Response Function Estimate Estimates of the uncertainty and error s ources in the FRF are documented in the literature (Seybert and Hamilton 1978; Seybert and Soenarko 1981; Schmidt 1985; Bendat and Piersol 2000; Pintelon and Schoukens 2001b; Pintelon et al. 2002). For this paper, two uncorrelated noise sources are as sumed to affect a si ngle-input/single-output system with a periodic and deterministic input signal, as described in Appendix B. Also from Appendix B, the FRF estimate is

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42 22 12 22 111212 22 12 22 121212ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆˆˆR IG C GCQ H H G Q GCQ (3.9) where 11ˆ G and 22ˆ G are the estimated autospectral densities of the signals from microphones 1 and 2, respectively, and 12ˆ C and 12ˆ Q are the coand quad-spectral density functions (i.e., 121212ˆˆˆ GCjQ). Equation (3.9) is commonly called the 3ˆ H estimate. Any phase bias can be eliminated using a sw itching technique, descri bed in Appendix B. The final estimate of the FRF is computed from the geometric average of the two interchanged measurements as ˆ ˆ ˆO SH H H, (3.10) where ˆOH and ˆSH are the FRF between the microphone s in their original and their interchanged locations respectively. The details on computing the estimate of th e FRF for this system model are given in Appendix B. The sample covariance matrix for 3ˆ H and the Jacobian matrix needed to propagate the uncertainty to the averaged FRF are also given in Appendix B. The uncertainty estimation requires an additio nal measurement with the pseudo-random source turned off to estimate the noise power spectrum.

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43 Table 3-1: Elemental bias and prec ision error sources for the TMM. Variable or Origin Error Source Error Estimator T RTD accuracy Manufacturer’s specifications or calibration accuracy Ambient temporal variations Minimize by conducting the test in limited amount of time Spatial variations Estimate by measuring the temperature at different locations along the waveguide Random variations Statistical methods ,sl Caliper accuracy Manufactur er’s specifications or calibration accuracy Acoustic centers Calibration or estimate as half microphone diameter Random variation Statistical methods Microphones Spatial averaging Minimize by using microphones with a diameter much smaller than the wavelength Impedance change of waveguide wall Minimize by using microphones with a diameter much smaller than the wavelength ˆ H Phase mismatch Correct for by using microphone switching Magnitude mismatch Correct for by calibrating each microphone and microphone switching A/D limitations Minimize by maximizing the significant bits Finite frequency resolution Not present for a periodic random input signal Random error Sample covariance matrix given in (Schultz, Sheplak and Cattafesta 2005) The reflection coefficient’s sensitivity to uncertainty in the FRF is described by 22ˆˆ cos2cos2cos 2 ˆˆˆˆˆ 12cos2sinRRR R RRIRIklsHklsRksH R HHHHksHks (3.11) 22ˆˆ cos2sin 2 ˆˆˆˆˆ 12cos2sinIRI R IRIRIHklsRksH R HHHHksHks (3.12)

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44 22ˆˆ sin2sin2cos 2 ˆˆˆˆˆ 12cos2sinRIR I RRIRIklsHklsRksH R HHHHksHks (3.13) and 22ˆˆ sin2sin 2 ˆˆˆˆˆ 12cos2sinIII I IRIRIHklsRksH R HHHHksHks (3.14) Consider the case when ksn which leads to ˆ 1nH As a result, the common denominator in Equations (3.11)-(3.14) equals zero, resulting in a singularity so that any uncertainty in the FRF will result in a large uncer tainty in the reflection coefficient. This result agrees with previous studies (Seybert and Soenar ko 1981; Bodn and bom 1986; bom and Bodn 1988). Equations (3.11)-(3.14) indicate that th e sensitivity to the uncertainty in ˆH is dependent on the value of ˆH and 00,00R. As ˆH approaches the limiting values of zero or infinity (i.e., when one of the microphones is located at a node), or as the magnitude of 00,00R approaches the limit of unity, the sensitivity will increase. This implies that the accurate measurement of the two extremes, sound-hard 00,001R and pressure release 00,001R boundaries, which possess cusps in the standing wave patterns, will show the largest sensitivities to uncertainty. The equations also show a periodic element to the uncertainty estimates that is dependent on the wavenumber and the locations of the microphones. Thus for a fixed set of micr ophone locations, the uncertainty estimates may vary versus frequency. The actual periodicity is complex to analyze because of the combinations of trigonometric functions present in the part ial derivatives.

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45 3.2.4 Microphone Locations This section addresses the effects of the uncertainty of the microphone locations on the reflection coefficient. The respective sens itivity coefficients for the distance between the specimen and the closest microphone l and for the microphone spacing s are 2R IR kR l (3.15) 2I RR kR l (3.16) 22 22ˆˆˆˆˆ sin2sin2sincos 2 ˆˆˆˆ 12cos2sinRRIRRI R RIRIHklsHHklsRHksHks R k s HHHksHks ,(3.17) and 22 22ˆˆˆˆˆ cos2cos2sincos 2 ˆˆˆˆ 12cos2sinRRIIRI I RIRIHklsHHklsRHksHks R k s HHHksHks .(3.18) The sensitivity coefficients for l and s are both directly proporti onal to the frequency via the wavenumber, emphasizing the difficulty of making accurate measurements at high frequency. Equations (3.17) and (3.18) have the same deno minator as Equations (3.11)(3.14), again showing that half-wavelength spacing ksn should be avoided. Again, the equations also show a periodic element to the uncertainty estimates that is dependent on the wavenumber and the locations of th e microphones as shown with the frequency response function derivatives and that analyz ing the periodicity would be even more involved because of the increased nu mber of trigonometric terms. 3.2.5 Temperature The random uncertainty in the temperat ure measurement can be handled using standard statistical procedures The effects of temporal va riations in the atmospheric

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46 conditions can be minimized by limiting the dura tion of the test. The spatial variation in the temperature of the waveguide can be ch aracterized by measuri ng the temperature at various locations and computing the standard deviation of the measurements, but this will be a crude estimate since the entire temperatur e is not measured. The temperature sensor for this study is mounted on the exterior wa ll to avoid undesired refl ections and scattering of the sound field inside the waveguide, and is found to give reliabl e estimate of the gas temperature if the wall is highly conductive. This wa s confirmed by comparing the measured temperature inside the waveguide to the outside surface metal temperature while the sound source was on for one experime ntal run. The total uncertainty in temperature is estimated from the root-sum -square of the individual uncertainties. The sensitivity coefficients of the reflection coefficient with respect to temperature are computed using the chain rule RRRR k TkT (3.19) and IIRR k TkT (3.20) where 22sin2sin2sin2 2 ˆˆˆˆ 12cos2sinR RIRIl AklsklBklsC R s s k HHHksHks (3.21) 22cos2cos2cos2 2 ˆˆˆˆ 12cos2sinI RIRIl AklsklBklsC R s s k HHHksHks (3.22) ˆ 12Rl AH s (3.23)

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47 22ˆˆ 1RIl B HH s (3.24) and ˆˆ sincosRRICRHksHks (3.25) Equations (3.19)-(3.25) reveal that the uncertainty in 00,00R is approximately proportional to the microphone spacing. Reducing the spacing between the microphones will reduce the sensitivity of the uncertainty in the reflection coefficient with respect to the wavenumber and temperature. Also, Equa tions (3.21) and (3.22) possess the same singularity as the other derivatives at ksn Again, the equations also show a periodic element to the uncertainty estimates that is dependent on the wavenumber and the locations of the microphones as shown before with the same difficulties. For the case with dispersion and dissipati on, the complex wavenumber is a function of the thermodynamic state (ambient temperat ure and pressure), the frequency, and the waveguide geometry (Morse and Ingard 1986; Blackstock 2000). The scaling of the uncertainty in 00,00R accounting for these effects is diffi cult to examine an alytically. If dissipation and dispersion are ne glected and an ideal gas is assumed, the wavenumber is given by Equation (2.4) and is onl y a function of temperature. Thus, the derivative of the wavenumber with respect to temperature is 2 02 g asR k k Tc (3.26) Equation (3.26) shows that the uncertainty will increase with frequency via the wavenumber and that the uncertainty is invers ely proportional to the square of the speed of sound.

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48 3.2.6 Normalized Acoustic Impedance Uncertainty For the uncertainty analysis, the normali zed specific acoustic impedance is treated as solely a function of th e reflection coefficient. The Jacobian matrix is 2 2 22 22 22 2 2 22 22 2221 41 11 21 41 11spacRI IR RIRI RI IR RIRIRR RR RRRR RR RR RRRR J. (3.27) Notice that each term has the same denomi nator and a singularity exists (i.e. when 2 210RIRR ) for a sound-hard boundary, 00,001 R This situation will be studied further in the section below. 3.3 Numerical Simulations Much of the observations in Section 3.2 have been previously reported in the literature (Seybert and Soen arko 1981; Bodn and bom 1986; bom and Bodn 1988). The main contribution of this chapter is to demonstrate how these uncertainty sources propagate and contribute to the overall uncertainty in 00,00R if they remain linear. But for typical experimental situati ons, the uncertainties cause nonlinear perturbations in the reflection coefficient and acoustic impedance. In order to demonstrate the uncertainty propagation, numerical experi ments on a sound-hard boundary and an “ideal” impedance sample are carried out using the analytical method outlined in Section 3.2 for the overall uncertainty estimate. Time-series data are simulated using E quations (2.2) and (2.23) by choosing a desired value of 00,00R and the resulting data are processed using the algorithms

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49 described in Section 3.2. The nominal values for the input parameters are given in Table 3-2. The test frequency of 5 kHz (0.60 ks ) is chosen because the uncertainties are bounded for this set of microphone locations. A parametric study of the effects of sensor signal-to-noise ratio (SNR) and uncertainties in temperature, microphone location, and spacing is completed in isolation, assuming th e perturbations remain linear. The relative uncertainties in the temperature, microphone location, and spacing are independently varied from 0.1% to 10% at a single freque ncy, while the other uncertainties are set to zero and the input signal is noise-free. The effect of the SNR is studied by varying the SNR from 30 dB to 70 dB while holding the ot her uncertainties to zero. The SNR for the numerical simulations is based on the power in the incident wave only at that frequency compared to the power in the noise signal at that frequency and is kept constant across the entire bandwidth. Next, the total uncertainty in 00,00R as a function of frequency is estimated from the case with the relative input uncertainties of 0.01% and 1% for a SNR of 40 dB. The estimated 95% confidence interval s are then compared to the results of the Monte Carlo simulation using 25,000 iterations. All the va riables are assumed to be normally distributed for the Monte Carlo simulation outlined in Figure 3-1 and the real and imaginary parts of the FRF are assumed to be correlated, as shown in Appendix B. The simulations used either a zero-mean periodic random signal for a broadband periodic source or a sinusoid for single-frequency excitation. The bandwidth chosen for the broadband simulations is 0 to 20 kHz. Spectral analys is is carried out using a sampling frequency of 51.2 kHz, with 1,024 samples per block and 1,000 blocks, yielding a frequency resolution of 50 Hz. In thes e simulations, the microphone spacing is not designed to avoid the situation where ksn or to maintain the inequality ksn

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50 (bom and Bodn 1988). This is acceptable since the goal of the simulations is to demonstrate that the uncertain ty analysis methods presente d earlier capture the correct behavior. In an actual experiment, multiple microphone spacings can be used to avoid the regions where ksn Table 3-2: Nominal values for input parameters of numeric simulations. Parameter Value l 32.1 mm s 20.6 mm T 23.8 C 3.3.1 Sound-Hard Sample The first specimen studied is a sound-ha rd boundary. To avoid the singularity present in the data reduction and uncertain ty expressions, the assumed value of the reflection coefficient is 00,000.999 R which gives a standing wave ratio (the ratio of the maximum to the minimum pressure amplitude al ong the axis of the waveguide) of greater than 60 dB. Figure 3-2 shows the absolute un certainty in the reflection coefficient as a function of the uncertainty in ,, ls and T at 5 kHz. Figure 3-3 shows the absolute uncertainty in the reflection coefficient as a function of the SNR. The results in these figures suggest that the dominant source of un certainty in the magnitude of the reflection coefficient is the random uncertainty in the FR F measurement for signal-to-noise ratios of 50 dB or lower. The dominant source of un certainty in the phase of the reflection coefficient is in the measurement of the distance between the specimen and the nearest microphone. Improvements in the measurement of the reflection coefficient could be obtained from improvements in the accuracy of the FRF measurements (reducing the noise in the system, increasing the number of averages) and the measurement of the distance between the specimen and the nearest microphone.

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51 Figure 3-1: Flow chart for the Monte Carlo methods.

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52 The estimated value of the reflection coeffi cient for the relative uncertainty in the measurement of the microphone location, th e microphone spacing, and the temperature each set to 1% and with a SN R of 40 dB is given in Fi gure 3-4(a) as a function of frequency. The uncertainty results for the multivariate method and the Monte Carlo simulation are shown in Figure 3-4(b). Note that the peak s in the uncertainty are at frequencies 8.4 and 16.7 kHz, where ksn and the frequencies where one of the microphones is at a node in the standing wave pa ttern are 1.6, 2.7, 4.9, 8.1, 8.2, 11.5, 13.5, 14.7, 18.0, and 18.8 kHz. The multivar iate method agrees with the Monte Carlo simulation within 5% for all frequencies except those corresponding to a node in the standing wave at a microphone lo cation or the singularity where ksn validating the multivariate method for very small component er rors. The true value only fell outside the estimated 95% confidence region for both th e multivariate method and the Monte Carlo 100 101 10-8 10-7 10-6 10-5 10-4 U|R| 100 101 10-2 10-1 100 101 102 U [deg]Relative Uncertainty [%] Figure 3-2: Absolute uncertainty of 00,00R due to the uncertainties in l s and T for 00,000.999 R at f =5 kHz. l s T.

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53 simulation five times for the magnitude and zer o times for the phase out of the total 400 frequency bins. The two methods also match at lower values of the input uncertainty, but such agreement is not universal for all acoustic materials, which is shown in the next section. Figure 3-4(a) shows that the estim ate of the reflection coefficient becomes nonphysical, i.e. 00,001 R at the two frequencies wher e the singularity occurs. The uncertainty in the estimate also increases to account for the singular ity and the confidence interval for 00,00R does include physical values for the estimate. 30 35 40 45 50 55 60 65 70 10-5 10-4 10-3 10-2 U|R| 30 35 40 45 50 55 60 65 70 10-4 10-3 10-2 10-1 U [deg]Signal-to-Noise Ratio [dB] Figure 3-3: Absolute uncertainty 00,00R due to the SNR for 00,000.999 R at f =5 kHz. 3.3.2 Ideal Impedance Model The second simulation corresponds to the ideal impedance model given in Figure 37 of the NASA CR-1999-209002 (Bielak, Pr emo and Hersh 1999), designed using Boeing’s Multi–Element Lining Optimization (MELO) program. The data provided in the NASA CR is limited to a frequency range of 500 Hz to 10 kHz and is extended to the

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54 0 2 4 6 8 10 12 14 16 18 20 0.98 0.99 1 1.01 1.02 |R| 0 2 4 6 8 10 12 14 16 18 20 -2 -1 0 1 2 Phase [deg]Freq [kHz] (a) 0 2 4 6 8 10 12 14 16 18 20 10-4 10-3 10-2 10-1 100 101 U|R| 0 2 4 6 8 10 12 14 16 18 20 10-1 100 101 102 103 104 U [deg]Freq [kHz] (b) Figure 3-4: Estimated value for the (a) reflect ion coefficient and (b) total uncertainty for the sound-hard boundary with 1% relative uncertainty for l, s and T and 40 dB SNR. Multivariate Method, Monte Carlo simulation. The two lines are indistinguishable at most frequencies.

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55 frequency range needed for this simulation by assuming that the first and last values are constant for the ranges of 0 to 500 Hz a nd 10 to 20 kHz, respec tively. The exact reflection coefficient and normalized impedan ce data are given in Figure 3-5. This specimen is chosen to determine the extent to which the uncertainties in a typical liner specimen scale in a manner similar to th at of a sound-hard boundary. The primary distinction between the two cas es is that there are no nodes in the standing wave pattern for this impedance sample. As a result, the coherence between the two microphone signals is expected to be near unity for al l frequencies assuming a reasonable SNR. Figure 3-6(a) shows the absolute uncertain ty in the reflection coefficient as a function of the uncertainty in ,, ls and T at 5 kHz and Figure 3-6(b) shows the absolute uncertainty in the normalized specific acous tic impedance. Figure 3-7(a) shows the absolute uncertainty in the reflection coeffi cient as a function of the SNR, and Figure 3-7(b) shows the absolute uncertainty in the normalized specific acoustic impedance. The results in these figures suggest that the dominant sources of uncertainty in the magnitude and the phase of the reflection coefficient are the microphone location and spacing. In contrast to the sound-hard boundary, there is no dominating uncertainty source for the total uncertainty in the ideal impedance model data. The estimated value of the reflection coeffi cient for the case with a SNR of 40 dB is included in Figure 3-5(a). The esti mates for the normalized specific acoustic impedance are included in Figure 3-5(b). The uncertainty results for the multivariate method and the Monte Carlo simulation are s hown in Figure 3-8(a) for the reflection coefficient and in Figure 3-8(b) for the nor malized specific acoustic impedance. The peaks in the uncertainty are at fr equencies 8.4 and 16.7 kHz, where ksn The

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56 0 2 4 6 8 10 12 14 16 18 20 0.2 0.3 0.4 0.5 |R| 0 2 4 6 8 10 12 14 16 18 20 -150 -100 -50 0 50 Freq [kHz] [deg] (a) 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 16 18 20 -1.5 -1 -0.5 0 0.5 1 Freq [kHz] (b) Figure 3-5: Ideal impedance model and estimat ed values, adapted from (Bielak, Premo and Hersh 1999), for (a) reflection coe fficient and (b) normalized specific acoustic impedance. Model value, Estimated value (40 dB SNR). The two lines ar e indistinguishable.

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57 100 101 10-4 10-3 10-2 10-1 10-1 100 U|R| 100 101 10-2 10-1 100 101 102 U [deg]Relative Uncertainty (a) 100 101 10-4 10-3 10-2 10-1 100 U 100 101 10-3 10-2 10-1 100 101 URelative Uncertainty (b) Figure 3-6: Absolute uncertainty of (a) 00,00R and (b) s pac due to the uncertainties in l s and T for the ideal impedance model at f =5 kHz. l s T.

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58 30 35 40 45 50 55 60 65 70 10-6 10-5 10-4 10-3 U|R| 30 35 40 45 50 55 60 65 70 10-3 10-2 10-1 100 U [deg]Signal-to-Noise Ratio [dB] (a) 30 35 40 45 50 55 60 65 70 10-5 10-4 10-3 10-2 U 30 35 40 45 50 55 60 65 70 10-5 10-4 10-3 10-2 USignal-to-Noise Ratio [dB] (b) Figure 3-7: Absolute uncertainty in (a) 00,00R and in (b) s pac due to the SNR for the ideal impedance model at f =5 kHz.

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59 average percent difference between the two methods is 5% for both the magnitude and phase for the reflection coefficient for the ca se with only 0.01% relative uncertainty in ,, ls and T, and the average percent difference is 2% for the normalized resistance and reactance. For the case with 1% relative uncertainty in ,, ls and T, large differences can be seen in the estimate of the uncertainty in the magnitude of the reflection coefficient at frequencies below 6 kHz. The multivariate method does not reproduce the local minima that the Monte Carlo simulations reveal, but the multivariate method estimates are conservative for this case. The average percent difference between the two methods increases to 75% for the magnitude of the refl ection coefficient, 14% for the phase of the reflection coefficient, 13% for the normali zed resistance, and 16% for the normalized reactance. These increases dem onstrate that uncertainties in ,, ls and T are causing nonlinear perturbations in bot h the reflection coefficient and the normalized acoustic impedance for the case with only 1% relative uncertainty. Thus, the multivariate method fails to give accurate values of the true uncertainty estimates. To increase the accuracy of the multivariate method, the multivariate Taylor series used in the derivations could be expanded to include as many terms as needed fo r the desired accuracy. The best option is to use numerical techniques such as the Mont e Carlo simulations used in this dissertation to propagate the uncertainty. The probability density function is plotted to further investigate the differences between the multivariate method and the Monte Carlo simulations for large uncertainties. This is done for the normalized specific acoustic impedance data and for a frequency of 5 kHz, since there is a large difference be tween the two methods and it avoids complications due to the microphone spacing (see Figure 3-8). Figure 3-9(a) shows the

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60 0 2 4 6 8 10 12 14 16 18 20 10-4 10-3 10-2 10-1 100 101 102 U|R| 0 2 4 6 8 10 12 14 16 18 20 10-2 10-1 100 101 102 103 U [deg]Freq [kHz] (a) 0 2 4 6 8 10 12 14 16 18 20 10-4 10-3 10-2 10-1 100 101 102 U 0 2 4 6 8 10 12 14 16 18 20 10-4 10-3 10-2 10-1 100 101 102 UFreq [kHz] (b) Figure 3-8: Total uncertainty in (a) 00,00R and (b) s pac as a function of frequency for the ideal impedance model. 0.01% Multivariate Method, 0.01% Monte Carlo simulation, 1% Multivariate Method, 1% Monte Carlo simulation. The two lines for the 0.01% relative uncertainty are indistinguishable at most frequencies.

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61 confidence region contours for the case w ith only 0.01% relative uncertainties in ,, ls and T, whereas Figure 3-9(b) is for the case with 1% relative uncerta inties. The figures show that as the uncertainties become larger and cau se nonlinear perturbations in the data reduction equation, the confiden ce region contours change fr om a normal distribution to an irregular “boomerang–shaped” distribution. Thus, the non linear effect invalidates the normal distribution assumption and the uncer tainty must be found from the actual computed distribution resulting from the M onte Carlo simulation. In general, the uncertainty cannot be summarized by the sample mean vector and the sample covariance matrix. The contour line in the joint probabi lity density function (pdf) that represents a probability of 0.95 should be found and used as the 95% confidence region estimate for the uncertainty. To find the uncertainty in the resistance and reactance due to 1% relative uncertainty in each input variable, 25,000 ite rations from the Monte Carlo simulation are used to estimate the joint pdf. The joint pdf is approximated by disc retizing the range of the resistance and reactance into 40 bins each, for a total of 1,600 bins, and is smoothed using a 2 bin 2 bin kernel. Next, 100 contours of constant joint probability density are found, and the joint pdf is integrated within each contour to find the total probability within that contour. Next, the contour corresponding to 95% c overage is found via interpolation. The quoted uncertainty is th en taken as the maximum and minimum values of the contour for each component, such as th e real and imaginary parts of the reflection coefficient or the resistance and reactance. The uncertainty estimates of the magnitude and phase of the reflection coefficient are found from the maximum and minimum values of the magnitude and phase for the contour co mputed from the real and imaginary parts of the reflection coefficients. For the case of the ideal impedance model with 1% relative

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62 2.197 2.198 2.199 2.2 2.201 2.202 0.094 0.096 0.098 0.1 0.102 0.104 0.106 95 85 75 65 55 45 35 25 15 (a) 2.1 2.15 2.2 2.25 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 15 25 35 45 55 65 75 85 95 (b) Figure 3-9: The confidence regi on contours for the resistance and reactance for the ideal impedance model at 5 kHz with only (a ) 0.01% relative unce rtainties and (b) 1% relative uncertainties in l, s, and T.

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63 uncertainty and a SNR of 40 dB for a freque ncy of 5 kHz, the 95% confidence region is given in Figure 3-10, along with the es timated 95% confidence region from the multivariate method and estimated and true values of the normalized impedance. This figure illustrates the difference in the pred icted uncertainty regions between the two methods and how much larger the Monte Carlo region is. The quoted uncertainty for this case is best given as a range si nce it is asymmetrical about the estimate. The estimate of the normalized resistance is 2.20 with a 95% confidence interval of 2.11,2.23 and the estimate of the reactance is 0.1 w ith a 95% confidence interval of 0.3,0.4 For comparison, the uncertainty estimates from the multivariate method are 0.03 for the resistance and 0.4 for the reactance. 2.1 2.15 2.2 2.25 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3-10: Confidence region of the ideal im pedance model at 5 kHz with 1% relative input uncertainty and 40 dB SNR. Monte Carlo confidence region, Multivariate confidence region, estimated impedance, true impedance, Monte Carlo method simulta neous confidence interval estimates, Multivariate method simultaneous confidence interval estimates.

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64 3.4 Experimental Methodology To demonstrate the multivariate method and the Monte Carlo method on experimental data, two experi mental setups are develope d, using two different size waveguides. The larger wave guide has a plane wave operatin g bandwidth up to 6.7 kHz, whereas the small waveguide has a plane wa ve operating bandwidth up to 20 kHz. A schematic of the experimental setup is show n in Figure 2-11. Each component of the experimental setup and the data acquisition and analysis routine will be discussed in turn. 3.4.1 Waveguides The larger waveguide is approximately 96 cm long and has a square cross-section measuring 25.4 mm on a side. The walls of the waveguide are cons tructed of 22.9 mmthick aluminum (type 6061-T6). The cuton frequencies for the higher-order modes, given in Table 2-2, show that the limiting bandwidth for the TMM is 6.7 kHz for this waveguide. The location of the microphone, l, and the microphone spacing, s, is measured before the experiment using digital calipers (with an accuracy of 0.05 mm ) The measurement is repeated 45 times and the data are used to compute the best estimates of the microphone location and spaci ng and the random uncertainty of the geometric center of the microphones. A bias uncertainty due to the difference between the geometric center and the ac oustic center is neglected sinc e over the entire operational frequency range of the large waveguide, the microphone diameter is assumed to be small compared to the wavelength and thus the microphones represent point measurements. The total uncertainty in the locations of the microphones is taken as the root-sum-square of the random uncertainty and the accuracy of the calipers. The microphone located closest to the specimen with a 95% confid ence interval estimate is located 32.00.8 mm

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65 from the specimen. The spacing between th e two microphones with a 95% confidence interval estimate is 20.71.1 mm The smaller waveguide is approximately 87 cm long and has a s quare cross-section measuring 8.5 mm on a side. The walls of th e waveguide are constructed out of at least 12.7-mm thick aluminum (type 6061-T6). Th e cut-on frequencies for the higher-order modes, given in Table 2-1, show that the limiting bandwidth for the TMM is 20 kHz for this waveguide. The location of the microphone, l, and the microphone spacing, s, is measured before the experiment using digital calipers. The measurement is repeated 45 times and the data are used to compute the best estimates of the microphone location and spacing and the random uncertainty. The uncer tainty in the acoustic centers of the microphones is estimated ad hoc to be 1.5 mm which is considered here because of the increased frequency range as compared to th e other waveguide. For this waveguide, the diameter of the microphones can no longer be considered small compared to the wavelength and the microphone measurements no longer represent a point measurement. The Helmholtz number (kd) is on the order of un ity at approximately 10 kHz The total uncertainty is taken as the root-sum-square of the random uncertainty the accuracy of the calipers, and the bias due to the acoustic ce nters. The microphone lo cated closest to the specimen with a 95% confidence inte rval estimate is located 38.12.0 mm from the specimen. The spacing between the two mi crophones with a 95% confidence interval estimate is 12.72.0 mm The majority of the uncertain ty in the microphone locations is due to the uncertainty in the acous tic centers of the microphones.

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66 3.4.2 Equipment Description The compression driver is a BMS 4590P, with an operating frequency range of 0.2 to 22 kHz, powered by a Techron 7540 power am plifier. The drive signal is generated by a Brel and Kjr Pulse Analyzer System, which also acquired and digitized the two microphone signals with a 16-bit digitizer. The measurement microphones are Brel and Kjr Type 4138 microphones (3. 18 mm diameter) and are inst alled into the waveguide with their protective grids attached to the microphone. The microphones are calibrated only for magnitude before mounting in the wa veguide using a Brel and Kjr Type 4228 Pistonphone. Atmospheric temperature is measured using a surface-mounted, 100platinum resistive thermal device (Ome ga SRTD-1) with an accuracy of 2K 3.4.3 Signal Processing For the large waveguide, the two microphone signals are sampled at a rate of 16.4 kHz with a record length of 62. 5 ms for a total of 1,000 spect ral averages. The frequency resolution is 16 Hz. For the small waveguide the two microphone si gnals are sampled at a rate of 65.5 kHz with a record length of 31.3 ms for a total of 1,000 spectral averages. The frequency resolution is 32 Hz. A peri odic pseudo-random signal is used as the excitation signal is to the compression driver. 3.4.4 Procedure The microphones are first calibrated. The excitation signal is applied, and the amplifier gain is adjusted such that the sound pressure level at the reference microphone is approximately 100-120 dB (ref.20Pa) for all frequency bins. Then the full-scale voltage on the two measurement channels of th e Pulse Analyzer System is adjusted to maximize the dynamic range of the data system The excitation signal is turned off and

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67 the microphone signals are measured to estimate the noise spectra (see Appendix B). The input and output signals for FRF estimation are assumed to contain uncorrelated noise and there the real and imaginary parts of the FRF may be correlated as shown in Appendix B. Next, the excitation signal is turned on and the tw o microphone signals are recorded with the microphones in their original positions and switched positions. The time-series data are used to compute the required spectra and ultimately ˆ H 00,00R and s pac via Equations (3.10), (2.27), and (2.21), respectively. For the temperature measurement, the ra ndom uncertainty is estimated from the standard deviation of at least 100 measuremen ts, and the bias uncertainty is estimated by the accuracy of the RTD (2 K). The total unc ertainty in temperature is computed from the root-sum-square of the random and bias uncertainties. The uncertainties in the reflection coeffici ent and the measured normalized acoustic impedance are estimated using both the multivariate method and a Monte Carlo simulation (see Figure 3-1). The input distributions for l, s and T are assumed to be independent Gaussian distributions and the input distribution for ˆ H is assumed to be a bivariate normal distribution computed from Appendix B. A specific form for the output distribution of the Monte Carlo simulation is not assumed as described previously at the end of Section 3.3.2. This approach is chos en because of its ability to handle the large perturbations that the uncer tainties in the temperatur e and the microphone locations represent. Results for four specimens are presented in Chapter 5.

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68 CHAPTER 4 MODAL DECOMPOSITION METHOD Modal decomposition methods presented in th e literature can be separated into two different schemes: correlation and direct me thods (bom 1989). Correlation approaches determine the modal amplitudes by measuring the temporal and spatial correlation of acoustic pressure inside the waveguide. Direct methods, however, use point measurements to compute the modal amplitude s from a system of equations derived from an analytical propagation m odel. Accurate propagation models exist for rectangular, square, or cylindrical ducts with rigid walls. However, multiple independent sources are required to resolve the acoustic properties of the test specimen, such as the reflection coefficients, mode scattering coefficients, and acoustic impedances This dissertation uses the latter approach and computes the m odal amplitudes by solving a system of linear equations. This method is also amenable to a least-squares solution for added robustness. Focusing now on prior research on direct methods, early work by Eversman (1970) investigated the energy flow of acoustic wave s in rectangular ducts but did not consider the decomposition of modal components. Moore (1972) was one of the first to investigate direct methods to determine the source distribution for ducted fans but limited his results to estimates of the sound pressure levels for each circumferential mode and neglected radial modes. Following this, Zinn et al. (1973) inve stigated measuring acoustic impedance for higher-order modes by adapting the standing-wave method. Yardley (1974) then added the effects of mean flow and reflected waves to determine the source distribution of a fan but did not e xpand the method to compute the reflection

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69 coefficient matrix. Yardley also suggested that the microphones should all be mounted flush to the waveguide or duct. Pickett et al. (1977) continued to improve the direct method by adding a discussion of optimum microphone locations but limited their algorithms to a deterministic system of equati ons. Only results at the fan blade passage frequency were reported. Moore (1979) c ontinued the analysis evolution by comparing integral algorithms for the solution of the deterministic set of equations to the leastsquares approach. He concluded that the deterministic system was susceptible to measurement noise, and the least-squares so lution provided robustness and approached the integral method solution in the limit of infinite measurement points. Again, his results were limited to estimates of the modal amplitudes. Subsequently, Kerschen and Johnston ( 1981) developed a direct technique for random signals, but restricted th e method to only incident waves. Pasqualini et al. (1985) concentrated their efforts on a transform sc heme for a direct method for circular or annular ducts only. A method for use with transient signals was then developed by Salikuddin and Ramakrishnan (S alikuddin 1987; Salikuddin and Ramakrishnan 1987). Continuing this line of work, bom (1989) ex tended the direct method to any type of signal by measuring the frequency response function between microphone pairs. bom noted difficulties associated with genera ting the necessary independent sources to calculate the reflection coefficient matrix. Akoum and Ville (1998) then developed and applied a direct method based on a Fourier-Lo mmel transform to the measurement of the reflection coefficient matrix at the baffled e nd of a pipe. They developed an apparatus for generating the necessary independent s ources by mounting a compression driver to the side of a circular wavegui de on a rotating ring. Their re sults were in good agreement

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70 with theoretical predictions for the normal mode, but they stated that discrepancies existed for the higher-order modes since all of the data were near the cut-on frequency. Most recently, Kraft et al. (2003) discusse d the development of a modal decomposition experiment using four microphones but did not provide any results. The contribution of this chapter is to ad apt a direct MDM based on a least-squares scheme to a square duct and to use simple sources to acquire the data necessary to estimate the entire reflection coefficient matr ix and the acoustic impedance at frequencies beyond the cut-on frequency of higher-order mode s. The outline of the chapter is as follows. The next section derives the da ta reduction procedure for estimating the complex modal amplitudes, the reflection coefficient matrix, and the acoustic impedance values from the measured data. Section 4.2 outlines the experimental procedure and analysis parameters. This section concl udes with a brief discussion concerning the sources of error. 4.1 Data Reduction Algorithm The MDM developed here is restricted to time-harmonic, linear, lossless acoustics without mean flow governed by the Helmholtz equation. The solution is given in Equation (2.5) in Chapter 2, but is repeated here for convenience as ,zzjkdjkd mnmnmn mnPxyAeBe, (4.1) where ,coscosmnmn x yxy ab (4.2) for a rigid-walled square duct. The dispersi on relation and an expr ession for the cut-on frequencies are given in Secti on 2.1.1 in Equation (2.8).

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71 4.1.1 Complex Modal Amplitudes The experimental procedure flush-mounts a number of microphone s in the sides of the waveguide, as in the TMM. The nu mber and locations of the microphones are selected to observe the desired modes. A test frequency is selected and the total number of propagating modes, is found from the equation for the cut-on frequency, Equation (2.8). The minimum number of microphone measurements required to uniquely determine the acoustic pressure is for this test frequency is 2 (bom 1989). Next, Equation (4.1) is written for each micr ophone measurement, summing only over the propagating modes for that frequency, to form a system of equations 11 221 11 2 22, ,zz zz zzjkdjkd mnmnmn mn jkdjkd mnmnmn mn jkdjkd mnmnmn mnPxyAeBe PxyAeBe PxyAeBe M, (4.3) where the subscript on P represents the mi crophone location and is the number of microphone measurements, which must be equal to or larger than 2 To decompose the sound field, the microphones should be located w ith some transverse separation and some axial separation. A simple way to confi gure the microphone locations is to group the microphones into two groups and locate each gr oup at a separate axial location. The system of equations can be compac tly expressed in matrix form as PW L, (4.4) where P is the 1 vector of measured complex acoustic pressure amplitudes, W is the 21 vector of the complex modal amplitudes given by

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72 A W B (4.5) and L is the 2 matrix of the coefficients from Equation (4.3), composed of the transverse function and the propagation exponential. The coefficient matrix has a special form; it is composed of two sub-matrices that are complex conjugates. As a result of this structure, the determinant of the L matrix has an imaginary part that is iden tically equal to zero. To avoid this problem, the matrix equation is transformed into a system of tw o real-valued matrix equations, each with a coefficient matrix that has a non-zero determinant RI RIPjPjWjW RILL, (4.6) where the subscripts “ R ” and “ I ” denote the real and imaginary parts, respectively (Rao 2002). The expression is rearranged by carryi ng out the multiplicati on and collecting the real and imaginary parts R R I IPW PW RI IRLL LL. (4.7) The solution to Equation (4.7) is found, for example, via Gaussian elimination for the deterministic case in which 2 For the overdetermined case in which 2 a least-squares solution to Equation (4.7) is desired for a robust solution, and this can be found by solving the normal equati ons (Chapra and Canale 2002) R R I IPW PW TT RIRIRI IRIRIRLLLLLL LLLLLL, (4.8) where the superscript T represents the transpose of the matrix (Chapra and Canale 2002).

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73 4.2.2 Reflection Coefficient Matrix With the existence of higher-order propa gating modes, an incident acoustic mode now may reflect as the same m ode and scatter into different modes. This increases the complexity of characterizing the specimen, as a single reflection coefficient no longer describes the acoustic interaction. Instead, a reflection coefficient matrix is defined as B A R, (4.9) where the size of R is and the vectors A and B are 1 (bom 1989; Akoum and Ville 1998). The elements of R are represented by ,mnqrR where the first index, mn, is the mode number for the refl ected mode and the second index, qr, is the mode number for the incident mode. The diagonal elements, mnmnR represent samemode reflection coefficients, wh ile the off-diagonal elements, ,mnqrR represent the mode scattering coefficients. To determine the unknown reflection coefficient matrix, a minimum of linearly independent so urce conditions must be measured (bom 1989; Akoum and Ville 1998). The additional vect ors of the incident and reflected complex modal amplitudes are combined together to form matrices such that 1212BBBAAA RLL (4.10) which can then be solved for the reflection coefficient matrix. The approach in this work, described below, to generate multiple independent sources is to place various restrictor plates between the waveguide and the comp ression driver, the purpose of which is to emphasize one of the modes. Previous re searchers placed the compression drivers perpendicular to the waveguide on a rotati ng ring and varied the location of the

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74 compression drivers relative to the micr ophones (Pasqualini, Ville and Belleval 1985; Akoum and Ville 1998; Blackstock 2000). 4.2.3 Acoustic Impedance The acoustic impedance ratio is define d only for same-mode reflections as (Blackstock 2000) 0 ,cos 1 1 cosspecimen tr mn mnmn mn mnmn mnZ R Z R (4.11) where s pecimenZ and 0 Z are the characteristic impedances of the specimen and medium, respectively, and tr mn is the angle of transmission for the ,mn mode. The acoustic impedance ratio is also called the ratio of oblique incidence wave impedance by (Dowling and Ffowcs-Williams 1983). The nor malized specific ac oustic impedance or the normalized characteristic impedance is obtained from Equation (4.11) as 0,cos 1 cos1tr mn specimen mnmn spac mnmnmnZ R ZR (4.12) Without further information concerning tr mn only the acoustic impedance ratio can be computed from the results of the MDM. Ho wever, locally reactive materials are desired for aeroacoustic applications as engine n acelle liners (Motsinger and Kraft 1991), and therefore are commonly tested (Jones et al. 2003; Jones et al 2004). A locally reactive material is a material whose impedance is independent of the angle of incidence and therefore is assumed to have a transmission angle of approximately zero (Morse 1981). In this case, Equation (4.12) simplifies to

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75 0,1 1 cos1specimen mnmn mnmnmnZ R ZR (4.13) which represents the normalized surface response impedance (Dowling and FfowcsWilliams 1983) and can be estimated from the MDM. To check the validity of the locally reactive assumption, the normalized sp ecific acoustic impeda nces, from Equation (4.13), for all modes at a given frequency should be equal. For the TMM, only plane waves are presen t and, hence, only the normal-incident specific acoustic impedance is determined. The angle of incidence for the (0,0) mode acoustic impedance is seen from Equation (2. 13) have normal incidence. Thus, the (0,0) mode acoustic impedances from Equations (4 .11)-(4.13) are identical to the estimate from the TMM and the two estimates can be compared. The higher-order modes assumed in the MDM can be thought of as plan e waves at an oblique angle of incidence, as discussed in Section 2.1.3. The effect of angle of incidence causes the acoustic impedance value to differ from the specifi c acoustic impedance value, and thus both estimates of impedance must be considered to fully characterize the specimen. 4.2.4 Acoustic Power In addition to the acoustic impedance ra tio, the absorption coefficient is an important parameter to characterize acoustic materials. The absorption coefficient, is defined as the amount of acoustic power ab sorbed by the specimen normalized by the incident power (ISO-10534-2: 1998 1998) and is given as 1ir r iiWW W WW (4.14) where iW and rW represent the power in the incide nt and reflected acoustic fields, respectively. In the case of the TMM, the absorption coefficient only considers the

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76 power contained in the plane wave mode, but in the MDM, the absorption coefficient will encompass the total power abso rbed in all the propagating modes. Equation (4.14) assumes that no acoustic power is transmitted through the waveguide into the surrounding environment, hence demonstrati ng the need for term inating the specimen with a rigid back plate and for insuring proper sealing of the waveguide. In this case, all the acoustic energy that is not absorbed by th e specimen and dissipated as heat is send back down the waveguide. Expressions for the incident and reflected powers are derived from integrating the acoustic intensity in the d-direction over the cross-section of the waveguide to obtain the total power, W, given by 001 Re 2aa dd SyxWfIdSPUdxdy (4.15) where dU is the acoustic velocity perturbation along the d-axis in the frequency domain and is found from Euler’s equation (Blackstock 2000) as 0d j P U ckd (4.16) where 0kc The orthogonal properties of the nor mal modes in the acoustic pressure solution given in Equation (4.1), and in the ac oustic velocity perturba tion solution, given in Equation (4.16), allow for the expression of the total power to be simplified and ultimately separated into two parts. Each part only contains the modal amplitudes for either the incident waves or the reflected waves. The resulting expressions for the incident and reflected powers are 2 2 00 08MN izmnmn mna WkA ck (4.17) and

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77 2 2 00 08MN rzmnmn mna WkB ck. (4.18) The full derivation is given in Appendix D. The absorption coefficient not only provides an estimate of the sound absorption capabilitie s of a material, but also provides a check on the measurement. The absorption coeffi cient is bounded between zero and unity, and values outside this range indicate a problem w ith the experimental setup and procedure. 4.2 Experimental Methodology To verify the data reduction routine outlined above and obtain acoustic impedance data beyond the cut-on frequency, an experiment al apparatus is developed. The actual results are presented in the next chapter along with the results for the TMM. The experimental procedure to acquire and redu ce the data is similar to the TMM. A compression driver is mounted at one end of a waveguide, and the test specimen is mounted at the other end. For the MDM, eight microphones are flush-mounted in the duct wall at two axial locations near the spec imen to resolve the incident and reflected waves. Fourier transforms of the phase-locked, digitized pr essure signals at each location are used to estimate the complex acoustic pre ssure and thus the modal coefficients and reflection coefficient matrix. A schematic of the experimental setup is shown in Figure 4-1, with eight microphones flush-mounted in the waveguide. Each component of the experimental setup and the data acquisition and analysis routine will be discussed in turn. 4.2.1 Waveguide The waveguide used in the measurements presented is approximately 96 cm long and has a square cross-section measuring 25. 4 mm on a side. The walls of the waveguide are constructed of 22.9 mm-thick aluminum (t ype 6061-T6). The cut-on frequencies for the higher-order modes, given in Table 4-1, show that the limiting bandwidth for the

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78 TMM is 6.7 kHz for this waveguide, as opposed to 13.5 kHz when the MDM is used with the first four modes. To resolve these four modes, ei ght microphones are placed in two groups of four microphones at two axial locatio ns. The placement is chosen such that each microphone is not located at the node line of any of the modes of interest and to achieve a sufficient signal-to-noise ratio. The locations of the eight microphones are provided in Table 4-2. The independent sources for the MDM are generated via the four different restrictor plates s hown in Figure 4-2, each one designed to emphasize one or more of the first four modes. ip rp Figure 4-1: Schematic of the experime ntal setup for the MDM (some microphone connections are left out for clarity). Table 4-1: Cut-on frequencies in kHz for the higher-order modes. m \ n 0 1 2 3 0 0 6.75 13.5 20.3 1 6.75 9.55 15.1 21.4 2 13.5 15.1 19.1 24.4 3 20.3 21.4 24.4 28.7 4.2.2 Equipment Description The compression driver is a BMS 4590P, with an operating frequency range of 0.2 to 22 kHz, powered by a Techron 7540 power amp lifier. The drive signal is generated by

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79 Table 4-2: Microphone m easurement locations ( a = 25.4 mm). Microphone x,y,d Location [mm] Microphone x,y,d Location [mm] 1 0.25,0,1.6aa 5 0.25,0,1.1aa 2 ,0.25,1.6aaa 6 ,0.25,1.1aaa 3 0.75,,1.6aaa 7 0.75,,1.1aaa 4 0,0.75,1.6aa 8 0,0.75,1.1aa (0,0) restrictor plate (1,1) restrictor plate (0,1) restrictor plate (1,0) restrictor plate a a Figure 4-2: Schematic of the four restrictor plates. (T he dotted line represents the waveguide duct cross-section.) a Brel and Kjr Pulse Analyzer System, wh ich also acquired and digitized the eight microphone signals with a 16-bit digitizer. The measurement microphones are Brel and Kjr Type 4138 microphones (3.18 mm diamete r) and are installed into the waveguide with their protective grids attached to the microphone. The microphones are calibrated only for magnitude before mounting in the waveguide. The phase mismatch between the eight microphones was measured in previous experiments, with each microphone flush mounted at the end of the large waveguide with a reference microphone, up to 6.7 kHz and was found to be no greater than 5 o. This error is found to be acceptable (less than 10 % uncertainty for the modal amplitudes and re flection coefficients) from the results of

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80 the numerical uncertainty studies in Section 4.2.4. Atmosphe ric temperature is measured using a 100platinum resistive thermal device with an accuracy of 2K 4.2.3 Signal Processing All eight microphone signals are measured a nd subsequently processed with a fast Fourier transform algorithm. The frequency re solution is 16 Hz with a frequency span from 0.3 to 13.5 kHz. The 1,000 linear aver ages are processed using a uniform window with no overlap. Leakage is eliminated by the use of a pseudo-random periodic signal to excite the compression driver. To ensure s ynchronous data acquisition, the sampling is triggered by the start of the generator signa l in a phase locked acquisition mode. The data are then processed using the MDM described above. 4.2.4 Numerical Study of Uncertainties The main sources of error for the MDM are the signal-to-noise ratio, microphone phase mismatch, uncertainties in the measur ements of the microphone locations, and the temperature. The frequency scaling of the un certainty in the computed values from the MDM is also important, as the goal of the MDM is to extend the frequency range of acoustic impedance testing. Numerical st udies have been conducted concerning the effects of the individual error sources and the frequency scali ng of the total error, and are only summarized here for brev ity; the results are given in Appendix E. These studies were performed for an approximate sound-hard te rmination, with four different vectors of incident complex modal amplitudes. The reflected modal amplitudes are computed from Equation (4.9) and the data are then used to calculate time-series data. The time-series data are then processed using the MDM de scribed above. The root-mean-square (rms) normalized error between the elements of the calculated reflection co efficient matrix and

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81 the modeled reflection coefficient matrix is us ed to gauge the uncertainty of the MDM. The numerical studies are performed at a fre quency of 12 kHz to avoid pressure nodes for the microphone locations listed in Table 4-2. The simulations varied the error introduced into the simulated input signals to th e MDM and computed the perturbed output reflection coefficient matrix for each of the error sources individually. These results showed that the MDM gives reliable and accu rate estimates (with ~10% uncertainty) for the complex modal coefficients and the reflec tion coefficient matrix. The influence of evanescent modes can be simulated to determ ine the magnitude of the bias error they cause if the amplitudes of the incident a nd reflected evanescent waver are known.

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82 CHAPTER 5 EXPERIMENTAL RESULTS FOR ACOUSTIC IMPEDANCE SPECIMENS Previous chapters have described tw o methods, the TMM and the MDM, for measuring acoustic impedance in detail, includ ing discussions about uncertainty sources. This chapter focuses on presenting the data fr om these methods when applied to various specimens. Five specimens are tested and pr esented, but some specimens are only tested with certain methods because of sample size issues described in Chapter 2. The five specimens are listed below with the methods used to test them. 1. Ceramic tubular honeycomb with 65% porosity* (CT65) (TMM, High frequency TMM, and MDM). 2. Ceramic tubular honeycomb with 73% por osity (CT73) (TMM, High frequency TMM, and MDM). 3. Rigid termination (TMM, High frequency TMM, and MDM). 4. SDOF liner (TMM and MDM). 5. Mode scattering specimen (MDM only). The first two specimens are chosen to re present a typical sound-soft boundary that consists of simple structures that can be easily modeled and does not represent a particularly hard measurement to make accura tely in terms of its uncertainty and are tested first to gain confidence in the methods The rigid termination is chosen to shake down the experiment rig since the uncertainty an alysis has show that this specimen is the most sensitive to input uncertainties and e rrors. The SDOF liner is representative of Porosity is the percentage of the open area in the te st surface versus the area of the entire test surface.

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83 actual flight hardware and the mode scat tering specimen is design to maximum the scattering of incident energy to investig ate what values of the mode scattering coefficients are possible. The rest of the ch apter presents the resu lts for each specimen, one at a time, in the order presented in the list above. All estimates are computed neglecting the effects of dissipation and disp ersion and all uncertainty estimates are the 95% confidence interval estimates made via the Monte Carlo method assuming an arbitrary distribution. 5.1 Ceramic Tubular Honeycomb with 65% Porosity The specimen chosen for the analysis is a ceramic tubular specimen with 65% porosity (CT65) shown in Figure 5-1 and each ceramic tubular cell has an estimate hydraulic diameter of approximately 0.443 mm The specimen is 56 mm long for both waveguides and is encased in at least 12.7 mm-thick alumi num, except for the test face, to prevent the loss of acoustic energy from the sides of the specimen. Approximately 10% of the test face of the large waveguide specimen is damaged due to cutting the material and placing the specimen in the moun ting fixture, whereas only approximately 2% of the small waveguide test face is dama ged. The results for the same material are given in Jones et al. (Jones, Watson and Parro tt 2004), but for a di fferent depth of 77.5 mm. Figure 5-1: Photograph of the CT65 material.

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84 5.1.1 TMM Results The results from the standard TMM are presented in Figure 5-2 for the reflection coefficient and Figure 5-3 for the normalized specific acoustic impedance, both with uncertainty estimates. The graphs for the coherence and the FRF are given in Appendix F. The coherence between the two microphones for the original and switched positions is above 0.99 for the entire bandwidth. The uncer tainty estimates of the magnitude of the reflection coefficient show a periodic increase and decrease in the confidence intervals along with an overall increase with frequency. The minimums are separated by the repeating pattern of approximately 600, 1, 000, and 1,000 Hz. The periodic structure suggests there are test conditions where the unc ertainty in the magnit ude of the reflection coefficient is only a weak function of the i nput uncertainties. The uncertainty estimates for the phase of the reflection coefficient and for the resistance and reactance show the same general increase in the confidence interval with frequency. The uncertainty in the phase of the reflection coefficient aro und 1.25 and 4.25 kHz, where phase wrapping occurs, appears unreliable, probably because the numerical techniques with the current discretization setting for approximation of the probabil ity density function could not resolve the phase wrapping. Another discre tization resolution issue is also apparent between 5.3 and 5.4 kHz in the reactance. In this narrow frequency range, the uncertainty estimate is jagged, unlike the estim ates over the remainde r of the bandwidth. Also, the asymmetrical uncertainty estimates at higher frequencies in the presented Monte Carlo simulation results ar e easily seen in Figure 5-3. For example, at 5.996 kHz the normalized specific acoustic impedance estimate is 4.50.6j but the uncertainty estimate for the normalized resistance is 2.7,5.1 and 2.7,1.6 for the normalized

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85 reactance. Also, the maximum uncertainty in the normalized specific acoustic impedance is at anti-resonance at 2.61 and 5.50 kHz and the minimum uncertainty appears at the resonance at 1.26 and 4. 24 kHz. At anti-resonance, the specimen acts as a high impedance specimen, like a soundhard boundary, and the results in Chapter 3 s howed that this condition is then most sensitive to the input uncertainti es. At resonance, there is a maximum of particle velocity in the ceramic tubes and thus, increased dissipation. This makes the specimen appear to be sound-soft and again from Chapter 3, le ss sensitive to input uncertainties. This supports the conclusion from the multivariate uncertainty analysis of the TMM that the uncertainty is a function of the specific acous tic impedance value of the specimen. The periodic structure of the uncertainty es timate of the magnitude of the reflection coefficient shows that there are frequencies wh ere the estimate is insensitive to the input uncertainties. This follows the trends displayed in the derivatives of the reflection coefficient from Sections 3.2.2-3.2.4, where the derivatives containe d a sine function. This could be used to an advantage in th e design of future waveguide to minimize the uncertainty in the estimates. The uncertainty estimates for the normalized resistance and reactance again show the maximum and mini mum at anti-resonance and resonance but the periodic structure is dimi nished and barely detectable This suggests that the uncertainty estimates for the normalized re sistance and reactance are less dependent on the input uncertainties that produced this structure.

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86 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-2: Reflecti on coefficient for CT65 for the TMM. Estimated value, Uncertainty estimates. 1 2 3 4 5 6 0 1 2 3 4 Freq [kHz] 1 2 3 4 5 6 -4 -2 0 2 Freq [kHz] Figure 5-3: Normalized specific acoustic impedance estimates for CT65 via TMM. Estimated value, Uncertainty estimates.

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87 5.1.2 High Frequency TMM Results The results from the high frequency TMM for a single specimen are presented in Figure 5-4 for the reflection coefficient and Figure 5-5 for the normalized specific acoustic impedance, both with uncertainty es timates. The graphs for the coherence and the FRF are given in Appendix F. The c oherence between the two microphones for the original and switched positions is above 0.9999 for the entire bandwidth. The uncertainty estimates of the magnitude of the reflecti on coefficient show a periodic increase and decrease in the confidence intervals along w ith an overall increase with frequency, the same as with the standard TMM results. The uncertainty estimates for the phase of the reflection coefficient and for the resistance a nd reactance show the same general increase in the confidence interval with frequency, again the same as with the standard TMM results. The difference is apparent for fr equencies above 10 kHz, where the uncertainty estimates become extremely unpredictable and large. In this high frequency range, the numerical technique can no longer resolve the confiden ce region, as the output uncertainty has become extremely sensitive to the input uncertainties. The uncertainty in the phase of the reflection coefficient cove rs the entire phase space from +180 to -180 degrees. The major contributor to the large and unpredictable confidence regions is the uncertainty in the microphone locations At 10 kHz, the wavelength is 34.4 mm and an uncertainty of 1 mmhas become approximately 3% of the wavelength. The percentage increases to approximately 6% at 20 kHz. To improve the measurement of the location of the two microphone centers, the distance could be meas ured with an advanced technique, such as describe in literature (Katz 2000).

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88 The estimated values for the magnitude of the reflection coefficient and normalized specific acoustic impedance show an anomaly in a small frequency range centered at 13.5 kHz. At this frequency, the microphone spaci ng is exactly equal to a wavelength and the TMM is subjected to a know n singularity, as shown in Ch apter 3. The uncertainty estimates at this frequency tends towards infi nity to help reveal the singularity and show the deficiency of the TMM. The data in th is range should be replaced by data from an additional measurement of the TMM with a di fferent microphone spacing, but this is not done since the goal of this dissertation is to illustrate the strength s and weaknesses of the TMM. 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-4: Reflection coefficient fo r CT65 for the high frequency TMM. Estimated value, Uncertainty estimates.

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89 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -4 -2 0 2 Freq [kHz] Figure 5-5: Normalized specific acoustic impedance estimates for CT65 via the high frequency TMM. Estimated value, Uncertainty estimates. 5.1.3 MDM Results and Comparison The standard TMM, the high frequency TMM and the MDM experimental results for the CT65 specimen are compared in this section, but first the MDM results for the incident and reflected pressure field and the power absorption coefficient are presented. The cut-on frequencies for the higher-order modes are 6.83 kHz for the (1,0) and (0,1) modes and 9.66 kHz for the (1,1) mode, base d on the measured temperature during the MDM measurements. Figure 5-6 shows the inci dent pressure field measured near the specimen for each of the restrictor plates and reveals that one of the sources generates a pressure level approximately 5-10 dB higher th an the other three sources for the higherorder modes. The exception is the plane wave mode, where the restrictor plate produces pressure levels only marginally higher than th e others. The data for the reflected pressure fields for each of the restrictor plates are presented in Figure 5-7, and this figure shows

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90 evidence of mode scattering, since there is no longer the same difference between the pressure amplitude of each of the sources, as shown in the incident pressure field. 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-6: Incident pressure field for the MDM for CT65. 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-7: Reflected pressure field for the MDM for CT65.

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91 The absorption coefficient is shown in Figure 5-8 for all four rest rictor plates. The data show that the total power absorbed is dependent on the modal content of the acoustic field, since the absorption coefficient va ries between the sources after the cut-on frequency for the first higher-ord er mode. The data also conf irms the experimental setup and measurement, since the computed value for the absorption coefficient remains bounded between zero and unity. 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 Freq [kHz] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-8: Absorption co efficient for CT65. The magnitude and phase of the reflecti on coefficients are shown in Figure 5-9 along with results from both TMM measuremen ts. The estimate of the plane wave reflection coefficient provided by the MDM ag rees with the TMM of the large waveguide to within its 95% confidence interval estimat es, which are not shown in the figures for clarity. The estimates from the small waveguide do not agree with the estimates from the large waveguide over the entire frequency range to within their 95% confidence intervals. A definite shift of the estimate of the small waveguide can be seen from the estimates for the large waveguide in both the reflection co efficient and the normal-incident normalized specific acoustic impedance values. The figure also shows that the estimate of the plane

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92 wave reflection coefficient by the MDM is a ffected by the same singularity that affects the high frequency TMM at 13.5 kHz. Figur e 5-10 through Figure 5-13 show the mode scattering coefficients estimated by the MDM. The magnitude of the mode scattering coefficients is less than 0.2 for all frequenc ies, except for the frequencies near the (1,1) mode cut-on or for coefficients going into th e plane wave mode near 13.5 kHz. Both of these exceptions are unreliable due to e ither the cut-on phenomenon or the microphone spacing issue. The results for the acoustic impedance ratio are shown in Figure 5-14, and Figure 5-15 shows the normalized specific acoustic impe dance. The test specimen is assumed to be a locally reactive material and thus the transmission angle is assumed to be normal to the surface. The data show a number of re sonant frequencies monotonically decaying in 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-9: Comparison of th e reflection coefficient estimates for CT65 via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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93 7 8 9 10 11 12 13 0 0.1 0.2 0.3 0.4 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,0) to (1,0) from (0,0) to (0,1) from (0,0) to (1,1) Figure 5-10: Mode scattering coefficients fo r CT65 from the (0,0) mode to the other propagating modes. 7 8 9 10 11 12 13 0 0.1 0.2 0.3 0.4 0.5 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,0) to (0,0) from (1,0) to (0,1) from (1,0) to (1,1) Figure 5-11: Mode scattering coefficients fo r CT65 from the (1,0) mode to the other propagating modes.

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94 7 8 9 10 11 12 13 0 0.1 0.2 0.3 0.4 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,1) to (0,0) from (0,1) to (1,0) from (0,1) to (1,1) Figure 5-12: Mode scattering coefficients fo r CT65 from the (0,1) mode to the other propagating modes. 10 10.5 11 11.5 12 12.5 13 0 0.1 0.2 0.3 |R|Freq [kHz] 10 10.5 11 11.5 12 12.5 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,1) to (0,0) from (1,1) to (1,0) from (1,1) to (0,1) Figure 5-13: Mode scattering coefficients fo r CT65 from the (1,1) mode to the other propagating modes.

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95 amplitude, which are 1.26, 4.24, 7.20, and 10. 3 kHz, as identified from the standard TMM and MDM results. If each ceramic cell is modeled as a rigid-wall waveguide with a rigid termination, the normalized specific acoustic impedance can be calculated at the entrance to the ceramic cells from (Kin sler, Frey, Coppens and Sanders 2000) 0cot s paccell j kl kc (5.1) where celll is the length of the specimen and the complex wavenumber, k, is given by Equation (2.30) and accounts for boundary la yer dissipation and dispersion. The calculated s pac from Equation (5.1) is shown in Fi gure 5-15. The comparisons between the experimental measured normalized speci fic acoustic impedance and the calculated normalized specific acoustic impedance are r easonable and provide phys ical insight into the mechanics of the ceramic tubular material as an acoustic liner. The estimate of the resonances between the model and the TMM in the large waveguide and the MDM match well, but the model underestimates the norma lized resistance possibly due to heat conduction within the specimen material. For the locally reactive a ssumption to be valid, the estimates for the normalized specific acoustic impedance should be identical, regardless of the mode. Uncertainty estim ates of the normalized specific acoustic impedance are needed to fully evaluate this assumption, but the re sults show reasonable agreement, except at the cut-on for the (0,1), (1,0), and (1,1) modes and for frequencies above 13 kHz. These frequency ranges should be investigated further, but these results suggest that the locally reactive assumption is reasonable. The unreliable results at the cut-on frequencies are affected by theoretical phase and angle of incidence for the mode cutting on. At the cut-on frequency, th e phase speed of the newly propagating mode

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96 approaches infinity and the incident angle a pproaches normal incidence. Physically, this situation is unrealizable and results in the i rregular behavior at the cut-on frequencies. Furthermore, even beyond the cut-on frequency the incidence angle is quite large; it is 67o for the (1,1) mode at 10.5 kHz, where in Figure 5-15 the data for the different modes appear to match. The MDM estimates of th e plane wave reflection coefficient and the normal-incidence acoustic impedance agree with the standard TMM estimate within the 95% confidence interval for the standard T MM in the large waveguide. Again, the TMM uncertainty estimates are not shown for clarity. 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 acFreq [kHz] 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 acFreq [kHz] Figure 5-14: Comparison of the acoustic impeda nce ratio estimates for CT65 via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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97 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 sp acFreq [kHz] 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 sp acFreq [kHz] Figure 5-15: Comparison of the normalized specific acoustic impedance estimates for CT65 via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, MDM (1,1) mode, and model. 5.2 Ceramic Tubular Honeycomb with 73% Porosity The multivariate method and the Monte Carlo method are now demonstrated on experimental data for a ceramic honeycomb te st specimen with a porosity of 73% shown in Figure 5-16 and each ceramic tubular cell has an estimate hydraulic diameter of approximately 1.1mm. The specimen is 51 mm long fo r both waveguides and is encased in 12.7 mm-thick aluminum, except for the 25.4 mm-by-25.4 mm test face, to prevent the loss of acoustic energy from the sides of the specimen. Approximately 15% of the test face of the large waveguide specimen is damaged due to cutting the material and placing the specimen in the mounting fixture, whereas approximately 12% of the small waveguide test face is damaged.

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98 Figure 5-16: Photograph of the CT73 material. 5.2.1 TMM Results The results from the standard TMM are pr esented in Figure 5-17 for the reflection coefficient and Figure 5-18 for the normalized specific acoustic impedance, both with uncertainty estimates. The graphs for the coherence and the FRF are given in Appendix F. The coherence between the two microphones for the original and switched positions is above 0.98 for the entire bandwidth. The un certainty estimates of both the reflection coefficient and normalized specific acoustic impedance show the same trends as the CT65 specimen results, such as a periodic structure with approximately the same frequency spacing between the minimums, incr easing uncertainty with frequency, and asymmetrical uncertainty intervals. The uncertainty estimates for the CT73 specimen appear smoother then the estimates for the CT65 specimen and are thus not impacted by discretization errors. Again, the maximu m uncertainty in the normalized specific acoustic impedance occurs at anti-resonance.

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99 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-17: Reflection coefficient for CT73 for the TMM. Estimated value, Uncertainty estimates. 1 2 3 4 5 6 0 2 4 6 Freq [kHz] 1 2 3 4 5 6 -4 -2 0 2 4 Freq [kHz] Figure 5-18: Normalized specific acoustic im pedance estimates for CT73 via the TMM. Estimated value, Uncertainty estimates.

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100 5.2.2 High Frequency TMM Results The results from the high frequency TMM for a single specimen are presented in Figure 5-19 for the reflection coefficient a nd Figure 5-20 for the normalized specific acoustic impedance, both with uncertainty es timates. The graphs for the coherence and the FRF are given in Appendix F. The c oherence between the two microphones for the original and switched positions is above 0.9999 for the entire bandwidth. The uncertainty estimates of the reflection coefficient and normalized acoustic impedance show the same behavior as with the CT65 specimen. The uncertainty estimates below 10 kHz appear reasonable, but beyond 10 kHz the uncertainty estimates are unpredictable because of the large input uncertainties in the location of the microphones as before with the CT65 high frequency TMM results. Again, a singularity in the TMM is present around 13.5 kHz, where the microphone spacing is equal to a half wavelength. 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-19: Reflection coefficient fo r CT73 for the high frequency TMM. Estimated value, Uncertainty estimates.

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101 2 4 6 8 10 12 14 16 18 20 0 2 4 6 Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -4 -2 0 2 4 Freq [kHz] Figure 5-20: Normalized specific acoustic impedance estimates for CT73 via the high frequency TMM. Estimated value, Uncertainty estimates. 5.2.3 MDM Results and Comparison The standard TMM, the high frequency T MM, and the MDM experimental results for the CT73 specimen are compared in this section, but first the MDM results for the incident and reflected pressure field and the power absorption coefficient are presented. The cut-on frequencies for the higher-order modes are 6.83 kHz for the (1,0) and (0,1) modes and 9.66 kHz for the (1,1) mode, base d on the measured temperature during the MDM measurements. Figure 5-21 shows the inci dent pressure field measured near the specimen for each of the restrictor plates a nd again, reveals that one of the sources generates a pressure level approximately 5-10 dB higher than the other three sources for the higher-order modes. The exception is the plane wave mode, where the restrictor plate produces pressure levels only marginally hi gher than the others. The data for the reflected pressure field for each of the restri ctor plates are presented in Figure 5-22, and

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102 again this figure shows eviden ce of mode scattering, since there is no longer the same difference between the pressure amplitude of eac h of the sources as shown in the incident pressure field. The absorption coefficient is shown in Figur e 5-23 for all four restrictor plates. Again, the data show that the total power ab sorbed is dependent on the modal content of the acoustic field, since the absorption coe fficient varies between the sources after the cut-on frequency for the first higher-order mode. The data also confirms the experimental setup and measurement, sinc e the computed value for the absorption coefficient remains bounded between zero and unity. 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-21: Incident pressure field for the MDM for CT73.

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103 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-22: Reflected pressure field for the MDM for CT73. 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 Freq [kHz] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-23: Absorption coefficient for CT73. The magnitude and phase of the reflecti on coefficients are shown in Figure 5-24 along with results from both TMM measuremen ts. The estimate of the plane wave reflection coefficient provided by the MDM ag rees with the TMM in the large waveguide to within its 95% confidence interval estimat es, which are not shown in the figures for clarity. The figure also shows that the estima te of the plane wave reflection coefficient

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104 by the MDM is affected by the same singularit y that affect the high frequency TMM at 13.5 kHz. Figure 5-25 through Figure 5-28 show the mode scattering coefficients estimated by the MDM. The magnitude of the mode scattering coefficients is less than 0.25 for all frequencies except for the freque ncies near the (1,1) mode cut-on or for coefficients going into the plane wave mode n ear 13.5 kHz. Both of these exceptions are unreliable due to either the cut-on phenome non or the microphone spacing issue. The figures also show that the most efficient sca ttering is from the plane wave mode into the (1,1) mode, with a maximum magnit ude of approximately 0.4. The results for the acoustic impedance ratio are shown in Figure 5-29, and Figure 5-30 shows the normalized specific acoustic impe dance. The test specimen is assumed to be a locally reactive material, and thus the transmission angle is assumed to be normal to 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-24: Comparison of the reflection co efficient estimates for CT73 via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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105 7 8 9 10 11 12 13 0 0.2 0.4 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,0) to (1,0) from (0,0) to (0,1) from (0,0) to (1,1) Figure 5-25: Mode scattering coefficients fo r CT73 from the (0,0) mode to the other propagating modes. 7 8 9 10 11 12 13 0 0.1 0.2 0.3 0.4 0.5 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,0) to (0,0) from (1,0) to (0,1) from (1,0) to (1,1) Figure 5-26: Mode scattering coefficients fo r CT73 from the (1,0) mode to the other propagating modes.

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106 7 8 9 10 11 12 13 0 0.1 0.2 0.3 0.4 0.5 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,1) to (0,0) from (0,1) to (1,0) from (0,1) to (1,1) Figure 5-27: Mode scattering coefficients fo r CT73 from the (0,1) mode to the other propagating modes. 10 10.5 11 11.5 12 12.5 13 0 0.2 0.4 0.6 |R|Freq [kHz] 10 10.5 11 11.5 12 12.5 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,1) to (0,0) from (1,1) to (1,0) from (1,1) to (0,1) Figure 5-28: Mode scattering coefficients fo r CT73 from the (1,1) mode to the other propagating modes.

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107 the surface. The normalized sp ecific acoustic impedance model given in Equation (5.1) is not used to model this specimen. The m odel is inadequate in capturing the physics of the specimen possibility due to either strong interactions between the ceramic cells at the test surface or due to inadequate sealing betw een the cells and the rigid termination. Still, the data in the two figures show a numb er of resonant frequencies monotonically decaying in amplitude, which are 1.36, 4.55, 7.62, and 10.9 kHz, as identified from the standard TMM and MDM results. If the ceram ic cells are modeled as an ideal quarterwave resonator, the first four resonant fre quencies are 1.69, 5.06, 8.43, and 11.8 kHz. The comparisons between the experimental resonant frequencies and the calculated resonant frequencies are reasona ble, and provide phys ical insight into the mechanics of the ceramic tubular material as an acoustic lin er. The results for th e normalized specific acoustic impedance show reasonable agreement, except at the cut-on for the (0,1), (1,0), and (1,1) modes and for frequencies above 10 kHz. These frequency ranges should be investigated further for the reasons described in Section 5.1.3, but these results suggest that the locally reactive assu mption is reasonable within the bandwidth up to 10 kHz. The MDM estimates of the plane wave refl ection coefficient and the normal incidence specific acoustic impedance agree with th e standard TMM in the large waveguide estimate to within the 95% confidence interval for the standard TMM as shown in Figure 5-18. Again, the TMM uncertainty estimates are not shown here for clarity.

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108 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 acFreq [kHz] 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 3 acFreq [kHz] Figure 5-29: Comparison of the acoustic impeda nce ratio estimates for CT73 via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode. 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 sp acFreq [kHz] 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 3 sp acFreq [kHz] Figure 5-30: Comparison of the normalized specific acoustic impedance estimates for CT73 via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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109 5.3 Rigid Termination The next specimen tested is a rigid term ination designed to simulate a sound-hard boundary. This is done by using a solid block of aluminum (type 6061-T6) with a polished test surface as shown in Figure 5-31. The specimen is approximately 31 mm thick for both waveguides. Only the reflection coefficient is consider ed for this specimen since the impedance should tend towards infinity for a rigid termination. Figure 5-31: Photograph of the rigid termination for the large waveguide. 5.3.1 TMM Results The results from the standard TMM are pr esented in Figure 5-32 for the reflection coefficient with uncertainty estimates. Th e graphs for the coherence and the FRF are given in Appendix F. The coherence betw een the two microphones for the original and switched positions is above 0.99 for the entire bandwidth, except at 1.63, 2,72, and 4.92 kHz. At these frequencies one of the mi crophones is located at a node in the standing wave and is only measuring noise. The uncer tainty estimates of the magnitude of the reflection coefficient do not st rongly show a periodic increase and decrease in the confidence intervals, but still show an overa ll increase with frequency. The uncertainty estimates for the phase of the reflection coe fficient show an almost linear relationship between the confidence interval and frequency. The uncertainty estimates for the phase

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110 of the reflection coefficient and for the resi stance and reactance show the same general increase in the confidence interval with fre quency. At some frequencies the confidence interval can be seen as asy mmetric, but not as much as in the results for the CT65 and CT73 specimens. The value for the magnitude for the reflection coefficient drops to a minimum of 0.96 at approximately 5 kHz, but no changes in the phase are apparent. This suggests that acoustic waves are sensing a diss ipation effect not included in the model of the sound field inside the waveguide. The sta nding wave ratio (SWR) is the ratio of the maximum amplitude in the pressure standing wave pattern to the minimum amplitude and can be computed in decibels from the reflection coefficient by 00,00 10 00,001 20log 1 R SWR R (5.2) Physically, the SWR will tend to infinity for an ideal rigid or sound hard termination. 1 2 3 4 5 6 0.9 0.92 0.94 0.96 0.98 1 |R|Freq [kHz] 1 2 3 4 5 6 -40 -20 0 20 40 [deg]Freq [kHz] Figure 5-32: Reflection coefficient for the rigid termination for the TMM. Estimated value, Uncertainty estimates.

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111 1 2 3 4 5 6 30 35 40 45 50 55 60 65 Freq [kHz]SWR [dB] Figure 5-33: Standing wave ratio for the rigid termination measured by the TMM. The SWR results are given in Figure 5-33 a nd show that the SWR is approximately 45dB, except in the frequency range around 5 kH z where the magnitude of the reflection coefficient drops as well. 5.3.2 High Frequency TMM Results The results from the high frequency T MM are presented in Figure 5-34 for the reflection coefficient with uncertainty estimat es. The graphs for the coherence and the FRF are given in Appendix F. The cohe rence between the two microphones for the original and switched positions is above 0. 9999 for the entire bandwidth, but there are drops in the coherence at discre te frequencies. The frequencies are 1.70, 2.27, 6.82, 8.51, 11.4, 12.0, 15.4, 16.0, 18.5, and 20 kHz. Again, these frequencies correspond to frequencies where one of the microphones is located at a node in the standing wave pattern. The uncertainty estimates for the refl ection coefficient show the same trends as with the standard TMM and the high frequency TMM of the other specimens. There is a periodic structure, and a gene ral increase in the confidence intervals with frequency. Above 10 kHz, the uncertainty estimates appear unpredictable due to the large

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112 uncertainty in the microphone locations and agai n at 13.5 kHz the singularity is present. Overall, the uncertainty estimates for this sp ecimen are larger than for the CT65 or CT73 specimens, confirming the analytical multivaria te scaling prediction. The SWR is shown in Figure 5-35, and again the drop in the SWR around 5 kHz is present. 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 |R|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-34: Reflection coefficient for the rigi d termination for the high frequency TMM. Estimated value, Uncertainty estimates. 2 4 6 8 10 12 14 16 18 20 10 15 20 25 30 35 40 45 Freq [kHz]SWR [dB] Figure 5-35: SWR for the rigid termination calculated from the high frequency TMM.

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113 5.3.3 MDM Results and Comparison The standard TMM, the high frequency TMM and the MDM experiment al results for the rigid specimen are compared in this section, but first the MDM results for the incident and reflected pressure field a nd the power absorption coefficient are presented. The cuton frequencies for the higher-order modes are 6.83 kHz for the (1,0) and (0,1) modes, and 9.66 kHz for the (1,1) mode, based on the measured temperature during the MDM measurements. Figure 5-37 shows the incide nt pressure field measured near the specimen for each of the restrictor plates. Fo r this measurement, th e combination of the (0,0), (1,0), (0,1), and (1,1) restrictor plat es did not yield an augmented matrix of the incident and reflected pressure amplitudes w ith a low condition numb er, the ratio of the largest to the smallest singular value in the singular value decomposition of a matrix (Chapra and Canale 2002). A condition number of unity woul d represent a well condition matrix that can be inverted easily. The condition number was greater than 20 for all frequencies beyond the cut-on freque ncy for the first higher-order mode. To accurately solve for the reflection coefficient matrix, another restrictor plate is designed and used to replace the (1,0) restrictor plat e. It is shown in Figure 5-36. The results reveals that one of the sources generates a pressure level approximately 5-10 dB higher than the other sources for the higher-order modes, except for during the use of the triangle restrictor plate, wher e the levels of both the (1,0) and (0,1) modes are higher. Still, for the plane wave mode, the restri ctor plates produce pr essure levels only approximately equal. The data for the reflected pressure fields for each of the restrictor plates are presented in Figure 5-38, and this figure shows evidence of mode scattering since there is no longer the same difference be tween the pressure amplitude of each of the sources as shown in the inci dent pressure field.

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114 The absorption coefficient is shown in Figur e 5-39 for all four restrictor plates. The data show that the total power absorbed is approximately zero fo r all frequencies and all restrictor plates. The re sults also show regions where the absorption coefficient is negative, indicating power ge neration. Physically this is impossible, since the termination is reactive only. Also, the majority of the nega tive power absorption coefficients are in the frequency range wh ere the data are affected by the singularity created when the microphone spacing is equa l to half a wavelength, and thus are Figure 5-36: Triangle restrictor plate. 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate plate (0,1) plate (1,1) plate Figure 5-37: Incident pressure field for the MDM for the rigid termination.

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115 unreliable. To confirm this conclusion, the measurement should be repeated with a different spacing between the microphone groups. The magnitude and phase of the reflecti on coefficients are shown in Figure 5-40 along with results from both TMM measuremen ts. The estimate of the plane wave 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate plate (0,1) plate (1,1) plate Figure 5-38: Reflected pressure field for the MDM for the rigid termination. 2 4 6 8 10 12 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Freq [kHz] (0,0) plate plate (1,0) plate (1,1) plate Figure 5-39: Power absorption coefficient fo r the rigid termination for the MDM.

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116 reflection coefficient provided by the MDM ag rees with the TMM in the large waveguide to within its 95% confidence interval estimat es, which are not shown in the figures for clarity. The figure also shows that the estima te of the plane wave reflection coefficient by the MDM is affected by the same singularit y that affects the hi gh frequency TMM at 13.5 kHz. The reduction in the magnitude of the reflection coefficient estimated by the high frequency TMM is easily seen to be greater than in either the standard TMM or the MDM in Figure 5-40. This suggests issues with either the small waveguide or the specimen. Figure 5-41 shows the SWR for th e plane wave mode. The approximate SWR is 40 dB and the drop in the SWR around 5 kHz is reduced. Figure 5-42 through Figure 5-45 show the mode scattering coefficients estimated by the MDM. The magnitude of the mode scattering coefficients is less th an 0.2 for all frequencies, except for the frequencies near the (1,1) mode cut-on or for coefficients going into the plane wave mode near 13.5 kHz. Both of these exceptions are unreliable due to either the cut-on phenomenon or the microphone spacing issue. The figures also show that there is no dominant mode of scatter, unlik e the CT65 or CT73 specimens.

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117 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 |R|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-40: Comparison of the reflection coe fficient estimates for the rigid termination via all three methods. TMM, high frequency TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode. 2 4 6 8 10 12 0 20 40 60 80 100 Freq [kHz]SWR [dB] Figure 5-41: SWR for the rigid termin ation calculated from the MDM.

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118 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,0) to (1,0) from (0,0) to (0,1) from (0,0) to (1,1) Figure 5-42: Mode scattering coefficients for ri gid termination from the (0,0) mode to the other propagating modes. 7 8 9 10 11 12 13 0 0.5 1 1.5 2 2.5 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,0) to (0,0) from (1,0) to (0,1) from (1,0) to (1,1) Figure 5-43: Mode scattering coefficients for ri gid termination from the (1,0) mode to the other propagating modes.

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119 7 8 9 10 11 12 13 0 0.5 1 1.5 2 2.5 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,1) to (0,0) from (0,1) to (1,0) from (0,1) to (1,1) Figure 5-44: Mode scattering coefficients for ri gid termination from the (0,1) mode to the other propagating modes. 10 10.5 11 11.5 12 12.5 13 0 0.5 1 1.5 |R|Freq [kHz] 10 10.5 11 11.5 12 12.5 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,1) to (0,0) from (1,1) to (1,0) from (1,1) to (0,1) Figure 5-45: Mode scattering coefficients for ri gid termination from the (1,1) mode to the other propagating modes.

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120 5.4 SDOF Liner The next specimen tested is a single degree-of-freedom liner donated from Pratt and Whitney Aircraft. The specimen is constr ucted of perforate f ace sheet adhered to a metal honeycomb substructure with an atypical pattern, all backed by a rigid plate. The specimen material is shown in Figure 5-46 with the perforate sheet removed. The perforate face sheet is 0.5 mm thick with 1.3 mm-diameter holes that cover approximately 7.3% of the surface. The circular holes are set in a 60o staggered pattern with a distance of 4.6 mm betw een the centers of two holes The honeycomb cells have an area of approximately 108 mm2, with a height of 19 mm. The maximum crosssectional dimension of the honeycomb is approx imately 20 mm. This dimension is larger then the length of a side of the small wavegui de, therefore this mate rial cannot be tested up to 20 kHz using the high frequency TMM. Th is demonstrates the need to increase the frequency range of acoustic impedance testi ng while maintaining a usable specimen size, and thus the results presented in this s ection are for only the standard TMM and the MDM. Figure 5-46: SDOF liner showing the irregu lar honeycomb and perforated face sheet. Perforate face sheet Honeycomb layer Rigid back plate

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121 5.4.1 TMM Results The results from the standard TMM are pr esented in Figure 5-47 for the reflection coefficient and Figure 5-48 for the normalized specific acoustic impedance, both with uncertainty estimates. The graphs for the coherence and the FRF are given in Appendix F. The coherence between the two microphone s for the original and switched positions shows decreases at three small frequency range s. The frequency ranges are 1.1-1.5, 3.63.9, and 5.2-5.4 kHz, but the coherence rema ins above 0.82 for the entire bandwidth of the measurement. The uncertainty estimat es of the magnitude of the reflection coefficient show an overall increase with fre quency. The results for the estimates of the reflection coefficient, the normalized speci fic acoustic impedance, and confidence intervals are jagged. The magnitude of the reflection coefficient also shows a gradual decrease in the range of 0.3 to 2.2 kHz to a minimum value of approximately 0.4. Afterwards, the magnitude of the reflection co efficient gradually increases to approach unity. At the minimum value in the magnit ude, the phase of the reflection coefficient goes through a 360 degree phase wrap. Also, at approximately 2.3 kHz the reactance passes, through zero indicating a resonance.

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122 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-47: Reflection coefficient fo r the SDOF specimen for the TMM. Estimated value, Uncertainty estimates. 1 2 3 4 5 6 0 1 2 3 4 Freq [kHz] 1 2 3 4 5 6 -10 -5 0 5 10 Freq [kHz] Figure 5-48: Normalized specific acoustic impedance estimates for the SDOF specimen via TMM. Estimated value, Uncertainty estimates.

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123 5.4.2 MDM Results and Comparison The standard TMM and the MDM experiment al results for the SDOF specimen are compared in this section, but first the MDM re sults for the incident and reflected pressure fields and the power absorption coefficient ar e presented. The cut-on frequencies for the higher-order modes are 6.83 kHz for the (1,0) and (0,1) modes, and 9.66 kHz for the (1,1) mode, based on the measured temperature du ring the MDM measurements. Figure 5-49 shows the incident pressure field measured near the specimen for each of the restrictor plates, and reveals that one of the sources generates a pressure level approximately 5-10 dB higher than the other three sources for th e higher-order modes. The exception is for the plane wave mode, where the restrictor plat e produces pressure levels only marginally higher than the others. The data for the reflec ted pressure fields for each of the restrictor plates are presented in Figure 5-50, and this figure shows less evidence of mode scattering than the previous specimens, since th ere is a significant difference in pressure amplitude of each for the sources, as show n in the incident pressure field. The absorption coefficient is shown in Figur e 5-51 for all four restrictor plates. The data show that the total power absorbed is almost independent on the modal content of the acoustic field for this specimen, sin ce the absorption coefficient does not vary much between the sources after the cut-on fre quency for the first higher-order mode. The data also confirms the experimental setup a nd measurement, since the computed value for the absorption coefficient remains bounded be tween zero and unity, except at the known singularity at 13.5 kHz, where the data are unreliable. The magnitude and argument of the reflecti on coefficients are shown in Figure 5-52 along with results from both TMM measurem ents. Unlike the other specimens, the estimate of the plane wave reflection coe fficient provided by the MDM does not agree

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124 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-49: Incident pressure field fo r the MDM for the SDOF specimen. 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-50: Reflected pressure field for the MDM for the SDOF specimen.

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125 with the TMM to within its 95% confidence interval estimat es, even though there appears to be a good agreement in the figures. The es timate via the MDM is ju st outside the edge of the confidence interval for the TMM the majority of the frequency range. The data only agree to within confidence intervals of the standard TMM results after approximately 4 kHz. The figure also shows that the estimate of the plane wave reflection coefficient by the MDM is affected by the same singularity that affects the high frequency TMM at 13.5 kHz. Figure 5-53 thr ough Figure 5-56 show the mode scattering coefficients estimated by the MDM. The magnit ude of the mode scattering coefficients is less than 0.2 for all frequencies, except around 10 kHz for the plane wave mode scattering into the (1,1) m ode. Unreliable results are also present around 13.5 kHz, as discussed before. The results for the acoustic impedance ratio are shown in Figure 5-57, and Figure 5-58 shows the normalized specific acoustic impe dance. The test specimen is assumed to be a locally reactive material and thus the transmission angle is assumed to be normal to the surface. The MDM results show an anti-resonance at ap proximately 9 kHz that is missed by the standard TMM because of the limited frequency range. This demonstrates the benefit of the MDM over the standard T MM. As with the reflection coefficient results, the results of the acoustic impeda nce ratio and normalized specific acoustic impedance do not agree with each other for the two methods until af ter 4 kHz. This suggests that test conditions, such as the m ounting of the specimen, may have varied in between tests. As for verifying the locally reactive assumption, the agreement between the results for the normalized specific acousti c impedance between the different modes is reasonable. The greatest difference in values is at the peak of the anti-resonance.

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126 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Freq [kHz] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-51: Power absorption coefficient for the SDOF specimen for the MDM. 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 |R|Freq [kHz] 2 4 6 8 10 12 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-52: Comparison of the reflection co efficient estimates for the SDOF specimen via the TMM and MDM. TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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127 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,0) to (1,0) from (0,0) to (0,1) from (0,0) to (1,1) Figure 5-53: Mode scattering coefficients for SDOF specimen from the (0,0) mode to the other propagating modes. 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,0) to (0,0) from (1,0) to (0,1) from (1,0) to (1,1) Figure 5-54: Mode scattering coefficients for SDOF specimen from the (1,0) mode to the other propagating modes.

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128 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,1) to (0,0) from (0,1) to (1,0) from (0,1) to (1,1) Figure 5-55: Mode scattering coefficients for SDOF specimen from the (0,1) mode to the other propagating modes. 10 10.5 11 11.5 12 12.5 13 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 10 10.5 11 11.5 12 12.5 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,1) to (0,0) from (1,1) to (1,0) from (1,1) to (0,1) Figure 5-56: Mode scattering coefficients for SDOF specimen from the (1,1) mode to the other propagating modes.

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129 2 4 6 8 10 12 0 10 20 30 40 50 acFreq [kHz] 2 4 6 8 10 12 -20 -10 0 10 20 30 acFreq [kHz] Figure 5-57: Comparison of the acoustic im pedance ratio estimates for the SDOF specimen via the TMM and MDM. TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode. 2 4 6 8 10 12 0 10 20 30 40 50 sp acFreq [kHz] 2 4 6 8 10 12 -20 -10 0 10 20 30 sp acFreq [kHz] Figure 5-58: Comparison of the normalized specific acoustic impedance estimates for the SDOF specimen via the TMM and MDM. TMM, MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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130 5.5 Mode Scattering Specimen The specimen measured is a specimen desi gned to promote mode scattering into the (1,1) mode. The specimen is construc ted of 38.1 mm thick aluminum (type 6061-T6) with 25.4 mm-deep square holes drilled out in two of the four quadrants, similar to the (1,1) restrictor plate. The two holes are then filled with Nomex, an acoustic absorption material, to present the propagating waves with a contrast of acoustic impedance. The specimen is shown in Figure 5-59. Since the pu rpose of this specimen is to scatter modal energy, only the MDM is used to measure the acoustic properties. Figure 5-59: Photograph of the mode scattering specimen. 5.5.1 MDM Results The MDM results for the incident and reflected pressure fields and the power absorption coefficient are presented. The cut-on frequencies for the higher-order modes are 6.83 kHz for the (1,0) and (0,1) modes, and 9.66 kHz for the (1,1) mode, based on the measured temperature during the MDM measurements. Figure 5-60 shows the incident pressure field measured near the specimen fo r each of the restrictor plates, and reveals that one of the sources generates a pressure level approximately 510 dB higher than the other three sources for the higher-order modes. The exception is the plane wave mode, where the restrictor plate produces pressure le vels only marginally higher than the others.

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131 The data for the reflected pressure field for each of the restrictor pl ates are presented in Figure 5-61, and this figure shows evidence of mode scattering, sinc e there is no longer the same difference between the pressure amp litude of each of the sources as shown in the incident pressure field, but there is a noticeable difference between the (1,0) and (0,1) restrictor plates and the other two restrictor plates for the (1 ,0) and (0,1) modes. Also, the amplitude of the (1,1) mode for all the restri ctor plates has increased compared to the incident pressure amplitude, most noticeably fo r the (0,0) restrictor plate. This offers evidence that the specimen is scattering pre dominantly into the desired (1,1) mode. The absorption coefficient is shown in Figur e 5-62 for all four restrictor plates. The data show that the total power absorbed is dependent on the modal content of the acoustic field, especially after the (1,1) mode cuts on. The absorption coefficient for the (0,0) restrictor plate falls dramatically af ter 9.5 kHz and remains lower than the other values. The energy from the (0,0) restrictor plate is not efficiently being absorbed and this helps to justify physically the increase in the reflection pressure amplitudes seen in Figure 5-61. The magnitude and argument of the refl ection coefficients are shown in Figure 5-63. The results show that the highest magn itude for the reflection coefficient is for the plane wave mode, but these results only s how the same mode reflections, which this specimen is not designed to maximize. Th e results of the phase of the reflection coefficients show that the wave is mostly reflected back in-phase, much like a sound-hard termination. The figure also shows that the estimate of the plane wave reflection coefficient by the MDM is affected by th e same singularity that affects the high

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132 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-60: Incident pressure field for th e MDM for the mode scattering specimen. 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (a) Mode (0,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (c) Mode (0,1)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (b) Mode (1,0)|P| [dB] 2 4 6 8 10 12 40 60 80 100 120 Freq [kHz] (d) Mode (1,1)|P| [dB] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-61: Reflected pressure field for the MDM for the mode scattering specimen.

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133 frequency TMM at 13.5 kHz. Figure 5-64 thr ough Figure 5-67 show the mode scattering coefficients estimated by the MDM. The largest magnitude for the mode scattering coefficients is for the plane wave mode scat tering into the (1,1) mode, with a value of unity or greater. Physically, this is showing that the specimen is promoting the conversion of energy from the (0,0) mode in to the (1,1) mode, as the design of the specimen is intended to do. There is also significant mode scattering between the (1,0) and (0,1) modes, with a value of 0.4 and almo st no phase change. The results still show the unreliable estimates due to either th e cut-on phenomenon or the microphone spacing issue, as explained before. The results for the acoustic impedance ratio are shown in Figure 5-68, and Figure 5-69 shows the normalized specific acoustic im pedance. The results for the normalized acoustic impedance show resonance at 8.6 and 12.4 kHz and anti-resonance at 9.6 and 13.4 kHz. The last values for the resonan ce and anti-resonance are unreliable because of the microphone spacing issue at 13.5 kHz. Actu ally, the resonances and anti-resonances may not exist at all, since the MDM is derived for a constant acoustic impedance 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 Freq [kHz] (0,0) plate (1,0) plate (0,1) plate (1,1) plate Figure 5-62: Power absorption coefficient fo r the mode scattering specimen for the MDM.

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134 termination mounted to the waveguide. Th e displayed values in Figure 5-68 and in Figure 5-69 represent an averaged value of th e acoustic impedances. The results for the normalized specific acoustic impedance show that the locally reactive assumption is not valid for this specimen. There are large diffe rences between the estimates for the various modes. 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 |R|Freq [kHz] 2 4 6 8 10 12 -180 -90 0 90 180 [deg]Freq [kHz] Figure 5-63: Comparison of the reflection coe fficient estimates for the mode scattering specimen via the MDM. MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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135 7 8 9 10 11 12 13 0 1 2 3 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,0) to (1,0) from (0,0) to (0,1) from (0,0) to (1,1) Figure 5-64: Mode scattering coefficients for the mode scattering specimen from the (0,0) mode to the other propagating modes. 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,0) to (0,0) from (1,0) to (0,1) from (1,0) to (1,1) Figure 5-65: Mode scattering coefficients for the mode scattering specimen from the (1,0) mode to the other propagating modes.

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136 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 |R|Freq [kHz] 7 8 9 10 11 12 13 -180 -90 0 90 180 [deg]Freq [kHz] from (0,1) to (0,0) from (0,1) to (1,0) from (0,1) to (1,1) Figure 5-66: Mode scattering coefficients for the mode scattering specimen from the (0,1) mode to the other propagating modes. 10 10.5 11 11.5 12 12.5 13 0 0.5 1 1.5 |R|Freq [kHz] 10 10.5 11 11.5 12 12.5 13 -180 -90 0 90 180 [deg]Freq [kHz] from (1,1) to (0,0) from (1,1) to (1,0) from (1,1) to (0,1) Figure 5-67: Mode scattering coefficients for the mode scattering specimen from the (1,1) mode to the other propagating modes.

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137 2 4 6 8 10 12 -2 0 2 4 6 acFreq [kHz] 2 4 6 8 10 12 -4 -2 0 2 acFreq [kHz] Figure 5-68: Comparison of the acoustic im pedance ratio estimates for the mode scattering specimen via the MDM. MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode. 2 4 6 8 10 12 -2 0 2 4 6 sp acFreq [kHz] 2 4 6 8 10 12 -4 -2 0 2 sp acFreq [kHz] Figure 5-69: Comparison of the normalized specific acoustic impedance estimates for the mode scattering specimen via the MDM. MDM (0,0) mode, MDM (1,0) mode, MDM (0,1) mode, and MDM (1,1) mode.

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138 CHAPTER 6 CONCLUSIONS AND FUTURE WORK This chapter summarizes the work presented in this dissertation. The technologies developed have been successful in es timating the accuracy of existing acoustic impedance measurement techniques and in ex tending the bandwidth of normal incident acoustic impedance testing. Conclusions and pot ential paths for future work are discussed for the two major impact areas: the T MM uncertainty analysis and the MDM. 6.1 TMM Uncertainty Analysis This dissertation demonstrated the multivariate uncertainty technique on the TMM for acoustic impedance testing compared to Monte Carlo methods. When all of the component uncertainties are very small 1% = or the specimen is sound-hard, the multivariate method matches the results from Monte Carlo method. When the component uncertainties assume values typical of current experimental configurations, they are large enough to violate the linear assumption. Hen ce, the perturbations caused on the output are nonlinear and also distort the output probability distribution from Gaussian behavior. The distortion of the out put probability distribution is thus a function of the acoustic impedance of the specimen itself. The actual probability density function is estimated numerically from Monte Carlo si mulations and is integrated to accurately estimate the uncertainty with re alistic nonlinear perturbations. This tends to increase the complexity of the uncertainty analysis program and requires more computation time than the multivariate method. This method was de monstrated and applied to experimental data for four material specimens: CT65, CT 73, a rigid termination, and a SDOF liner.

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139 The Monte Carlo results demonstrated the potential for asymmetric uncertainty estimates and the general Monte Carlo is recommende d as the method for accurate uncertainty estimates. The main contribution of this research is thus a verified systematic framework to estimate frequency-dependent uncertainties in the complex reflection and normalized acoustic impedance using a multiavariate approach and a Monte Carlo approach. Presumably, this tool will be useful to a ssess the suitability of candidate acoustic liner materials. Future improvements to the uncertainty analysis can include changing the uncertainty estimation program to compute the uncertainty of the unwrapped phase, increasing the number of Monte Carlo iterati ons, and increasing the number of discrete bins used to approximate the probability de nsity function. Increasing the number of iterations will increase the demand on computer resources and will exceed those available on a typical desktop computer. Also, multip le microphone spacings should be used to avoid the troublesome singul arity that occurs when ksn when the spacing is equal to a half wavelength. This si ngularity was demonstrated to have a large impact on the results in the data from the high frequenc y TMM. To improve the accuracy of the computed acoustic impedance, the uncertain ty in the locations of the microphones and the uncertainty in the measure of the te mperature should be improved, along with a rigorous investigation of the periodic structures in the uncertainty estimates for the reflection coefficients for specimens CT65 and CT73. Understanding the conditions at the minimums in the uncertainty estimates c ould lead to reducing the sensitivity of the estimated reflection coefficient and acoustic im pedance values to the uncertainty in the input quantities. The accuracy of the mi crophone locations can be improved by using a

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140 method that measures the locations of the microphones installed in the waveguide and accounts for the acoustic centers (Katz 2000). For the high frequency TMM, spatial averaging of the microphone measurements ma y present an error source that can be reduced with probe tips attached to the microphones (Franzoni and Elliott 1998) or by using microphones with a smaller diameter, such as MEMS-based microphones (Arnold et al. 2003). The uncertainty estimates fr om the TMM in the small waveguide showed large confidence intervals in the data starting at approximately 12 kHz. The results also demonstrated the difficulties of creating a te rmination that could accurately represent a sound-hard boundary. Improvements in the refl ection coefficient might be attainable through the development of an improved m ounting system for the specimen and the microphones. Other improvement may include the use of probe tips on the microphones to reduce the effects of spatial averaging, especially for the high frequency TMM. The comparison between the data for the small waveguide and the large waveguide demonstrated that the specimen size can im pacted the estimated acoustic impedance values. For the same material, the two meas urements did not agree with each other to with in the uncertainty estimates possibly due to local variations in the material. In order to definitively define the local variation in the materials, a statistical number (>10) of different specimens of a single material type should be test ed in both waveguides. Then the statistical analysis of th e estimated acoustic impedance values for a given waveguide can be used to characterize th e local variations in the mate rial and the averaged values can be compared between the two waveguides. 6.2 Modal Decomposition Method The MDM offers the ability to extend th e bandwidth of acoustic impedance testing by accommodating the propagation of higher-ord er modes and maintaining a larger

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141 specimen size to avoid sample size effects. Th is also gives the ability to obtain acoustic impedance data at oblique angles of incidence, with the primary drawback being increased complexity as compared to the TMM. A direct MDM procedure was developed for a square duct using the first four modes. This dissert ation restricted the maximum number of propagating modes to four for a proof-of-concep t demonstration. The routines presented can easily be e xpanded to handle more propagating modes through the addition of measurement microphone s. As the bandwidth of the MDM is increased, spatial averaging of the microphone measurements may present an error source that can be reduced with probe tips att ached to the microphones (Franzoni and Elliott 1998) or by using microphones with a sma ller diameter, such as MEMS-based microphones (Arnold, Nishida, Cattafesta and Sheplak 2003). Also, diffraction effects of the sound scattering off the protective grid s of the microphones can introduce error (Underbrink 2002). To achieve the 20 kH z bandwidth goal for the 25.4 mm-square waveguide, modes up to and including the (2 ,2) mode must be accounted for. This requires 18 microphone measurements in order to resolve the nine propagating modes, and will require nine independent source conditions. The additional microphones could be placed at new axial locations around the perimeter of the waveguide. The additional axial locations will also help in reducing the effects of the singularity when the existing spacing equals a half wavelength. The additi onal source conditions c ould be created with additional restrictor plates. The routines can also be applied to rectangular and cylindrical ducts through substi tution of the correct transver se factor, but a square duct was chosen because geometry allows for th e largest cross-section for a given cut-on

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142 frequency for the first highe r-order mode and allows fo r the microphones to be easily flush-mounted to the duct walls. The results presented show that the MDM can accurately resolve the sound field inside the waveguide and decompose the m odal coefficients, except near the cut-on frequencies, and provide estimates for the co mplete reflection coefficient matrix. The MDM results were consistent with the TMM in the large waveguide results to within the 95% confidence interval estimates for the T MM results, except for the SDOF specimen. To fully characterize the MDM, uncertainty estimates are needed along with results of repeated testing on existing specimens. The comparison should also be conducted with other testing facilities, such as the NASA La ngley Research Center. The inclusion of the uncertainty estimates would allow for statemen ts to be made about the equality of the estimates of the specific acoustic impedan ce for the different modes, and thus would allow for statements about the validity of th e locally reactive assump tion. Another goal of future work should focus on extendi ng the frequency range of the MDM up to approximately 20 kHz, as is desi red for aeroacoustic testing.

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143 APPENDIX A VISCOTHERMAL LOSSES The propagation of sound in a real flui d involves a loss in amplitude due to viscosity and heat transfer within the fluid and surroundings. In many cases, the absorption of sound in the bulk fluid can be neglected, especially for a bounded domain where the bulk losses are small compared to the boundary layer losses (Kirchhoff 1869; Weston 1953; Tijdeman 1975; Blackstock 2000). An example of such a case is the propagation of an acoustic wave within a pipe or duct that has a ch aracteristic length of its cross-section larger than the boundary layer thickness but an area not so large that the mainstream bulk losses are important. Quantitiv ely, this restricti on can be written as (Blackstock 2000) 2 0 22 c L (A.1) where is the viscous boundary layer thickness and L is the hydraulic diameter. The two rectangular waveguides described in this dissertation satisfy this condition for the bandwidth of plane waves. Under this restri ction, the acoustic flow can be separated and treated as two distinct regions. Thus the at tenuation of the amplitude of the sound can be found by solving fluid dynamic equations in the boundary layer and using the velocity and temperature profiles to estimate a wall shear stress and a heat flux. The wall shear stress and heat flux are then used as sinks for momentum and energy in the derivation of the fluid dynamic equations for the mainst ream. Also, since the boundary layers are small compared to the hydraulic diameter and only occupy a small percentage of the

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144 cross-sectional area, the duc t boundary can be unwrapped an d the problem treated as a two dimensional problem with an axia l and a transverse coordinate. First, the geometry and boundary conditions are presented to define the problem to be solved. Then the Navier-Stokes equatio ns are presented and nondimensionalized to determine the important parameters governing th e solution. The velocity and temperature profiles are then found for the boundary layer and the wall shear stress and heat flux are computed. The wall shear stress and heat flux are used to derive the fluid dynamic equations for the mainstream, which are then used to estimate the axial attenuation of the amplitude of the sound wave propagating down the duct. A.1 Nondimensionalization and Lineariz ation of the Navier-Stokes Equations The problem to solve is a purely oscilla ting flow of a thermally and calorically perfect gas over a flat plate, as shown in Fi gure A-1. The problem is assumed to be twodimensional, with constant thermodynami c transport properties, and governed by the Navier-Stokes equations, neglecting bulk vi scosity and molecular relaxation. The conservation equations are 0,uv txy (A.2) 22 22, 3uuupuuuv uv txyxxyxxy (A.3) 22 22, 3vvvpvvuv uv txyyxyyxy (A.4) 22 22,pTTTTTppp cuvuv txyxytxy (A.5) and

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145 ,gas p RT (A.6) where is the density, uv are the flow velocities in the x y -directions respectively, p is the pressure, T is the temperature, is the dynamic viscosity, is the nonlinear viscous dissipation function, pc is the specific heat at constant pressure, and g as R is the idea gas constant for the fluid, in most cases air. Other thermodynamic relations that will come in handy are 2 0 0 0,gas p cRT (A.7) and 1,gasvRc (A.8) where pvcc is the ratio of specific heats, vc is the specific heat at constant volume and the 0 subscript references the m ean isentropic condition. Figure A-1: Oscillating fl ow over a flat plate. Now the appropriate scales must be c hosen to nondimensionalize the dependent and independent variables of the governing equa tions. Also, at this stage the dependent variables will be decomposed in to a mean and a perturbation. The appropriate time scale for this problem is 1 where is the angular frequency, since the flow is oscillatory.

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146 The oscillatory nature of the flow provides an the length scale for the x -direction as 0c The length scale for the y-direction should be representa tive of the cross-section and given by the hydraulic diameter, 4perimeterLAl and allow the nondimensionalization to apply to an arbitrary cross-section. The dependent variables are nondimensionalized by their mean isentropic state and are assumed to be decomposed into a mean and a perturbation. The flow velocity components are only composed of a perturbation and therefore are nondimensiona lized by the isentropic speed of sound. Thus, the new nondimensional variables are *, t t (A.9) 0, c x x (A.10) *, yLy (A.11) 01, p pp (A.12) 01, (A.13) 01 TTT (A.14) 0, ucu (A.15) and 0. vcv (A.16) When Equations (A.9)-(A.16) are substituted into the governing equations, Equations (A.2)-(A.6), the perturbations of the dependent variables are to be assumed small such that the resulting equations can be linearized.

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147 A.1.1 Continuity Beginning with the continuity equation, E quation (A.2) is transformed as follows: ***** 00000 ** 0111 0,cucv c t Ly x ******* 00 000 *** 00. uuvv c c tcxLy Now, neglecting the higher-o rder terms and dividing by 0 *** 0 ***0.c uv txLy Substituting the definition of the wavenumber, 0kc the nondimensionalized continuity equation becomes *** ***1 0.uv txkLy (A.17) A.1.2 x -direction Momentum Equation The x -direction momentum equation, Equati on (A.3), is nondimensionalized as follows: *** 000 ***** 00000 * 0 2*2*** 0 0000 22 * *** 000 0111 1 3 cucucu cucv c t Ly x pp cucucucv ccc Ly c Ly xxx x Simplifying and neglecting th e higher-order terms,

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148 2 **2*2*** 00 00000 ***22*2*** 0000. 3cc upuuuv cpcc tcxcxLycxcxLy Simplifying and recalling the de finition of the wavenumber, **2*2*** 2 0 000 **2*2*2*** 0. 3c upuuuv cpkLkL tcxLxyLxxy Now divide the equation by 00c to get **2*2*** 2 0 *2*2*2*2*** 0001 3p upuuuv kLkL tcxLxycLxxy **2*2*** 2 0 *2*2*2*22*** 0001 3p upuuuv kLLkL tcxLxyLcxxy where is the kinematic viscosity. The Stokes number, 2SL can be recognized and the expression simplified farther with th e help of Equation (A.7) when solved for 0 p to achieve 2 00 **2*2*** 2 *2*2*2*22*** 00111 3 c upuuuv kLkLkL tcxSxySxxy or finally, **2*2*** 22 **2*2*22***11111 3 upuuuv kLkL txSxySxxkLy (A.18) A.1.3 y-direction Momentum Equation The y -direction momentum equation, Equati on (A.4), is nondimensionalized as follows:

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149 *** 000 ***** 00000 * 0 2*2*** 0 0000 22 *** * 0 0111 1 3 cvcvcv cucv c t Ly x pp cvcvcucv c LyLyLy c Ly x x Simplifying and neglecting higher-order terms, 2 **2*2*** 000 0000 ***22*2*** 00. 3 pcc vpvvuv ccc tLycxLyLycxLy Simplifying and recalling the definition of the wavenumber 2 **2*2*** 000 00 **2*2*2*** 00, 3pcc vpvvuv cLL tLyLcxyLLycxy **2*2*** 2 000 00 **2*2*22***. 3pcc vpvvuv ckLkL tLyLxyLyxy Now, dividing the equation by 00c and recalling the definiti on of the Stokes number, **2*2*** 2 000 **2*2*22*** 000000, 3pcc vpvvuv kLkL tcLycLxycLyxy **2*2*** 2 0 **2*2*22*** 001 3p vpvvuv kLkL tcLyLxyLyxy **2*2*** 2 0 **2*2*22*** 00111 3p vpvvuv kLkL tcLySxySyxy The expression simplified further with the help of Equation (A.7) when solved for 0 p to achieve

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150 2 00 **2*2*** 2 **2*2*22*** 00111 3c vpvvuv kLkL tcLySxySyxy **2*2*** 2 0 **2*2*22***1111 3c vpvvuv kLkL tLySxySyxy or finally **2*2*** 2 **2*2*22***11111 3vpvvkLuv kL tkLySxySyxkLy (A.19) A.1.4 Thermal Energy Equation The nondimensionalization of the thermal en ergy equation, Equation (A.5), starts by neglecting the viscous dissi pation term. Since the vi scous dissipation term is nonlinear, the resulting term after the substi tution of Equations (A.12)-(A.16) will be a higher-order term and is therefore neglected as part of the linearizat ion. The remaining steps for the thermal energy equation are as follows: *** 000 *** 000 * 0 2*2*** 0000 22 * 0 0 0 *111 1 1111 1 .pTTTTTT ccucv c tLy x TTTTpppp u c t c Ly x x pp v Ly Neglecting the higher-order terms and simplifying, 2 *2*2** 0000 **22*2* 01 .pTTTp TcTp tcxLyt

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151 Now dividing the expression by 00 pTc 2 *2*2** 0 **22*2* 00001 .ppp TTTp tccxLyTct Simplifying, 2 *2*2** 0 *2*2*2* 00001 ,ppp TTTp L tcLcxycTt *2*2** 2 0 *2*2*2* 001 .ppp TTTp kL tcLxycTt Now, simplify by recalling the definition of the Prandtl number, Prpc and the Stokes number and substituting in the equation of state, Equation (A.6), to get *2*2** 2 *2*2*2*1 Prgas pR TTTp kL tSxyct Using the thermodynamics relation given in Equation (A.8), the expression can be simplified to *2*2** 2 *2*2*2*1 1 Prv pc TTTp kL tSxyct or finally *2*2** 2 *2*2*2*11 PrTTTp kL tSxyt (A.20) A.1.5 Equation of State for an Ideal Gas The equation of state, Equation (A.6), is nondimensionalized as follows: *** 000111.gas p pRTT

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152 Divide the expression by the equation of state for the isentropic state, 000 gas p RT to get *** 000 000111 ,gas gas p pRTT pRT ***111, p T *****11. p TT Finally, neglecting the higher-order terms, ***. p T (A.21) A.1.6 Summary of the Nondimensional, Linearized Equations The nondimensional, linear equations are summarized below for oscillatory flow over a flat plate assuming a time-harmonic solution. They are *** ***1 0,uv txkLy (A.22) 2 **2*2*** 2 **2*2*22***1111 3 kL upuuuv kL txSxySxxkLy (A.23) **2*2*** 2 **2*2*22***11111 3vpvvkLuv kL tkLySxySyxkLy (A.24) *2*2** 2 *2*2*2*11 PrTTTp kL tSxyt (A.25) and ***. p T (A.26)

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153 A.2 Boundary Layer Solution The flow in the boundary layer is driven by an oscillating pressure gradient in the mainstream, but the thickness of the boundary layer is assumed to be much smaller than the dimensions of the cross-section. This a ssumption allows the problem to be reduced to a simple model of oscillating flow over a flat plate, but assuming that the boundary layer thickness is much smaller than the wa velength allows for the assumption of incompressible flow in the boundary layer. With this assumption the driving pressure wave appears to propagate through the bounda ry layer instantaneously. This simplifies the continuity equation, Equation (A.22), to ** **1 0.uv xkLy (A.27) The solution in the boundary layer is simp lified further by the use of the boundary layer assumption, which states that the changes in the transverse direction are more important than changes along the axial direction. Mathematically, this is represented by ,nn nnyx (A.28) and thus the axial derivative can be neglected. Also, the transverse velocity component is assumed much smaller than the axial veloci ty component, and is thus neglected. The above assumptions allow the governing equations to be simplified. Equations (A.23)-(A.25) become **2* **2*211 ,upu txSy (A.29) *11 0, p kLy (A.30)

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154 and *2** *2*2*11 Pr TTp tSyt (A.31) Equation (A.30) shows that the pressure field does not vary within the boundary layer and is only a function of the axial coordinate. This is consist with the assumptions made above, and thus the axial pressure gradient is determined by the mainstream pressure gradient, denoted by p and now is a known parameter with respect to the boundary layer solution. Therefore the pr oblem is reduced to solving *2* **2*211 ,p uu txSy (A.32) and *2* *2*2*11 Pr p TT tSyt (A.33) for *u and *T with the boundary conditions ****000, uyTy (A.34) and ****. uyTyfinite (A.35) A time-harmonic solution is assumed and is represented by ** *, ,jt jt p pe TTe and **.jtuue The boundary layer problem to solve becomes

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155 2 *2*211 ,dp u ju dxSy (A.36) 2 2*211 Pr T jTjp Sy (A.37) with **000, uyTy (A.38) and **. uyTyfinite (A.39) The solutions for the velocity and temp erature profiles are easily found from Equations (A.36) and (A.37) with the boundary conditions defined in Equations (A.38) and (A.39). The velocity profile is *1 2 *1 1.j Sydp uyje dx (A.40) The boundary layer thickness is found by solving for the *y location where the velocity is equal to 99% of the mainstream velocity, **0.99 uy The boundary layer thickness is given as *1 2ln0.01, S (A.41) or in dimensional terms 6.5. (A.42) Equation (A.42) shows that the boundary laye r thickness is inversely proportional to the frequency and that the thickest boundary layers will be at low frequencies. At 100 Hz, the boundary layer thickness for air is 0.45 mm. This is much smaller than the hydraulic

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156 diameter for the two waveguides described in this dissertation (8.5 mm and 25.4 mm) and verifies the small boundary layer assumption. The temperature profile is *1 Pr 21 1.j SyType (A.43) The thermal boundary layer thickness is found by solving for the *y location where the temperature is equal to 99% of the mainstream temperature, **0.99 Ty The thermal boundary thickness is *1 2ln0.01, PrthS (A.44) or in dimensional terms 6.5. Pr Prth (A.45) Equation (A.45) shows that the thermal boundary layer thickness is re lated to the viscous boundary thickness by a factor of Pr and also has the same dependence on frequency. At 100 Hz, the thermal boundary layer thickness for air is 0.54 mm and again this is small as compared to the hydraulic diameter for the two waveguides. In summary, the full velocity and temperature profiles are *1 ** 2 *1 1,j Sy jtdp uyjee dx (A.46) and *1 Pr ** 21 1.j Sy jtTypee (A.47)

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157 A.2.1 Wall Shear Stress Now that the velocity profile is known, the wall shear stress can be found. In dimensional terms, the wall shear stress is given as 0,w yu y (A.48) which can be nondimensionalized by substituti ng in the definitions given in Equations (A.11) and (A.15). This results in ** 0 0.w yc u Ly By defining the nondimensional shear stress as 0,w wc L (A.49) Equation (A.48) becomes ** * 0.w yu y (A.50) By evaluating the derivative w ith the aid of the velocity pr ofile in Equation (A.46), the wall shear stress is ** **11 22jt w p dp jj SS e dxx (A.51) This represents the shear force applied to the wall due to the fluid motion. This will be used in the derivation of the governing equatio ns for the mainstream flow to account for the viscous losses.

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158 A.2.2 Wall Heat Flux The heat flux at the wall is found from the temperature profile and Fourier’s heat conduction law. The wall heat flux is given as 0,w yT q y (A.52) or substituting in the definitions of the nondimensional temperature and y-coordinate, ** 0 0.w yT T q Ly Defining the nondimensional wall heat flux as 0,w wq q T L (A.53) the nondimensional Fourier’s he at conduction law becomes ** * 0.w yT q y (A.54) By evaluating the derivative with the aid of temperature pr ofile in Equation (A.47), the wall heat flux is ** *1111 PrPr, 22jt wjj qSpeSp (A.55) which represents the heat flux through the wall in the transverse direction. This will be used in the derivation of the governing equatio ns for the mainstream flow to account for heat conduction. A.3 Mainstream Flow The mainstream flow is modeled with the same assumptions previously stated in this appendix, with the exception of incompre ssibility and the addition of inviscid flow

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159 with the losses due to the bounda ry layer handled with an a pplied external shear stress and heat flux at the boundaries. The propaga ting waves are assumed to be plane waves, thus requiring that 0 v throughout the entire flow field. The governing equations will be modified to include the new loss term for the axial momentum equation and the thermal energy equations. A control volume anal ysis is easily used to construct the new governing equations, and since th e boundary layer thicknesses are small compared to the hydraulic diameter, the cross-sect ion of the mainstream flow is assumed to be the same as the cross-section of the waveguide In the mainstream, the continuity equation is given as 0, u tx (A.56) or if linearized and nondimensionalized as ** **0. u tx (A.57) A.3.1 Axial Momentum The two-dimensional control volume is s een in Figure A-2, showing the external forces and the velocities th at cross the boundaries. Appl ying an Eulerian view of Newton’s Second Law to the control volume to obtain 22.wperimeter xxdxxxdxuAdx uAuApApAldx t Note that p now denotes the mainstream pressure a nd that the infinity subscript has been dropped for convenience. The expression can be simplified as follows: 22,wperimeter xdxx xdxxu AdxAuAuApApldx t 224. 4perimeter xdxxxdxx wuu pp l u tdxdxA

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160 Now, taking the limit as 0 dx to convert the second and third terms to derivatives to obtain 24 .wu u p txxL Expand the first two terms and simplify with th e use of the continui ty equation given in Equation (A.56) as follows: 4 ,wu uup uuu ttxxxL 0 from continuity4 ,wu uup uu txtxxL 1442443 4 .wuup u txxL Now the expression is linearized and nondimensionalized as follows: ** 0 00 **** 0 000 ** 001 4 11,wpp cucu c cu cc LL t xx ** 00 00 **2 04.wpc up c tcxL Now divide by 00c and recall Equation (A.7) to obtain 0** 0 *2*2 004,wp up tcxL 02 00 ** *2*2 04,wc up tcxL

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161 ** **214wup txS (A.58) Now substitute in Equation (A.51) fo r the wall shear stress to obtain *** ***122 1. upp j txSx (A.59) Equation (A.59) is the axial momentum equation for the mainstream flow which includes a correction term for the viscous losses due to the presence of the boundary layer. The equation shows that force provided by the driv ing pressure gradient is split between the time rate of change of momentum and the losses at the boundary layer. w pxdx p xw y x uxdx ux w pxdx p xw y x uxdx ux Figure A-2: Control volume showing the ex ternal forces and flows crossing the boundaries. A.3.2 Energy Equation The two-dimensional control volume is s een in Figure A-3, showing the external heat fluxes, pressure work boundaries, and the velocities that cr oss the boundaries. Applying an Eulerian view of conversion of energy to the control volume to obtain ,wperimeter xxdxxxdxeAdxueAueApuApuAqldx t where e is the internal energy. The expression can be simplified as follows: ,wperimeter xdxxxdxx A dxeAueAueApuApuqldx t

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162 .perimeter xdxxxdxx wueuepupu l eq tdxdxA Taking the limit as 0dx to convert the second and third terms to derivatives to obtain 4, 4perimeter wl euepu q txxA 4 .weuepu q txxL Expanding the derivatives and using continuity as follows: 4 ,wupu ee eueq ttxxxL 0from continuity4 ,wupu ee ueq txtxxL 1442443 4 .wpu ee uq txxL For this situation, the internal energy is given by 22 eu, and is to simplify the expression for energy as follows: 224 22 ,wuu pu uq txxL 224 22 ,wuu pu uuupq ttxxxxL 4 ,wuupu uuuuupq ttxxxxL 4 .wuupu uuupq txtxxxL Recall that for inviscid flow, the momentum equation in the x -direction is given by

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163 0, uup u txx which is used to simplify the energy equation to 4 ,wu upq txxL or switching to the materi al derivative denoted by D Dt 4 .wDu p q DtxL Recall continuity in material derivative form as 1 Du Dtx and substitute into the expression for the energy equation to obtain 4 .wDpD q DtDtL Now note that material derivative of the pressure can be expressed as pp DD DppD DtDtDtDt or more conveniently as p D pDDp DtDtDt Substituting this expression into the energy equation gives 4 ,wp D DDp q DtDtDtL

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164 4 ,wp D DDp q DtDtDtL 4 .wp D Dp q DtDtL This expression can be simplified farther by noting the definition of enthalpy as p h (A.60) thus reducing the en ergy equation to 4 .wDhDp q DtDtL For a thermally and calorically perf ect gas, enthalpy is given as ,phcT (A.61) thus allowing the energy equation to be written in terms of temperature as 4 ,pwDTDp cq DtDtL 4 .ppwTTpp ccuuq txtxL Now the energy equation is nondimensi onalized and linearized as follows: ** 00 *** 000 * 0 ** 00 ** 0 0 * 011 11 11 4 ,pp wTTTT cccu c t x pppp T cuq c LL t x

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165 ** 0 000 **24.pwT Tp Tcpq ttL Now divide the expression by 00 pTc to obtain ** 00 **2 00001 4,w pppT Tp q tTctTcL ** 0 **2 0001 4.w ppp Tp q tTctcL Using the equation of state for a perfect gas, the thermodynamic relation given in Equation (A.8), and the definition of the ratio of specific heats, the Stokes number and the Prandtl number, the energy equation is simplified as follows: ** **21 4, Prw pTRp q tctL ** **21 1 4, Prv w pc Tp q tctS and finally ** **21 1 4. PrwTp q ttS (A.62) Substituting in Equation (A.55) to put the energy equation in terms of only the field variables results in ** **21 111 4Pr, Pr 2 Tpj Sp ttS or ** **1 11 22. Pr Tpj p tt S (A.63)

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166 Equation (A.63) is the thermal energy equati on for the mainstream flow including losses due to heat conduction at the wall. pxdx p xwq y x uxdxux wq pxdx p xwq y x uxdxux wq Figure A-3: Control volume showing the exte rnal heat fluxes and flows crossing the boundaries. A.3.3 Summary of the Mainstream Flow Equations The system of equations governing the mainstre am flow is given here for reference. The equations are ** **0, u tx (A.64) ** **122 11, up j tSx (A.65) ** **1 122 1, Pr Tp jp ttS (A.66) and ***. p T (A.67) These are the equations that need to be solv ed to determine the decay in the amplitude and the shift in phase speed due to ther moviscous losses for a sound wave that propagates down a duct.

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167 A.3.4 Mainstream Flow Wave Equation The equations summarized in Section A.3.3 are combined to form a single wave equation for the mainstream flow in terms of the acoustic pressure. To start, the equation of state, Equation (A.67), is used to eliminate from continuity, Equation (A.64). The new form of the continuity equation is *** ***0, pTu ttx which is rearranged to become *** ***. Tpu ttx Now this expression is used to replace *T in the energy equation, Equation (A.66). The result is *** ***1 122 1, Pr pup jp txtS or **** ****1 122 1, Pr uppp jp xtttS ** **1 122 1. Pr up jp xtS (A.68) Now the system of four equations with four unknowns has been reduced to a system of two equations, Equations (A.65) and (A.68), with only two unknowns. To combine these two equations into a single wave equation, di fferentiate Equation (A.65) with respect to x and derivate Equation (A .68) with respect to *t This yields

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168 2*2* ***2122 11, up j x tSx and 2*2** ***2*1 122 1. Pr upp j x ttSt Now subtract the two equations to get 2*2** *2*2*1 122122 0111, Pr p pp jj SxtSt and simplify as 2*2** *2*2*1 2222 111. Pr p pp jj SxtSt (A.69) Equation (A.69) is the lossy, nondimensional wa ve equation for the mainstream flow that accounts for the losses due to the boundary laye r but neglects losses in the bulk fluid. A.3.5 Dissipation and Dispersion Relations To find the dissipation and dispersion relati ons, the pressure is assumed to have the form ***,jxjt s p Pee (A.70) where is the nondimensional propagation constant and s P is a constant that determines the amplitude of the sound wave. Next, substitute Equation (A.70) into the wave equation, Equation (A.69), and solve for the propagation constant as follows; 2 *2*1 2222 111, Pr jjpjpjjp SS 21 2222 1111, Pr jj SS

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169 so finally, when choosing the positive squa re root, the propagation constant is 1 22 11 Pr 22 11 j S j S (A.71) The nondimensional propagation constant can be simplified further by expanding the two square roots into seri es. The series expansions re quire that the second term in each of the square roots have a magnitude of less than one for the series to converge. This can be expressed mathematically as 1 22 11, Pr j S and 22 11. j S In order for these requirements to be satisfied, 1/ S will have to be a small parameter. The two expressions can be simplified using th e definition of the Stokes number, and this results in 2 21 8 Pr f L (A.72) and 28 f L (A.73) respectively. Table A-1 gives the minimum frequencies to satisfy the two requirements for the series expansion for the two waveguide s presented in this dissertation for air at

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170 20 Co. The results in the table show that the minimum frequencies are easily exceeded in any practical experiment for this dissertation a nd therefore the series e xpansion is valid. Table A-1: Minimum frequency required for series expansion for the two waveguides for air at 20C. Requirement 8.5 Lmm 25.4 Lmm (A.72) 0.12 Hz 0.014 Hz (A.73) 0.6 Hz 0.06 Hz The expression for the propagation consta nt, Equation (A.71), is expanded using the series expansions for the two square roots. This results in 2 211 122122 111 28 PrPr 122322 111. 28 jj SS jj SS K K This is simplified as follows: 11 122122122122 11111. 2222 PrPr jjjj SSSS K Since 1 S is a small parameter, terms of order 21 S or higher are neglected. This simplifies the propagation constant to 1 122122 111, 22 Pr jj SS or to 1 2 111. Pr j S (A.74)

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171 To extract the dissipation and dispersion relations, the pressure solution must be considered and dimensionalized. Starting wi th Equation (A.70) and substituting in the dimensional variables and expanding yields 000,RIjxx cc jt s p pPee or 000,IRxjx cc jt s p pPeee where ReR and ImI. This equation must be equal to ,jx xjt c p consteee Thus, by inspection, the equations for the dissipation constant, and phase speed, c, are determined to be 0,Ic and 01 .Rcc Substituting in the imaginary part of into the expression for the dissipation constant is 021 1, Pr Sc (A.75) or 021 1. Pr Lc (A.76)

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172 This is the equation for the dissipation constant presented in Section 2.2.2. Substituting in the real part of into the expression for c, the phase speed is given as 01 21 11 Pr cc S Assuming that the second term in the denominator is small, the expression is expanded into a series as 021 11. Pr cc S K Only the first two terms in the series are retained, thus 021 11, Pr cc S (A.77) or 021 11 Pr cc L (A.78) Again, this is the dispersion relation gi ven for the phase in Section 2.2.2.

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173 APPENDIX B RANDOM UNCERTAINTY ESTIMATES FOR THE FREQUENCY RESPONSE FUNCTION Uncertainty estimation is an important part of any measurement, but it is often neglected for complex-valued or multivariate data (e.g., vectors). This appendix presents a methodology for estimating the uncertainty in multivariate experimental data and applies it to the measurement of the fr equency response function obtained using a periodic random input signal. This multivar iate uncertainty method is an extension of classical uncertainty methods used for scalar variables and tracks the correlation between all variates along with the sample varian ce instead of just tracking the standard uncertainty. The method is used in this diss ertation to propagate the sample covariance matrix from spectral density estimates to the uncertainty in the frequency response function estimate for two different system m odels. The first model considers the case when only the output signal is corrupted by noise, while the second model examines when both the input and output signals are co rrupted by uncorrelated noise sources. The results for the single-noise model are ve rified by comparing them to published expressions in the literature, while the result s for the two-noise mode l are verified using a direct computation of the sta tistics. Finally, the method is applied to experimental data from two microphone measurements within an acoustic waveguide. The random uncertainty estimates in the frequency re sponse function from the multivariate method agree well with the results from a dire ct computation of the statistics.

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174 B.1 Introduction Linear, time-invariant systems occur in many engineering applications and are completely characterized by a frequency re sponse function (FRF). Examples of applications include digital and analog filters and acoustic impedance measurements. Measurements of the FRF are commonly perf ormed to test unknown systems and verify analytical models, but knowledge of the uncertainty in the estimated FRF is also desirable if not essential. The advent of inexpensiv e and powerful microprocessors has made the computation of the FRF using the Fast Fourie r Transform (FFT) routine. Early efforts used a Gaussian random noise input, whic h prompted research on the associated measurement uncertainty in the FRF (Be ndat and Piersol 2000), spectral leakage (Schmidt 1985; Bendat and Piersol 2000), the development of specialized window functions (Gade and Herlufsen 1987), and the us e of periodic input si gnals to eliminate leakage (Pintelon and Schoukens 2001a; Pint elon and Schoukens 2001b; Pintelon, Rolain and Moer 2002; Schoukens et al. 2003). Wh ere possible, the use of a periodic random (or pseudo-random) noise input is acknowledged as the preferred approach to eliminate bias errors in the FRF associated with spect ral leakage, but the random errors associated with such an input differ from the correspondi ng white noise input case. Indeed, modern spectrum analyzers now incorporate these feat ures but do not provide an estimate of the measurement uncertainty. The contribution of this appendix is to demonstrate a multivariate statistical methodology for estimating the random uncertainty in the frequency response function using a periodic random noise input. Two different system models are considered, one with noise onl y on the output signal and the other with uncorrelated noise on both the input and output signals.

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175 The outline of the appendix is as follows. Section B.2 contains a brief review of classical uncertainty analysis, which sets the stage for a discussion of multivariate methods. The section also includes a simple demonstration of the multivariate method by converting the uncertainty in r eal and imaginary parts of a complex variable to magnitude and phase. Section B.3 discusses and provide s expressions for the random uncertainty in the FRF estimate using a periodic random input for the two system models mentioned above to discern the effects of input and output noise. Th e results for the single-noise model are verified by comparing them to published expressions in the literature (Bendat and Piersol 2000), while the resu lts for the two-noise model are verified using a direct computation of the statistics. The multivariate uncertainty method is then applied to data from an acoustic application that requires the measurement of the FRF between two microphones. B.2 Uncertainty Analysis Experimental data analysis consists of tw o parts: estimating the measured quantity and the corresponding uncertainty. The estimat e of the measured quantity, called the measurand is an estimate of the true value of th e quantity. The uncertainty quantifies the estimated accuracy in terms of a confiden ce interval (Kline a nd McClintock 1953). Many articles and books have been pub lished outlining methods to estimate uncertainty in experimental data. One of the first publications, by Kline and McClintock in 1953 (Kline and McClintock 1953), was revisited in 1983 by the ASME Symposium on Uncertainty Analysis and published in th e Transactions of the ASME in 1985 (Kline 1985). Currently, the many texts ava ilable on the subject include the Guide to the Expression of Uncertain ty in Measurements published by ISO (1995) (referred to as the ISO Guide), the NIST Technical Note 1297 by Taylor and Kuyatt (Taylor and Kuyatt

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176 1994), and Experimentation and Uncertainty Analysis for Engineers by Coleman and Steele (Coleman and Steele 1999). All these pu blications prescribe essentially the same procedure, which is summarized in the next se ction, except they vary slightly in the philosophy of uncertainty source cl assification. This dissertation will cl assify uncertainty sources as random and bias (Kline and McClintock 1953; Kline 1985; Coleman and Steele 1999) as opposed to Type A and T ype B (Taylor and Kuyatt 1994; 1995). The approaches to uncertainty analysis described in (Kline and McClintock 1953; Kline 1985; Taylor and Kuyatt 1994; 1995; Co leman and Steele 1999) are limited to scalar or real-valued data. These methods do not apply to data that are multi-dimensional or multivariate Multivariate data have multiple possibly correlated, components. Examples of multivariate data include measurements of vector quantities and complexvalued data such as the FRF. Relevant ex amples using multivariate uncertainty analysis that parallel classical work are presented in references (Ridler and Salter 2002; Willink and Hall 2002; Hall 2003; Hall 2004). B.2.1 Classical Uncertainty Analysis The classical uncertain ty method described in (Kline and McClintock 1953; Kline 1985; Taylor and Kuyatt 1994; 1995; Coleman an d Steele 1999) estimates the uncertainty associated with a data reduction equation usi ng a first-order Taylor series expansion. Thus, the uncertainties of the input variable s must be small enough such that they do not violate the local linear appr oximation. The uncertainty propagation equation for the standard uncertainty or sa mple standard deviation, ru is 1 2 1112iijnnn rixijxx iijiuuu (B.1)

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177 where ixu is the standard uncertainty or sample standard deviation of the thi input variable, ijxxuis the sample covariance between the thi and thj input variables, and i ir x is the sensitivity coefficient. Th e confidence interval is estimated by multiplying ru by a coverage factor, k, that is a function of the distribution of the variable and the confidence level desired. Methods for computing the coverage factor based on the t -distribution are given in (Taylo r and Kuyatt 1994; 1995; Coleman and Steele 1999). Moffat provides an extension to the above classical method by eliminating the requirement of computing the derivative s analytically via nu merical approximations (Moffat 1985). Another subtlety in the classi cal method is that the underlying statistical distributions of the input vari ables are not propagated in the analysis. Therefore, one must assume a form of the distribution in order to complete the uncertainty analysis and estimate the confidence interval. Monte Carlo methods offer an alternative to assuming a distribution but are computati onally expensive (Coleman and Steele 1999). In spite of these issues, classical uncertainty analys is provides a way to estimate a confidence interval for experimental data and can be used before any measurements are taken for experimental design purposes. B.2.2 Multivariate Uncertainty Analysis Multivariate uncertainty analysis exte nds classical methods to multivariate problems via systematic use of the correlati on between variates both in the input and output variables. The multivariate method prov ides greater insight into uncertainty than the approach outlined in (Pintelon and Schoukens 2001a; Pintelon and Schoukens 2001b), where complex statistics are used. Th e complex statistics approach represents

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178 the sample standard deviation of a complex number with a single real-valued number and does not treat the real and imaginary parts as separate variates. Thus, confidence regions cannot be estimated from complex statistics. Finally, the complex statistics approach does not offer a methodology to propagate the uncertainty through a data reduction routine like the multivariate method does. For these reasons, this paper focuses on the use of the multivariate method instead of the complex statistics approach. B.2.2.1 Fundamentals Before outlining the procedure for multivariate uncertainty analysis, some general information is needed. First, a multivariate problem with p variates will have 2 p uncertainty components, but not all the com ponents are independent due to the symmetry of the covariance matrix (Johnson and Wicher n 2002). Thus, the covariance matrix will have only 12 pp independent elements. The task of the multivariate uncertainty analysis is to propagate th e covariance matrix through the data reduction equation. The result is another covariance matrix that repres ents the variation in the calculated output variables. Complex-valued data can be thought of as bivariate because each variable has two parts. The two parts of any complex variab le can be represented in either real and imaginary parts or magnitude and phase pol ar form. All complex computations are performed with the real and imaginary parts and converted into magnitude and phase if desired. The real and imaginary axes extend to infinity in both directions as compared to the magnitude and phase axes, in which the magn itude is constrained to be a positive real number, and the phase lies between 180 o and 180 o. This forces the use of modular arithmetic that can influence the prediction of the uncertainty (Rid ler and Salter 2002).

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179 In addition to complex-valued variables, another difference compared to the classical method is that a conf idence interval is now extende d to multidimensional space. For bivariate data, the confidence interval is extended to a confidence area. The shape of the confidence area is a function of the un certainty in each of the variates and the correlation between them. For a multivariate normally distributed variable, the confidence region is defined by the probability statement (Johnson and Wichern 2002) ,1,Prob1 1effeff pp effp F p 1xxsxx, (B.2) where x is a vector representing the multivariate variable, x is the sample mean vector, s is the sample covariance matrix, ,1,effppF is the statistic of the F distribution with p 1eff p degrees of freedom and a probability 1 1 is the level of significance, p is the number of variates, and eff is the effective number of degrees of freedom in the measurements. For a single random variable, the effective number of degrees of freedom is the number of measur ements less one. Otherwise, Willink and Hall discuss in detail how to estimate the effective number of degrees of freedom (Willink and Hall 2002). The method estimates the effec tive degrees of freedom by matching the generalized variances for all the input vari ates and reduces to the Welch-Satterhwaite method for a univariate problem. The WelchSatterhwaite is the method recommended to estimate the effective number of degr ees of freedom in (Taylor and Kuyatt 1994; 1995). Application of Equation (B.2) for a comp lex variable shows that the confidence region is an ellipse. If R x and I x are the real and imaginary parts of a complex variable, respectively, then the conf idence region is given by

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180 22 2 ,1,21 1eff RRIIeff RRRRIIII pp xxxxeffp xxxxxxxx F uuuup (B.3) where R xu and I xu are the sample standard deviations of R x and I x respectively, and is the correlation coefficient, defined as RIRRII xxExxxx uu (B.4) where E is the expectation operator. Fr om Equation (B.3), the simultaneous uncertainty bounds on the real and imaginary parts are given by the projections of the ellipse onto the respective axis and the corr elation determines the orientation of the ellipse. For the case of no correlation, 0 the axes of the ellipse are aligned with the real and imaginary axes. For the case of perfect correlation, 1 the ellipse collapses to a line. This is intuitive because if the two variates are perfectly correlated, then only knowledge of one of them is requ ired to determine the other variable. If the entire confidence regi on is not desired, the simulta neous confidence interval estimates of the uncertainty for each variate can be computed from ncfnUku (B.5) where nu is the estimate of the sample standard deviation for the thn variate, and cfk is the coverage factor given by ,1,1effeff cfpp effp kF p (B.6) The F -distribution is necessary to accommodate the correlated multiple dimensions since the population distribution is assumed to be a multivariate normal distribution (Johnson

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181 and Wichern 2002; Ridler and Salter 2002). For the remainder of this paper, the coverage factor will be com puted using two variates for complex data and the number of spectral records minus one. T hus the uncertainty reported fo r one of the variates would be nn x U and a corresponding expression can be written for each variate. B.2.2.2 Multivariate uncertainty propagation The task of the multivariate method for uncertainty analysis is to propagate the uncertainty estimates through a data reducti on equation. The difference versus the classical method is that the multivariate method simultaneously computes the uncertainty estimates for each variate along with the corr elation between them, while the univariate approach only estimates the uncertainty for a si ngle variable. Consider a generalized data reduction equation of the form rrx vv, (B.7) where r v is the real-valued vector of output variates, and x v is the real-valued vector containing all input variates. Note that any data reduction equation that has multiple input variables can be recast in the form of a single input variable with multiple variates. The first step is to create a separate expression for each output variate 1,...,prr and then form the Jacobian matrix 111 12 222 12 12 p p ppp prrr x xx rrr x xx rrr x xx J L L MMOM L (B.8)

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182 where the subscript denotes the variate. Th e uncertainty propaga tion equation now takes the form (Ridler and Salter 2002; Willi nk and Hall 2002; Hall 2003; Hall 2004) T rx sJsJ (B.9) where rs is the sample covariance matrix for the output variable, and xs is the sample covariance matrix for the input variable. If Equation (B.9) is used for a univariate output, the result identically matches the expression given in Equation (B.1) for the classical uncertainty analysis method. The limitations and remedies of the multivariate uncertainty analysis are the same as those for the classi cal uncertainty analysis discussed above, and include linearization and nu merical approximations. B.2.2.3 Application: Converting uncertai nty from real and imaginary parts to magnitude and phase The multivariate uncertainty method is now demonstrated for converting complexvalued data from real and imaginary parts to magnitude and phase. For this example, the true mean value is 0.644435j truexje and the population covariance matrix is ,,0.010.0021 cov, 0.00210.0049trueRtrueIxx (B.10) The population distribution is a bivariate norm al distribution. The sample covariance matrix is estimated from ten random data sa mples, and the uncertainty is propagated to the magnitude and phase using Equation (B.9). Also, each of the ten data points are converted to a polar represen tation, and the output sample covariance matrix of the magnitude and phase are calculated directly The estimates of the output sample covariance matrix are then comp ared to illustrate the effectiveness of both methods. To conclude the example, the coverage factor is computed and the confidence region is plotted using Equation (B.3), along with a scatter plot of the data.

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183 The sample means of the real and imagin ary parts of the data are 4.01 and 3.04, respectively. The raw data ar e plotted in Figure B-1 along w ith the sample mean value, the population mean value and the estimate of the 95% confidence region around the sample mean value. The computed coverage factor from Equation (B .6) is 3.08. Notice that the major and minor ellipse axes are not parallel to the respective coordinate axes due to the correlation between the two variat es. The figure shows that the estimated confidence region contains the true value of the population mean. 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 2.95 3 3.05 3.1 3.15 3.2 xRxI Figure B-1: A plot of the raw data and estimates for a randomly generated complex variable. data points, estimated mean value, true value, 95% confidence region. Now that the sample covariance matrix and the uncertainty region are computed and verified for the real and imaginary part s, the uncertainty can be propagated to the magnitude and phase. The Jacobian matrix becomes 2222 2222 RI RI RIRI IR RIRI RIxx xx xx x xxx xx xx x xxx xx J. (B.11)

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184 The Jacobian matrix is evaluated at the sample mean values and Equation (B.9) is used to propagate the standard uncertainty The propagated sample covariance matrix computed is 3 ,0.7830.0322 10 0.03220.0130polarpropagated s, (B.12) and the sample covariance matrix computed di rectly from the 10 sample data points in polar form is 3 ,0.7820.0335 10 0.03350.0130polardirect s. (B.13) The element 1,1 in Equations (B.12) and (B.13) represents the variance in the magnitude, and element 2,2 represents the variance in the phase. The off-diagonal elements give the covariance between the magnitude and phase. The difference between any two corresponding elements in the two esti mates for the sample covariance matrix is less than 5%. The data, the mean values, and the uncertainty estimates are shown in Figure B-2, where again the true value is co ntained within the c onfidence region. The computed coverage factor from Equation (B.6 ) applied to the estimates of the standard uncertainty to compute the 95% confidence estim ates is 3.08. The estimated values for the magnitude and phase with uncertainties are 5.030.09x and 0.6480.011radx This simple example demonstrates the usefulness of the multivariate uncertainty analysis and provi des some insight concerning the terms and concepts described above.

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185 4.8 4.9 5 5.1 5.2 0.63 0.64 0.65 0.66 0.67 MagnitudePhase [rad] Figure B-2: A plot of the raw data and estimates in polar form for a randomly generated complex variable. data points, estimated mean value, true value, 95% confidence region. B.3 Frequency Response Function Estimates The FRF is ubiquitous in engineering systems, yet the measurement and estimation of its uncertainty is nontrivial There are many factors to consider when measuring and estimating a nonparametric FRF, such as analog-to-digital sampling settings, FFT settings, the assumption of a system model describing the wa y noise enters the signals, etc. Previous researchers have studied t echniques for reducing the error in the FRF but few have studied the uncertai nty in the final estimate. Bendat and Piersol (Bendat and Piersol 2000), Schmidt (Schmidt 1985) and Pint elon et al. (Pintelon, Rolain and Moer 2002) have derived expressions for the uncerta inty in the FRF for some special cases. This paper, however, provides a systematic fr amework to extend their analysis to other cases involving periodic determin istic inputs and to propagate the resulting uncertainty to derived quantities. This section analyzes th e random uncertainty for two system models and the data reduction equations for estimating the FRF using the multivariate uncertainty analysis framework. The first model is a single-input/single-output (SISO) system with

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186 Gaussian noise added only to the output signal, which is relevant for system identification applications involving a noise -free input signal and a (perhaps) noisy output sensor signal. The expressions derived by the multiv ariate method are compared to the results given in Bendat and Piersol (Bendat and Piersol 2000). The second model is another SISO system, but uncorrelated Gaussian noise signals are added to both the input and output signals. The results of this case ar e compared to numerical simulations designed to verify the derived un certainty expressions. The systems studied in this dissertation are assumed to be excited by a periodic random input signal. This type of signal, which is standard in most modern spectrum analyzers, is tailored to the parameters chos en for the spectral analysis and designed to prevent spectral leakage. An example of this type is called pseudo-random noise (Randall 1987). This signal is actually deterministic and consists of a finite summation of discrete sine waves at exact bin fre quencies for the spectral analysis, but each component has a random, uniformly distributed phase angle. The probability density function for the pseudo-random signal approach es a Gaussian distribution as the number of components is increased. In practice, the distribution can be approximated as Gaussian if 400 or more discrete sine waves are used. The periodic ity of the input signal prevents any bias error in the estimated spect rum due to spectral leakage when a uniform or boxcar window is used; thus the remainder of this appendix assumes that there is no bias uncertainty due to leakage in the spectral estimates. B.3.1 Output Noise Only System Model The first system model is illustrated in Figure B-3, where x is the pseudo-random input signal, v is the noise free output of the system, n is a zero-mean, Gaussian noise

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187 signal, y is the measured output, Hf is the FRF of the linear system, and f is the frequency. For this case, the noise signal m odels all measurement noise and also includes unmodeled system inputs. This model is ap propriate for linear systems where noise in the input signal is negligible such as when an input signa l is the output from a function generator. Figure B-3: System model with output noise only. The time-domain output signal is the sum of the noise-free output and the noise v yttnthtxtnt (B.14) where t is time, ht is the impulse response function, and “*” denotes the convolution. The Fourier transform representation is YfVfNfHfXfNf (B.15) Equation (B.15) and the Fourier transform of the input signal X are substituted into the definitions of the autoand cross-spectrum density functions (Bendat and Piersol 2000). Then the spectra functions are substituted into convenient forms defining the population variance and covariance, which are 22var A EAA %% (B.16) and cov, A BEABAB %% %% (B.17)

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188 Here, A % and B % are random complex variables and A and B are the corresponding mean values. The resulting expre ssions are expanded, and the e xpectation operator is applied to each term, noting that the noise Nf is the only random variable and is uncorrelated with x and v. Then the equations 0RIENN (B.18) and 22ˆ 4RInnT ENENG (B.19) from Section 9.1 of Bendat and Piersol, are used to simplify the expectation operations of the Gaussian noise signal (B endat and Piersol 2000), where ˆnnG is the autopower spectral density of the noise signal, and T is the record length. The sample variance and sample covariance of the mean are obtained by simp ly dividing the variance and covariance by the number of sample records, recn. The resulting sample covariance matrix between the spectral components is 2 22 22 22 220000 ˆ ˆ ˆˆˆˆ 1 ˆˆ 11 0 ˆˆˆˆ ,,, ˆˆˆˆ ˆˆ 11 00 2 ˆˆˆˆ ˆˆ 11 00 2xyyy xyyyxyxyyyxy recrecrec xxyyxyxy xyyyxyxyxxyy recrec xyyyxyxyxxyy recrecG GCGQ nnn GGCQ GCGG nn GQGG nn s,(B.20) where 2 2ˆˆˆ ˆxyxyxxyyGGG is the estimate of the or dinary coherence function, ˆxxG and ˆyyG are the estimates of the power spectral densities of the input and output signals,

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189 respectively, and ˆxyC and ˆxyQ are the estimates of the coand quad-spectral densities. The cross-spectral density, ˆxyG, is given by ˆˆˆxyxyxyGCjQ A convenient feature of this model is that the sample covariance matrix can be estimated from just the measurement of the input signa l and the output signal. Sin ce the input signal is noisefree, there is no variation in its power spect ral density and it is uncorrelated with any other spectral quantity. This is reflected by the row and column of zeros is Equation (B.20). The other two zeros in the sample c ovariance matrix show that the coand quadspectral densities are also uncorrelated to each other. The remaining nonzero elements quantify the variance, if on th e diagonal, and the covariance, if off the diagonal. The uncertainty in the FRF is found by using a multivariate method to propagate the uncertainty from the spectral estimates give n in Equation (B.20) to the FRF estimator. The unbiased estimator of the FRF for this sy stem model is (Vold et al. 1984; Bendat and Piersol 2000) 1ˆ ˆ ˆxy xxG H G (B.21) or, using real and imaginary forms that are more convenient for the multivariate method, 1, 1,ˆ ˆ ˆ ˆ ˆ ˆxy R xx I xy xxC H G H Q G (B.22) With the data reduction equation defi ned, the Jacobian matrix becomes

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190 11111 2 1111 2ˆ 1 00 ˆˆ ˆ 1 00 ˆˆxy RRRR xxyyxyxy xxxx H IIII xy xxyyxyxy xxxxC HHHH GGCQ GG HHHH Q GGCQ GG J (B.23) Thus, the sample covariance matrix for the estimate of the FRF is given by Equation (B.9) as 2 1,1, 2ˆ ˆ 1 0 ˆ 2 ˆˆ ˆ ˆ 1 0 ˆ 2xy yy rec xx RI xy yy rec xxG n G HH G n G s (B.24) Equation (B.24) gives the standard uncertainty that is used to propagate the uncertainty in the FRF through any subsequent data reduc tion equation. Using Equations (B.9) and (B.11) and simplifying gives the polar form of the uncertainty as 2 2 1 2 11 2 21 ˆ 0 2 ˆˆ 1 0 2xy recxy xy recxyH n HH n s (B.25) The square roots of the diagonal terms in E quation (B.25) exactly match those given in Table 9.6 in Bendat and Piersol (Bendat and Piersol 2000), thus validating the multivariate technique used to derive those e xpressions. This approach is extended in the next section to a more complex system mode l where expressions for the uncertainty are not available in the literature. Equation (B.25) shows that th e uncertainty in the FRF is related to the number of spectral averages, the ordinary coherence func tion, and the magnitude of the FRF itself. Increasing the value of the ordinary c oherence function will lower the standard

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191 uncertainty in the FRF. This can be accomplished by designing the measurement to minimize unmodeled dynamics, to reduce any non linearities and to reduce noise sources. Increasing the number of averages will also decrease the sample covariance matrix but only reduces the standard uncertainties as 1recn B.3.2 Uncorrelated Input/Output Noise System Model The second model, with uncorrelated noise added to the input a nd output signals, is shown in Figure B-4. An example of such a system is the measurement of the mechanical impedance or admittance of a structure. Here, where u is the pseudo-random input signal, v is the noise free output of the system, m and n are uncorrelated, zeromean, Gaussian noise signals, x is the measured input signal, y is the measured output signal, and Hf is the FRF of the system. Ag ain, the noise signals account for measurement noise and unmodeled dynamics. Th is model is appropriate for systems in which an input signal is supplie d to an actuator that excites the system, and the outputs of the actuator (instead of th e function generator) and the sy stem are both measured. Figure B-4: System model with uncorrelated input/output noise. The time-domain input signal is x tutmt (B.26) and the Fourier transfor m representation is

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192 X fUfMf (B.27) The corresponding output signals are v yttnthtutnt (B.28) and YfVfNfHfUfNf (B.29) The expressions for the random uncertainty a nd the variance are derived in a similar manner as before. One important note for th is case is the assumption of uncorrelated noise sources. Since m and n are assumed to be uncorrelate d, the cross-spectral density function between them is identically zero. Th is fact is used to simplify the variance and covariance expressions along with the expressi ons for the expectati ons of two zero-mean, Gaussian signals given in Equations (B.18) and (B.19), with corresponding version for m, and with ˆ 0 4RRIImnT EMNEMNC (B.30) and ˆ 0 4RIIRmnT EMNEMNQ (B.31) all provided in Section 9.1 of (Bendat and Piersol 2000), where ˆmnC and ˆmnQ are the coand quad-spectral density of the two noise signals. The simplified result for the estimate of the sample covariance matrix is

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193 ˆˆˆ ˆˆˆˆ 2 0 ˆˆˆ ˆˆˆˆ 2 0 ˆˆˆˆ ,,, ˆˆˆˆ 0 2 ˆˆˆˆ 0 2mmxxmm mmxymmxy nnyynn nnxynnxy xxyyxyxy mmxynnxy mmxynnxyGGG GCGQ nnn GGG GCGQ nnn GGCQ GCGC nnn GQGQ nnn s ,(B.32) where ˆˆˆˆˆˆnnxxmmyymmnnGGGGGG ˆmmG is an estimate of the power spectral density of the input noise signal, and ˆnnG is an estimate of the power spectral density of the output noise signal. The zero elements (1,2) and (2,1) in Equation (B.32) show that the estimates of the power spectral density fo r the input signal and output signal are uncorrelated. Similarly, the zero elements (3,4) and (4,3) in Equa tion (B.32) show that the coand quad-spectral densi ties are again uncorrelated. Note that this system is not completely characterized by measurements of just the input and output signals, because estimates of ˆmmG and ˆnnG are also required. In practice, these can be estimated by either measuring the input and output signals when the source is turned off, thus setting 0 u or using application-specific noise models. With no input signal, the measured quantities arise solely due to the noise. Inherent to th is approach is the assumption that the addition of the input signal does not change the estimates of the noise power spectra. To propagate the uncertainty to the FRF, a form of the FRF must be chosen. If the definitions of the spectral estimate are substituted into Equation (B.21), the result is 11 ˆ ˆ 1 ˆmm uuHH G G (B.33)

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194 where H is the true FRF. The result is clearl y biased by the noise-to -signal ratio of the input (Vold, Crowley and Rocklin 1984; Bendat and Piersol 2000) Another common estimate of the FRF is 2ˆ ˆ ˆyy yxG H G (B.34) and for this case 2ˆ H becomes 2ˆ ˆ 1 ˆnn vvG HH G (B.35) which is biased by the noise-to-signal ratio of the output (Vold, Crowley and Rocklin 1984; Bendat and Piersol 2000). A third estimate for the FRF uses a geom etric average (Vold, Crowley and Rocklin 1984) 312ˆˆˆ HHH (B.36) Substituting in the definitions for 1ˆ H, 2ˆ H, and the spectral dens ities, Equation (B.36) becomes 3ˆ 1 ˆ ˆ ˆ 1 ˆnn vv mm uuG G HH G G (B.37) Equation (B.37) shows that 3ˆ H is also biased, but if the input and output noise-to-signal ratios are either both small or non-negligib le but the same order of magnitude, these biases tend to cancel each other, providing a be tter estimate of the FRF in the case with two uncorrelated noise sources (Herlufsen 1984).

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195 One other choice for an estimate of th e FRF is described in (Pintelon and Schoukens 2001b; Pintelon and Schoukens 2001a) ˆ ˆ ˆfftY H X (B.38) which is the ratio of the Fourier transform co efficients. Substituting in the definitions of ˆ X and ˆ Y as given in Equations (B.15) an d (B.27), Equation (B.38) becomes ˆ 1 ˆ ˆ ˆ 1 ˆfftN V HH M U (B.39) Equation (B.39) shows that there are two disa dvantages to this estimate as compared to 3ˆ H. First, the bias term in ˆfftH is complex, and thus both the magnitude and phase of the estimate are biased. The bias term of 3ˆ H is real-valued and th erefore only affects the magnitude, leaving the phase unbiased. Second, the bias on 3ˆ H will be smaller because of the square root function. For these reasons, 3ˆ H is used in this dissertation. The resulting form of 3ˆ H for the multivariate uncertainty analysis is 22 3, 3, 22ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆˆˆyy xy xxxyxy R I yy xy xxxyxyG C GCQ H H G Q GCQ (B.40) The Jacobian matrix for 3ˆ H is

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196 32 33 2 2 33 2ˆˆˆˆˆˆˆˆ ˆˆˆ ˆˆ ˆˆ 2 ˆˆˆ 2 ˆˆˆˆˆˆˆˆ ˆˆˆ ˆˆ ˆˆ 2 ˆˆˆ 2yyxyxyyyxyyyxyxy xxxxxx xxxy xyxy xxyyxy H yyxyxyyyxyxyyyxy xxxxxx xxxy xyxy xxyyxyGCCGQGCQ GGG GG GG GGG GQQGCQGC GGG GG GG GGG J ,(B.41) where 22ˆˆˆxyxyxyGCQ Equation (B.41) reveals that the uncertainty in 3ˆ H scales with ˆ ˆyy xxG G Thus, the uncertainty increases at resona nce, where the output is large for a small input. While the opposite may appear to be true at anti-resonance, in this case the output signal is low, and this measurement cond ition is also often dominated by noise. The Jacobian matrix in Equation (B.41) and the sample covariance matrix in Equation (B.32) are cumbersome and do not le nd themselves to directly propagating the uncertainties analytically. A suitable option is to evaluate each term numerically and then use matrix multiplication as required by Equation (B.9). The 95% simultaneous confidence intervals for each variate are then estimated by taking the square root of the diagonal element and multiplying by the covera ge factor computed from Equation (B.6), with two variates and the correct effective degrees of freedom. A numerical simulation is performed to ve rify the expressions derived in this section. A normalized two degree of freedom system model is chosen for the numerical simulations to represent a system with a known FRF. The model of the FRF is 2 1 11 2 2 23 223312 1212 ff j ff H f fff jj f fff (B.42)

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197 where f is the frequency, 1 f is the anti-resonance frequency, 2 f and 3 f are the resonance frequencies, 1 is the anti-resonan ce damping ratio, and 2 and 3 are the damping ratios associated with the two re sonance frequencies. The values of the parameters chosen for the simulations are 120Hz f 210Hz f 330Hz f 10.05 20.2 and 30.05 as a matter of convenience. The spectral analysis is performed with 1,000 sample records of da ta with 1,024 samples each, and the Nyquist frequency is set to 128 Hz. The pseudo-ra ndom noise signal is constructed as the summation of 512 sine wave components at th e bin frequencies with the dc component set to zero; thus the uncerta inty of the dc component is ignored. Each sine wave component has unit amplitude and the power in the input noise signal is 22.5610 units squared and the power in th e output noise signals is 32.510 units squared. The overall signal-to-noise ratio is thus 40 dB for the input signal, x and is 45 dB for the output signal, y A uniform window with no overlap is used for the spectral analysis. The procedure of the simulation is as follo ws. First, one block of the noise-free pseudo-random noise signal is generated and passed through the system FRF in the frequency domain to determine the noise-fre e output. After applyi ng an inverse Fourier transform to both signals, independent, zero-m ean, Gaussian noise signals are added to both noise-free signals in the ti me domain. The spectral quantities are then computed and 3ˆ His estimated using Equation (B.40). This procedure is repeated for each of the 1,000 sample records. Finally, the uncertainties ar e computed using the results of this section via the multivariate method, as well as directly from the statistics of the 1,000 sample records.

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198 The results of the FRF from the simulation are shown in Figure B-5. The results for this simulation are limited to the frequency range of 0 to 60 Hz to avoid the bias error described in Equation (B.37). Since the mode led system attenuates the output signal at higher frequencies, the signal-to -noise ratio of the output si gnal will decrease, as the power in the noise signal is assumed to be constant across all fre quencies. By limiting the results to 60 Hz, the bias error in the estimated FRF is less than 0.15%. The uncertainty estimates are not shown in Figure B-5 for clarity but are plotted in Figure B-6 using both the multivariate method and the calcul ated statistics. The distribution of the 1,000 raw averages is a bivari ate normal distribution; t hus the confidence region is assumed to be symmetric and is computed by estimating the sample covariance matrix and applying the coverage factor to the square of the diagonal elements. The value of the 95% coverage factor is 2.45, as is computed from Equation (B.6). The two methods to estimate the uncertainty are essentially indi stinguishable in Figur e B-6. The largest difference between the two uncertainty methods is 5% for the magnitude and 21.110 degrees for the phase, and the average di fference is 1.5% for the magnitude and 31.310 degrees for the phase. The true value of th e FRF falls outside the uncertainty range at only 5 frequency bins for the magnitude and only 7 frequency bins for the phase. Under the assumption that for a linear system, each frequency bin is independent, the estimated value of the FRF is consistent wi th a 95% confidence interval.

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199 0 10 20 30 40 50 60 0 0.5 1 1.5 2 2.5 Magnitue 0 10 20 30 40 50 60 -200 -150 -100 -50 0 Phase [deg]f [Hz] Figure B-5: Bode plot of the true FRF and the experimental estimate. True FRF, FRF Estimate. The true and esti mated FRF are indistinguishable. 0 10 20 30 40 50 60 0 0.5 1 1.5 x 10-3 Magnitue 0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 Phase [deg]f [Hz] Figure B-6: Magnitude a nd phase plot of the uncertainty estimates. Direct Computation, Multivariate Method. The tw o methods to estimate the uncertainty are essentia lly indistinguishable.

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200 The plot of the uncertainty estimates shows that the maximum uncertainty in the magnitude is at the resonance frequencie s, and the minimum is at anti-resonance frequency. Conversely, the ma ximum uncertainty in the phase of the FRF is at the antiresonance. At both resonance and anti-resonance, one of the transducers will have problematic signal-to-noise ra tios. Figure B-6 also shows that the uncertainty in the phase angle continues to increase as the frequency is increased This reveals that at high frequencies the uncertainty in the phase a ngle may be dominant and may determine the accuracy and number of spectral averages need ed to obtain the desired uncertainty in the FRF estimate. B.4 Application: Measurement of th e FRF Between Two Microphones in a Waveguide The multivariate uncertainty method is now demonstrated on actual experimental data in an important acoustic application. The Two-Micr ophone Method (TMM) is the standard technique for measuring the specific acoustic impedance of a material specimen (ASTM-E1050-98 1998; ISO-10534-2:1998 1998). Th is method uses a waveguide with a compression driver mounted at one end, wh ile the specimen is mounted at the other end. Two microphones are flushmounted to the side of the waveguide at two different axial locations. The compression driver is typically excited with a broadband signal, such as a pseudo-random noise signal, to produce plane traveli ng waves within the waveguide over a limited frequency range. Th e incident waves reflect off the specimen mounted at the end and create a standing wave pattern. The FRF is measured between the two microphones and a data reduction equatio n is then used to compute the acoustic properties of the specimen from the FRF and a few other measurements, such as the temperature and the locations of the microphones. To estim ate the uncertainty in the

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201 computed acoustic properties, such as the complex-valued reflection coefficient, the uncertainty in the FRF must first be k nown. This section will estimate the random uncertainty using the multivariate method and vi a direct computation of the statistics. The two measured acoustic signals are assumed to be corrupted by uncorrelated Gaussian noise. One of the microphones is assi gned to be the input signal, and the other is assigned to be the output signal. Therefor e, both the input and output signals contain noise, and the appropriate system mo del is shown in Figure B-4. The measurement of the FRF is subject to random and bias errors, the latter of which is primarily due to calibration errors in the two measurement channels when a periodic excitation signal is used. To rem ove this bias, measurements are taken in original and switched positions (AST M-E1050-98 1998; ISO-10534-2:1998 1998). The original FRF estimate, ˆOH, and the switched FRF estimate, ˆSH, are geometrically averaged to remove the calib ration bias, resulting in ˆ ˆ ˆO SH H H (B.43) An appropriate form of the real and imaginary part of ˆ H for the multivariate method is 1 4 22 1 4 22 1 4 22 1 4 22ˆˆ cos ˆˆ ˆ ˆˆ sin ˆˆOO RI SS RI OO RI SS RIHH HH H HH HH (B.44) where

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202 11ˆˆ tantan ˆˆ 2OS II OS RRHH HH (B.45) The uncertainty in ˆ H can be estimated by calculating the uncertainty in the original and switched estimates and then propagating the result to ˆ H using the multivariate method. The input sample covariance matrix is formed by applying Equations (B.32) and (B.41) to each of the FRF in the original, ˆO H s, and switched positions, ˆS H s, to form their respective sample covariance matrices as out lined in Section B.3.2. The two sample covariance matrices are formed into a single input covariance matrix as ˆ ˆ ˆO SH H H s0 s 0s, (B.46) where 0 is the zero matrix. The Jacobian for Equation (B.44) is 11 22 1,11,2 1,31,4 3355 11 222222 ˆ 11 22 2,12,2 2,32,4 3355 11 222222ˆˆ ˆˆˆˆˆˆ 2222 ˆˆ ˆˆˆˆˆˆ 2222OO OSOSSS H OO OSOSSSHH AA AA HHHHHH J HH AA AA HHHHHH (B.47) where 22ˆˆˆOOO RIHHH 22ˆˆˆSSS RIHHH and 1,12,1 1,22,2 1,32,3 1,42,4ˆˆˆˆ cossin,cossin, ˆˆˆˆ cossin,cossin, ˆˆˆˆ cossin,cossin, ˆˆˆˆ cossin,cossin.OOOO RIIR OOOO IRRI SSSS RIIR SSSS IRRIAHHAHH AHHAHH AHHAHH AHHAHH (B.48) The waveguide used in this experiment ha s a cross-section of 25.4 mm by 25.4 mm and a usable frequency range of 0.5 to 6. 7 kHz. The acoustic pressure signals are

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203 measured using two Brel and Kjr Type 4138 microphones and a Brel and Kjr Pulse Analyzer data acquisition system. The two microphone signals are sa mpled at a rate of 16,384 Hz with a record length of 0.125 s for a total of 100 spectral averages. A periodic pseudo-random excitation signal is generate d by the Pulse system and amplified with a Techron Model 7540 power amplifier before application to the BMS 4590P compression driver. The microphones are calibrated with a Brel and Kjr Type 4228 Pistonphone. The microphone that is initially mounted furt hest from the specimen is considered the reference signal and is denoted the input signal, x The excitation signal is then applied, and the amplifier gain is adjusted such th at the sound pressure le vel at the reference microphone is approximately 120 dB (reference 20Pa) for all frequency bins. Then the full-scale voltage on the two measurement ch annels of the Pulse system is adjusted to maximize the dynamic range of the data system The excitation signal is turned off and the microphone signals are measured to estimate the noise spectra. The excitation signal is turned on and the two microphone signals are recorded with th e microphones in their original positions and switched positions. The time-series data are used to comput e the power spectra and the cross-spectra between the two microphones fo r the original position a nd switched positions. The spectra are used to compute 3ˆ H using Equation (B.40) and the sample covariance matrices using Equation (B.41). The computed FRF is shown in Figure B-7. The uncertainty in the spectral estimates is pr opagated to the magnitude and phase of the averaged FRF via the multivariate method using Equation (B.9) with Equations (B.46) and (B.47). The value of the 95% coverage factor for the averaged FRF is 2.50, as computed from Equation (B.6) using 1recn The computation of the effective degrees

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204 of freedom, as described by (Willink and Hall 2002), would limit the effective degrees of freedom of the averaged FRF between the le ast amount of degrees of freedom for an input variable (99) and the sum of all the degrees of freedom for all input va riables (198). Within this range, the 95% coverage factor will only change by a maximum of 1%, and thus this change is neglected for this de monstration. The confidence intervals are computed from Equation (B.5). Then the 100 sample estimates of the FRF are used to compute the sample covariance matrix between the magnitude and phase. The estimates of the uncertainty in the magnitude and phase are shown in Figure B-8. The uncertainty estimates agree well with each other except for at 1.65, 2.70, and 4.90 kHz, where one of the microphone locations corresponds to a node in the standing wave pattern. When this occurs, one of the microphones is measuring a small acoustic pressure and the signal is dominated by the measurement noise. The value of the FRF w ill theoretically tend towards zero if the output microphone is at the node and will tend towa rds infinity if the input microphone is at the node. In both cases, the uncertainty in the FRF becomes large. The average difference between the two es timates of the simultaneous confidence intervals is 10% for the magnitude and 11% for the phase angle, and the maximum difference is 0.04 for the magnitude and 0.05 degrees for the phase angle. Given the small number of records, these differences are deemed small enough to validate the multivariate uncertainty analysis.

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205 0 1 2 3 4 5 6 10-3 10-2 10-1 100 101 102 103 Magnitude 0 1 2 3 4 5 6 -200 -150 -100 -50 0 50 Freq [kHz]Phase [deg] Figure B-7: The experimentally measur ed FRF between the two microphones. 1 2 3 4 5 6 10-5 10-4 10-3 10-2 10-1 100 Magnitude 1 2 3 4 5 6 10-4 10-3 10-2 10-1 100 Freq [kHz]Phase [deg] Figure B-8: Comparison for the uncertainty estimated by the multivariate method and by the direct statistics. Direct statistics, Multivariate method. The two methods to estimate the uncertainty are essentially in distinguishable.

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206 B.5 Conclusions An experimental measurement consists of two parts, an estimate of the measured quantity and an estimate of the uncertainty. The uncertainty allows users to determine whether or not the estimate is accurate enough for their needs. Classical methods for uncertainty analysis are restricted to scalar quantities and are not applicable to complexvalued FRF estimates, an important quantity in linear, time-invariant dynamic systems. The multivariate method extends the techniques of the classical method to problems with any number of variates or dimensions. This paper applies the multiv ariate method to the nonparametric measurement of the FRF. Two system models were considered, one with only noise in the output signal and the other with uncorrelate d noise sources in both the input and output signals. The sample covari ance matrices were derived for both cases for the spectral density estimates. The results s howed that, in the first model, all required information is contained in the measurement of the input and output signals, while the second model required an extra measurement to estimate the power spectra of the two noise signals. The sample covariance matri ces were then propagated to the magnitude and phase of the FRF. For the first model, the derived expressions were identical to published expressions in Bendat and Piersol [1]. The second model was verified by numerical simulations, which showed that the multivariate method yielded uncertainty estimates consistent with the direct computation of the statistics from the sample records. Finally, this appendix demonstrated the multivariate method on real experimental data involving the frequency response estimation between two microphones in an acoustic waveguide. The estimate of the unc ertainty by the multivariate method yielded consistent results with the dire ct computation of the statistics from the sample records.

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207 The results demonstrate that the multivariate method can be applied to experimental data that are multivariate in nature and provide reliable estimates of measurement uncertainty.

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208 APPENDIX C FREQUENCY RESPONSE FUNCTION BI AS UNCERTAINTY ESTIMATES C.1 Bias Uncertainty The bias uncertainty is the part of the total uncertainty that is constant for every measurement (Coleman and Steele 1999). For an example with ten length measurements of a rod, the bias uncertainty would be the ac curacy of the ruler si nce the accuracy of the ruler did not change during the ten measuremen ts. Sources of bias uncertainty are more complex for the FRF than just the accuracy of a ruler. Three sources of bias uncertainty for the FRF are the total accuracy of the analogto-digital converter, spectral leakage due to the finite frequency resolution of discrete -time spectral analysis, and time delays that may be present between the two measurement ch annels. The bias uncertainty due to the accuracy of the analog-to-digital converter can be minimized by using the proper range for the device with regards to the input signal, thus maximizing the significant number of bits. In most cases where the range has been adjusted, the bias uncertainty because of the accuracy of the analog-to-digital converter can be neglected. The bias uncertainty due to a time delay in the output signal as compared to the input signal has been developed by Seybert and Hamilton (Seybert and Hamilt on 1978) and by Schmidt (Schmidt 1985). Physically, a time delay in one of the measurem ent channels compared to the other is an effect of a phase mismatch between the re sponses of the two measurement channels, including phase mismatches in the transducers. Both papers conclude d that the bias error is dependent on the parameter T where is the time delay and T is the record length. Thus the bias error can be minimized by keepi ng the time delay short as compared to the

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209 record length. Schmidt also stated that the time delay bias error is a function of the window function used and gives expressions for a rectangular window and an arbitrary window (Schmidt 1985). The bias uncertainty due to spectral leakage is more difficult to quantify than any other uncertainty source for the FRF. The FR F is computed through the use of ensemble averaging and a FFT algorithm. The advantage to this procedure is the ability to use digital computers for the processing, but the disadvantage is that the signal must be approximated. Only discrete samples and fi nite record lengths can be used in the computation. This gives rise to the finite frequency resolution in the estimate for the FRF, and in the coherence as well as for other spectral measurements. The bias uncertainty can be minimized through the us e of properly designed window functions, but there still is a necessity to estimate a value for the bias uncertainty. Bendat and Piersol (Bendat and Piersol 2000) and Schmidt (Schmidt 1985) have derived expressions for the bias uncertainty of the FRF and coherence functions for the cases with a continuous window function and white noise excitation, but ther e are no expressions given for other excitation signals other than wh ite noise. A particularly useful signal is a periodic random noise signal. This signal is designed specifically for an FFT analyzer that will be measuring the output signal of a linear system. The periodic random noise signal is a summation of discrete tones or frequencies which also are exact bin frequencies for the FFT analyzer. This allows for the signal to be pe riodic within a single record and allows for the use of a rectangular or boxcar window. Spectral leakage is not possible since no frequency conten t exists in frequencies other than the bin frequencies, therefore eliminating any bias uncertainty in the FRF at the exact bin frequencies. There

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210 still would be a bias uncert ainty in the FRF as a result of insufficient frequency resolutions, which would limit the ability to reso lve fine scale detail in the FRF, such as peaks in the response for resonant absorber s, for example the SDOF liner discussed earlier. Usually, sufficient frequency reso lution can be achieved through an iterative process, but still a properly de signed input signal and window function should be used to avoid bias error due to spectral leakage. To illustrate the conclusion that no bias e rror due to spectral leakage is present when a properly designed periodic random input signal is used, numerical simulations are preformed. These simulations first choose an analytical form for the FRF of the system model and a bandwidth for the analysis. Next, the remainder of the FFT analysis parameters are chosen and the input periodic random signal is constructed. The output signal of the modeled system is then co mputed by using the input signal and the analytical FRF model. The output signal and th e input signal are then reduced to estimate the FRF. The estimated FRF is then compared to the analytical FRF at the bin frequencies and a root-mean-square (rms) erro r estimate is made for the magnitude and phase from the two FRFs to provide a represen tative number of the av eraged error in the estimate of the FRF at each bin. This will he lp to illustrate the order of magnitude for the error and thus to illustrate wh ether or not bias error is present. Next, a nonlinear leastsquares fit is used to estimate the parameters of the FRF from the simulated data. In most cases the estimated parameters would have a residual imaginary part that is over four orders of magnitude smaller than the real part This residual imaginary is neglected since the parameters of the FRF are known to be real-valued.

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211 The analysis parameters are chosen with c onsideration to the experimental goals for acoustic impedance testing. The bandwidth of interest is from 0 to 20 kHz. To accommodate the bandwidth and maintain an integer bin width, the sampling frequency chosen is 65.536 kHz with 4,096 points used per FFT block. This provides a frequency resolution of 16 Hz, which is small enough to distinguish the small scale detail of the FRF, including resonant peaks. The Nyquist frequency for this sampling frequency is 32.768 kHz, which is beyond the required 20 kHz for the bandwidth. The data in the range from 20 kHz to the Nyquist frequency is not used in the computation of the rms errors and for the nonlinear least-squares fit si nce it is beyond the bandwid th of interest. The first system model chosen is a se cond-order system represented by a springmass-damper system. The analytical expression for the FRF for this model is 211 12nnH k ff j f f (B.49) where f is the frequency, n f is the natural frequency, is the damping ratio and kis the spring constant. Evalua ting equation (B.49) for 0 f shows that 01 Hfk and the spring constant can easily be found from the DC response of the system. Therefore in these simulations the FRF has been normalized by the DC response in order to isolate the parameters responsible for th e dynamic response. Thus the form for the FRF used in these simulations is 21 0 12nnH Hf ff j f f (B.50) For these simulations, the natural frequency is varied in ten steps from 12 kHz to 12.008 kHz. The first chosen frequency corresponde d exactly to a bin frequency and the last

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212 chosen frequency corresponded to exactly half way in between two bin frequencies. The damping ratio is chosen to be 0.01 in orde r to prevent the response from becoming too large at the natural frequency, which prevents numerical problems. The results for the ten simulations are given in Table C-1. Th e estimated natural frequencies and damping ratios matched the true values, and the rms errors for magnitude and phase were on the order of 1210 and 1310, respectively. Both of the rms errors are on the order of numerical accuracy that can be expected from an FFT-based algorithm. The results show that bias error is not present. Table C-1: Simulation results for the second-order system. RMS Error fn [Hz] est fn [Hz] est magnitudephase [rad] 12000.0 0.01 12000.0 0.010 1E-12 5E-13 12000.8 0.01 12000.8 0.010 2E-12 5E-13 12001.6 0.01 12001.6 0.010 1E-12 5E-13 12002.4 0.01 12002.4 0.010 1E-12 5E-13 12003.2 0.01 12003.2 0.010 1E-12 5E-13 12004.0 0.01 12004.0 0.010 2E-12 5E-13 12004.8 0.01 12004.8 0.010 1E-12 5E-13 12005.6 0.01 12005.6 0.010 2E-12 5E-13 12006.4 0.01 12006.4 0.010 1E-12 5E-13 12007.2 0.01 12007.2 0.010 1E-12 5E-13 12008.0 0.01 12008.0 0.010 1E-12 5E-13 Any real measurement of the FRF would al so be influenced by random noise that would not be periodic within the record leng th. If a rectangular window is used this would lead to spectral leakage of the random noise component. To understand how this would affect the results for the previous simu lations, another simulation is performed, but this time a small random noise signal is added to the output signal before the computations of the estimated FRF. For th is case a single natural frequency of 12.008 kHz is chosen and the damping ratio is left at 0.01. Only a single time record is used so

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213 that the effective number of averages is one. The random noise signal that is added to the output signal has a total vari ance of 0.1 units-squared. Th is yielded a signal-to-noise ration (SNR) of approximately 40-50 dB for each bin based on the output signal for the flat response region in the FRF before the resonance peak. The results for the simulation are shown in Figure C-1. The figure shows that the estimated FRF is in good agreement with the an alytical form of the FRF. The rms errors for the magnitude and phase of the FRF are 0. 007 and 0.3 rads, respec tively. This is a significant increase from the rm s errors that were found before. The rms uncertainty estimates due to the random components, as described in Appendix B, are 1E-7 and 2E-8 radians for the magnitude and phase of th e FRF, respectively. These uncertainty estimates only partially explai n the increase in th e rms errors, therefore the random noise did contribute a bias error because of spectral leakage. To verify the results, the 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 |H| Analytic Measured 0 2 4 6 8 10 12 14 16 18 20 -200 -150 -100 -50 0 50 [deg]f [kHz] Figure C-1: FRF for the si mulation with random noise.

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214 simulation is rerun again, but this time with 1,000 spectral av erages. The rms errors are 2E-4 and 2E-4 radians for the magnitude and phase, respectively. These results demonstrate that as the number of averages is increased, the rms error decreases as expected, since spectral averag ing is known to reduce the eff ects of random noise. Thus spectral averages also help to reduce any sp ectral leakage and thus bias error due to random noise. C.2 Conclusions The FRF is an important estimator for the response of many engineering systems and thus its sources of error should be understood. The two major categories for uncertainty are random uncertainty and bias uncertainty. Estimates for the random uncertainty were developed in Appendix B, wh ereas the bias uncertainty was dealt with in this appendix. The bias uncertainty has three sources which can be eliminated, minimized, or corrected. The bias error due to the accuracy of the analog-to-digital converter can be minimized by choosing the proper range of the device to match the measured signal. The bias error due to a time delay can be minimized by keeping the time delay small compared to the record le ngth and also can be corrected for with analytical expressions (Seybert and Hamilt on 1978; Schmidt 1985). The major focus of this appendix was with the bias error due to spectral leakage. In general, there are expressions that can be used to estimat e the uncertainty (Sch midt 1985; Bendat and Piersol 2000), but one special case was shown not to have any bias error. The bias error due to spectral leakage is e liminated if the input signal is a periodic within the record length.

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215 APPENDIX D SOUND POWER FOR WAVES PROP AGATING IN A WAVEGUIDE This appendix documents the detailed deri vation of the formulas in Chapter 4 for the power contained in the incident and reflect ed waves propagating in a rectangular duct. Recall the solution to the wave equation fo r pressure in the frequency domain as ,,coscoszmnzmnjkdjkd mnmn mnmn PxyAeBe ab (D.1) and the solution for particle velocity in the d-direction as 0djPjP U dckd (D.2) The derivative is given by ,, ,,,, ,coscos coscoszmnzmn zmnzmnjkdjkd zmnmnzmnmn mn jkdjkd zmnmnmn mnPmn xyjkAejkBe dab mn jkxyAeBe ab (D.3) Thus, ,,, 01 coscoszmnzmnjkdjkd dzmnmnmn mnmn UkxyAeBe ckab (D.4) The acoustic intensity in the d-direction in the frequency domain is given by *1 Re 2dd I PU (D.5) Making the substitutions and simplifying,

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216 ,, ,,* 01 Recoscos 2 1 coscoszmnzmn zmnzmnjkdjkd dmnmn mn jkdjkd zmnmnmn mnmn IxyAeBe ab mn kxyAeBe ckab (D.6) ,, ,,0 * ,1 Recoscos 2 coscoszmnzmn zmnzmnjkdjkd d mnmn mn jkdjkd zmnmnmn mnmn IxyAeBe ckab mn kxyAeBe ab (D.7) ,, ,,* 0 *1 Recoscoscos 2 coszqrzqr zmnzmndzqr mnqr jkdjkd jkdjkd mnmnqrqrmqn I kxxy ckaab r yAeBeAeBe b (D.8) Now, integrate over the cro ss-section area to find the total sound power. Thus, dd SWfIdS (D.9) ,, ,,* 00 0 *1 Recoscos 2 coscoszqrzqr zmnzmnab dzqr xy mnqr jkdjkd jkdjkd mnmnqrqrmq Wfkxx ckaa nr yyAeBeAeBedydx bb (D.10) ,, ,,* 0 0 01 Recoscos 2 coscoszqrzqr zmnzmna dzqr x mnqr b jkdjkd jkdjkd mnmnqrqr ymq Wfkxxdx ckaa nr yydyAeBeAeBe bb (D.11) Using the orthogonal condition, the two integrals are only nonzero when mq and nr Thus the quadruple summation simplifies to a double summation as ,, ,,* 0 *1 Re 222zmnzmn zmnzmnjkdjkd dzmnmnmn mn jkdjkd mnmnab WfkAeBe ck AeBe (D.12)

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217 or ,, ,,* 0 *Re 8zmnzmn zmnzmnjkdjkd dzmnmnmn mn jkdjkd mnmnab WfkAeBe ck AeBe (D.13) Continuing to simplify ,, ,,* 0 **Re 8zmnzmn zmnzmnjkdjkd dzmnmnmn mn jkdjkd mnmnab WfkAeBe ck AeBe (D.14) ,,,, ,,,,** 0 **Re 8zmnzmnzmnzmn zmnzmnzmnzmnjkdjkdjkdjkd dzmnmnmnmnmn mn jkdjkdjkdjkd mnmnmnmnab WfkAeAeBeBe ck AeBeAeBe ,(D.15) ,, ,,,,22 0 **Re 8zmnzmn zmnzmnzmnzmnjkdjkd dzmnmnmn mn jkdjkdjkdjkd mnmnmnmnab WfkAeBe ck AeBeAeBe (D.16) ,, ,,,,22 0 **Re 8zmnzmn zmnzmnzmnzmnjkdjkd dzmnmnmn mn jkdjkdjkdjkd mnmnmnmnab WfkAeBe ck AeBeAeBe (D.17) ,,,,22 0 **Re 8zmnzmnzmnzmndzmnmnmn mn jkdjkdjkdjkd mnmnmnmnab WfkAB ck AeBeAeBe .. (D.18) At this point, an assumption regarding th e wavenumber is needed. This derivation progresses under the assumption of isentropic flow, thus the wavenumber is real-valued and Eq. (D.14) can be simplified further. First, consider the following case:

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218 ** ** 121211221122 1212122112121221 1212122112121221 12212 zzzzxjyxjyxjyxjy x xyyjxyjxyxxyyjxyjxy xxyyjxyjxyxxyyjxyjxy jxyxy (D.19) which is purely imaginary. This is of the same form as the last two terms in the summation in Eq. (D.14). Thus, taking the re al part of those terms results in zero contribution to the sum and Eq. (D.14) is simplified to 22 08dzmnmnmn mnab WfkAB ck, (D.20) which can easily be broken into the incident and reflected components as 2 08izmnmn mnab WfkA ck (D.21) and 2 08rzmnmn mnab WfkB ck. (D.22) To satisfy the first law of thermodynamics, the inequality, riWW must hold.

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219 APPENDIX E MODAL DECOMPOSITION METHOD NUMERICAL ERROR STUDY The main sources of error for the MDM are the signal-to-noise ratio, microphone phase mismatching, and uncertainties in the measurement of the microphone locations and the temperature. The frequency scaling of the uncertainty in the MDM is also important, as the goal of the MDM is to extend the frequency range of acoustic impedance testing. To this end, numerical studies are made of the effects of the individual error sources and the frequency scal ing of the total error. These studies are done for the waveguide described in the pape r and for a sound-hard termination modeled by a reflection coefficient matrix 20.990.010.010.01 0.010.950.010.01 0.010.010.950.01 0.020.010.010.90trueje Ro. (E.1) Along with the chosen value of trueR, four different vectors of incident complex modal amplitudes are chosen as 77.47.2 12 21.650.4 341212 0.51.6 ,, 0.50.5 0.50.5 1212 0.50.5 1.60.5 0.51.6jj truetrue jj truetrueAeAe AeAe oo oo, (E.2)

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220 with all amplitudes given in Pascals. The reflected modal amplitudes are computed using Equation (4.9). The data are then used to calculate time-series da ta based on Equations (2.2) and (2.5). The time-series data are then processed using the MDM. The power contained in each error-free and noise-free si gnal is given in Table E-1. The ambient temperature used for the error studies is 27. 7 C, thus the cut-on frequencies for the higher-order modes are 6.83 kHz for the (1,0) and (0,1) modes and 9.66 kHz for the (1,1) mode. Table E-1: Power in Pa2 for all signals from all simulation sources. Signal Source 1 Source 2 Source 3 Source 4 111,, x yd 158.5 143.0 136.8 146.2 221,, x yd 198.1 224.1 181.1 212.5 331,, x yd 200.4 218.8 226.6 187.0 441,, x yd 190.6 167.1 208.4 204.6 112,, x yd 262.1 270.9 274.5 232.6 222,, x yd 285.4 273.3 294.8 318.6 332,, x yd 241.2 233.1 229.7 212.9 442,, x yd 289.1 302.5 280.7 322.5

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221 The root-mean-square (rms) normalized er ror between the true values of the complex modal amplitude and the calculated values and between elements of the calculated reflection coefficient matrix and trueR is used as the test metric to gauge the success of the MDM. The rms normalized erro r is defined separately for the magnitude and phase as 22 001calctruecalctrue MN mnmnmnmn m truetrue mn mnmnAABB AB (E.3) and 22 001calctruecalctrue MN mnmnmnmn p truetrue mn mnmnAABB AB (E.4) for the complex modal coefficients and as 2 2 001calctrue mnmn true mn mnRR R R (E.5) and 2 2 001calctrue mnmn true mn mnRR R R, (E.6) for the reflection coefficient matrix. The num erical studies are perf ormed at a frequency of 12 kHz to avoid pressure nodes for the micr ophone locations listed in Section 3.4.1 of this dissertation. For the error-free/noise-f ree simulation, the rms normalized error in the modal coefficients is on the order of 1410 for the magnitude and 1310 for the phase, and for the reflection coefficient matrix the rms normalized error is on the order of 1310 for

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222 the magnitude and 1110 for the phase. The results offer a baseline for comparison to the studies with perturbations introduced to model the error sources. E.1 Signal-to-Noise Ratio To test the sensitivity of the MDM to noi se in the measurement signals, white noise is added to the simulated data signals. The power in the noise component is varied from 410 to 1 2Pato yield a range of signal-to-noise ra tios (SNR) representative of realistic values. The results for the rms normalized e rror for the modal coefficients are shown in Figure E-1, and Figure E-2 show s the results for the reflection coefficient matrix. To better model an actual experiment, averaging of the measured complex acoustic pressure is added to the routine. For this si mulation, the noise power is fixed at 2210 Pa and the number of averages is varied from no averag es to 10,000 averages. The results in Figure E-3 for the complex modal amplitudes and in Figure E-4 for the reflection coefficient matrix show that the rms normalized error can be significantly reduc ed with the use of large numbers of averages. E.2 Microphone Phase Mismatch The phase mismatch between the micr ophones is studied by introducing a phase offset into one of the microphone signals and then processing the data. The measurement location chosen is the microphone located at 442,, x yd. The phase offset applied is varied from 0.01 to 10 degrees. The results for the complex modal coefficients are given in Figure E-5, and Figure E-6 shows the result s for the reflection coefficient matrix. The results show that a phase offset of order uni ty increases the error in the phase of the reflection coefficient, but reliable estimates of the complex modal amplitudes and the reflection coefficient can be obtained within an order of magnitude of 010. If the

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223 application of the data requires higher accuracy, the microphones should be phase calibrated before use. 10-4 10-3 10-2 10-1 100 10-5 10-4 10-4 10-3 10-2 10-1 Noise Power [Pa2]Magnitude 10-4 10-3 10-2 10-1 100 10-5 10-4 10-4 10-3 10-2 10-1 Noise Power [Pa2]Phase Figure E-1: The rms normalized error for the modal coefficients versus noise power added to the signals. Source 1, Source 2, Source 3, Source 4.

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224 10-4 10-3 10-2 10-1 100 10-2 10-1 100 101 Noise Power [Pa2]Magnitude 10-4 10-3 10-2 10-1 100 10-1 100 101 102 Noise Power [Pa2]Phase Figure E-2: The rms normalized error for the reflection coefficient matrix versus noise power. 100 101 102 103 104 10-6 10-5 10-4 10-3 10-2 # of AveragesMagnitude 100 101 102 103 104 10-6 10-5 10-4 10-3 10-2 # of AveragesPhase Figure E-3: The rms normalized error versus th e number of averages for a noise power of 0.01 Pa2. Source 1, Source 2, Source 3, Source 4.

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225 100 101 102 103 104 10-3 10-2 10-1 # of AveragesMagnitude 100 101 102 103 104 10-2 10-1 100 101 # of AveragesPhase Figure E-4: The rms normalized error for the reflection coefficient versus the number of averages for a noise power of 0.01 Pa2. 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 Phase Offset [deg]Magnitude 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 101 102 Phase Offset [deg]Phase Figure E-5: The rms normalized error for the modal coefficients versus a phase error applied to microphone 4 in group 2 for each source. Source 1, Source 2, Source 3, Source 4.

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226 10-2 10-1 100 101 10-3 10-2 10-1 100 101 Phase Offset [deg]Magnitude 10-2 10-1 100 101 10-3 10-2 10-1 100 101 102 Phase Offset [deg]Phase Figure E-6: The rms normalized error for the re flection coefficient matrix versus a phase error applied to microphone 4 in group 2 for all sources. E.3 Microphone Locations Error in the location of the microphone m easurements is studied by applying an error to one of the microphone locations befo re data reduction with the MDM. The applied location error is varied from 0. 001 to 1 mm and is applied to the microphone located at 111,, x yd. The results for the rms normalized error are show in Figure E-7 for the complex modal amplitudes and in Figure E8 for the reflection coefficient matrix. The error in the phase of both the complex modal amplitudes and the reflection coefficient matrix is larger than in the magni tude. Physically, this is realistic, as a location error can also be m odeled as a phase error for pr opagating waves. Combined with studies of the uncertainty in the T MM, this error source for the MDM will be a strong function of frequency.

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227 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 Microphone Location Error [mm]Magnitude 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100 Microphone Location Error [mm]Phase Figure E-7: The rms normalized error for th e modal coefficients versus a microphone location error applied to microphone 1 in group 1 for each source. Source 1, Source 2, Source 3, Source 4. 10-3 10-2 10-1 100 10-4 10-3 10-2 10-1 100 101 Microphone Location Error [mm]Magnitude 10-3 10-2 10-1 100 10-2 10-1 100 101 102 Microphone Location Error [mm]Phase Figure E-8: The rms normalized error for th e reflection coefficient matrix versus a microphone location error applied to mi crophone 1 in group 1 for all sources.

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228 E.4 Speed of Sound The last error source studied individually is errors in the estimate for the speed of sound. Under the assumption of lossless acousti cs with an ideal gas, the speed of sound is solely related to the temperature vi a the isentropic speed of sound equation 0 g ascRT (E.7) where is the ratio of specific heats, g as R is the ideal gas constant, and T is the absolute temperature. With these assumptions and this equation, studying the er ror in the speed of sound is reduced to studying the error in the measurement of temperature. This is done by applying a perturbation to temperature be fore processing with the MDM. The perturbation is varied from 0.01 to 10 K a nd the results are given in Figure E-9 for the complex modal amplitudes and in Figure E-10 for the reflection coefficient matrix. The 10-2 10-1 100 101 10-5 10-4 10-3 10-2 10-1 100 Temperature Error [K]Magnitude 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 101 Temperature Error [K]Phase Figure E-9: The rms normalized error for the modal coefficients versus a temperature error. Source 1, Source 2, Source 3, Source 4.

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229 10-2 10-1 100 101 10-5 10-4 10-3 10-2 10-1 Temperature Error [K]Magnitude 10-2 10-1 100 101 10-3 10-2 10-1 100 101 102 Temperature Error [K]Phase Figure E-10: The rms normalized error for th e reflection coefficient matrix versus a temperature error. E.5 Frequency Knowledge of the frequency scaling of th e error sources is desirable, since the MDM is not applied only to a single frequenc y. For this study, the error in the input measurements is held constant and the simulation is swept across the frequency range of 1 to 12 kHz in steps of 1 kHz. The errors used in this study are designed to resemble uncertainty levels to which an actual experi mental measurement may be subject. The noise power added to the signals is 2210 Pa, processed with 1,000 averages. The applied phase offset is 0.1 degree, the applied lo cation error to a single microphone location is 0.01 mm, and the applied temper ature error is 1 K. The re sults are shown in Figure E-11 for the complex modal amplitudes and in Fi gure E-12 for the reflection coefficient

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230 1 2 3 4 5 6 7 8 9 10 11 12 13 0 0.01 0.02 0.03 Freq [kHz]Magnitude 1 2 3 4 5 6 7 8 9 10 11 12 13 0 0.2 0.4 0.6 0.8 Freq [kHz]Phase Figure E-11: The rms normalized error for th e modal coefficients versus frequency. Source 1, Source 2, Source 3, Source 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 0 0.05 0.1 0.15 0.2 Freq [kHz]Magnitude 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 Freq [kHz]Phase Figure E-12: The rms normalized error for the reflection coefficient matrix versus frequency.

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231 matrix. The results for the frequency ranges of 1 to 6 kHz only have plane wave propagation, whereas higher-order modes are pr esent in the remaining frequency ranges. The results show that the rms normalized error increases dramatically in the vicinity of a cut-on frequency. E.6 Conclusions The results for the error st udy presented here show that the MDM has the potential to provide accurate and reli able results for acoustic impedance measurements, except near the cut-on frequencies. The actual expe rimental uncertainty values will still be needed for experimental data to concisely de fine the accuracy of any measurements made with this technique and to evaluate it compared to the requirements of the application. The accuracy of the results from the MDM could be improved upon by increasing the accuracy of the measurement instruments. The largest improvements to the accuracy can be had from improvements in the measur ements of the microphone locations and temperature and from improvements in th e phase matching between the microphones. The temperature measurement can be improved by calibrating the measurement device before use to achieve uncertainties on the order of 0.1 K, the microphone location measurements can be improved by using a high resolution measurement system to measure the locations, and the phase matc hing of the microphones can be improved by phase calibrating before use. Since this er ror study was performed with simulated timeseries data, the interesting behavior of the rm s error near the cut-on frequencies is due to the data reduction algorithm. Further studies are necessary to determine the exact cause of the rise in the rms error.

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232

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233 APPENDIX F AUXILIARY GRAPHS This appendix presents aux iliary data graphs from the experimental measurements of the different specimens. F.1 CT65 1 2 3 4 5 6 0.998 0.9985 0.999 0.9995 1 Freq [kHz]Coherence Orginal Position Switched Position Figure F-1: Ordinary cohe rence function for the TMM measurement of CT65.

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234 1 2 3 4 5 6 0 1 2 3 |H3|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Figure F-2: The measured FRF for CT65 for the TMM. Averaged, Original, and Switched. 2 4 6 8 10 12 14 16 18 20 0.9999 0.99995 1 Freq [kHz]Coherence Original Position Switched Position Figure F-3: Ordinary cohere nce function for the high frequency TMM measurement of CT65.

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235 2 4 6 8 10 12 14 16 18 20 0 1 2 3 |H3|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -360 -270 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Figure F-4: The measured FRF for CT65 for the high frequency TMM. Averaged, Original, and Switched. F.2 CT73 1 2 3 4 5 6 0.98 0.985 0.99 0.995 1 Freq [kHz]Coherence Original Position Switched Position Figure F-5: Ordinary cohe rence function for the TMM measurement of CT73.

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236 1 2 3 4 5 6 0 2 4 6 |H3|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Figure F-6: The measured FRF for CT73 for the TMM. Averaged, Original, and Switched. 0 2 4 6 8 10 12 14 16 18 20 0.9999 0.99995 1 Freq [kHz]Coherence Original Position Switched Position Figure F-7: Ordinary cohere nce function for the high frequency TMM measurement of CT73.

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237 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 |H3|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -360 -270 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Figure F-8: The measured FRF for CT73 for the high frequency TMM. Averaged, Original, and Switched. F.3 Rigid Termination 1 2 3 4 5 6 0.4 0.5 0.6 0.7 0.8 0.9 1 Freq [kHz]Coherence Original Position Switched Position Figure F-9: Ordinary coherence functi on between the two microphones for the TMM measurement of the rigid termination.

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238 1 2 3 4 5 6 0 50 100 150 |H3|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Averaged Original Switched Figure F-10: The measured FRF for the rigid termination for the TMM. 2 4 6 8 10 12 14 16 18 20 0.9999 0.99995 1 Freq [kHz]Coherence Original Position Switched Position Figure F-11: Ordinary coherence function fo r the high frequency TMM measurement of the rigid termination.

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239 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 |H3|Freq [kHz] 2 4 6 8 10 12 14 16 18 20 -360 -270 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Averaged Original Switched Figure F-12: The measured FRF for the rigid termination for the high frequency TMM. F.4 SDOF Liner 1 2 3 4 5 6 0.8 0.85 0.9 0.95 1 Freq [kHz]Coherence Original Position Switched Position Figure F-13: Ordinary coherence function for the TMM measurement of the SDOF specimen.

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240 1 2 3 4 5 6 0 5 10 15 20 |H3|Freq [kHz] 1 2 3 4 5 6 -180 -90 0 90 180 arg(H3) [deg]Freq [kHz] Figure F-14: The measured FRF for the SDOF specimen for the TMM.

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241 APPENDIX G COMPUTER CODES G.1 TMM Program Files G.1.1 TMM Program Readme File The file format for the input files for tmm v5.m are given in this document. Both of the files are to be tab delimited text files. Note the convention that microphone 1 is the microphone farther from the test specimen a nd microphone 2 is the microphone closer to the test specimen. The first file is to c ontain the data on the e xperimental setup (from LabVIEW) as such: tube number (0 = 1 inch PWT, 1 = 8.5 mm PWT) # of effective averages mic location 2 [m] (closer to the specimen) uncertainty in mic location 2 number of measurements of l mic spacing [m] uncertainty in mic spacing number of measurements of s Temp [C] uncertainty in Temp [C] number of measurements of T press [kPa] uncertainty in press [kPa] number of measurements of P The second file is to contain the measured data. Each entry should be a column array with each column containing the foll owing information in the given order (from Pulse). The uncertainty analysis used in the tmmv4.m code assumes that there is no biased error in the measurem ent of the frequency responsefunction. This is done by

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242 using a periodic or pseudo random signal for th e source. Gmm and Gnn are estimates of the noise spectrum from micr ophone 1 and 2 respectively. freq[Hz] Gxxo Gyyo Cxyo Qxyo Re[TFo] Im[TFo] coho Gxxs Gyys Cxys Qxys Re[TFs] Im[TFs] cohs Gmm Gnn SPLref[dBspl] G.1.2 Pulse to MATLAB Conversion Program % pulsereadv6.m % % Todd Schultz 9/2/2005 % % % This program is designed to call the BKfiles.m script generated from the % B&K Pulse system to read in the data files and then to assign useful % variable names to the required data from the tmmv6.m script file. Then % the data is saved to the current directory in a mat file formate for use % by the tmmv6 script. % % Inputs % This file must be run in the same folder as the BKfiles.m and the text data files. % Must pull of the frequency array, frequency response for both the original and switched and % the coherence for both the original and switched and the reference sound pressure level. % % Outputs % f = frequency array [Hz] % w = angular frequency array [rad/s] % navg = effective number of spectral averages % Gxxo = power spectrum of mic 1 % Gyyo = power spectrum of mic 2 % Cxyo = co-spectrum % Qxyo = quad-spectrum % Gxyo = Cxyo+j*Qxyo; % h12o = frequency response function with the mics in the original position % coho = coherence with the mics in the original position % Gxxs = power spectrum of mic 1 % Gyys = power spectrum of mic 2 % Cxys = co-spectrum % Qxys = quad-spectrum % Gxys = Cxys+j*Qxys; % h12s = frequency response function with the mics in the switched position % cohs = coherence with the mics in the switched position % Gmm = noise power spectrum of mic 1 % Gnn = noise power spectrum of mic 2 % splref = SPL at the reference mic [dB re 20*10^-6 Pa]

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243 %% Program %% call BKfiles.m to retrieve the data BKFiles %pause(1) %% extract the required data splrefo = Group3.Function3.DatasetSection.Data.'; splrefs = Group4.Function3.DatasetSection.Data.'; splref = (splrefo + splrefs)/2; navg = Group3.Function1.SpecialSection.Tags.AverageNumber; f = Group3.Function1.DatasetSection.X_axis; w = 2*pi*f; Gxxo = Group3.Function1.DatasetSection.Data.'; Gyyo = Group3.Function2.DatasetSection.Data.'; Gxyo = Group3.Function5.DatasetSection.Data.'; Cxyo = real(Gxyo); Qxyo = imag(Gxyo); coho = Group3.Function4.DatasetSection.Data.'; h12o = Group3.Function6.DatasetSection.Data.'; Gxxs = Group4.Function1.DatasetSection.Data.'; Gyys = Group4.Function2.DatasetSection.Data.'; Gxys = Group4.Function5.DatasetSection.Data.'; Cxys = real(Gxys); Qxys = imag(Gxys); cohs = Group4.Function4.DatasetSection.Data.'; h12s = Group4.Function6.DatasetSection.Data.'; Gmm = Group8.Function1.DatasetSection.Data.'; Gnn = Group8.Function2.DatasetSection.Data.'; clear BKFilenames BKIndex Group3 Group4 Group5 Group6 Group7 Group8 %% Resize the data arrays I = find(f>=300 & f<=20000); f = f(I); w = w(I); Gxxo = Gxxo(I); Gyyo = Gyyo(I); Gxyo = Gxyo(I); Cxyo = Cxyo(I); Qxyo = Qxyo(I); coho = coho(I); h12o = h12o(I); Gxxs = Gxxs(I); Gyys = Gyys(I); Gxys = Gxys(I); Cxys = Cxys(I); Qxys = Qxys(I); cohs = cohs(I); h12s = h12s(I); Gmm = Gmm(I); Gnn = Gnn(I); splref = splref(I); %% save data to mat file for convience later save pulsedata f w Gxxo Gyyo Gxyo Cxyo Q xyo coho h12o Gxxs Gyys Gxys Cxys Qxys cohs h12s Gmm Gnn splref

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244 G.1.3 TMM Main Program %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% %% tmmv6.m % % Two Microphone Method % Todd Schultz 9/2/2005 % v6 4/27/2006 *updated uncertainty calculation for magnitude % and phase of R for the Monte Carlo method % This file imports the data from the correct sources and preforms the % calculations to estimate the reflection cofficient and normalized acoustic % impedance and the corresponding uncertainties. This code assumes that % there is no bais error in the measurement of the frequency response % function. This is done by using a periodic random signal as the source. % % Assumptions: 1. exp(jwt) sign convention % 2. no mean flow % 3. mic 1 is farther from the sample % 4. mic 2 is closer to the sample % 5. no bias error for H12 % % trig = logical trigger for dispersion and attentuation % 0 = off (analytical method used for the uncertainty of r) % 1 = on (attentuation and dispersion are not accounted for % in the uncertainty analysis which still uses the % analytical method) % trig_uncert = logical trigger for the uncertianty method % 0 = linear multivariate method (Doesn't account for % attentuation and dispersion.) % 1 = Monte Carlo simulation assuming normal output % distribution % 2 = Monte Carlo simulation with arbitrary output % distribution % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% clear all ; close all ; pack; %% Input file names disp( 'Did you run pulsereadv6 or is the data in the proper text file format?' ) disp( '0 = pulsereadv6' ) disp( '1 = text file' ) trig_pulse = input( 'Choose now.\n' ); fname1 = input( 'Input the file name for the experimental setup information file.\n' 's' ); if trig_pulse == 1 fname2 = input( 'Input the file name for the data file.\n' 's' );

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245 end fname3 = input( 'Input the file name for the output file (no extension).\n' 's' ); pic = input( 'Press 0 if you would like the figures saved or press 1 otherwise.\n' ); disp( 'Input whether dispersion and dissipation should be accounted for.' ) disp( '0 = NO dispersion and dissipation' ) disp( '1 = Dispersion and dissipation included' ) trig = input( 'Choose now.\n' ); disp( 'Input the method for the uncertainty estimates.' ) disp( '0 = Linear multivariate method (Doesn''t included dispersion and dissipation)' ) disp( '1 = Monte Carlo simulation assuming normal output distribution' ) disp( '2 = Monte Carlo simulation assuming arbitrary output distribution' ) trig_uncert = input( 'Choose now.\n' ); disp( 'Thank you. Your answers are being computed now.' ) % trig_pulse = 1 % fname1 = 'inputjsv1.txt' % fname2 = 'ideal40db.txt' % fname3 = 'ideal40' % pic = 0 % trig = 0 % trig_uncert = 2 iter = 25000; % number of iterations %% Read in experimental setup information A = dlmread(fname1, '\t' ); % Read in set up file tube = A(1); % Tube number navg = A(2); % Number of averages used l = A(3); % mic location closer to the specimen [m] Ul = A(4); % standard Uncertainty in d [m] nl = A(5); % # of measurements of l s = A(6); % mic spacing [m] Us = A(7); % standard Uncertainty in s [m] ns = A(8); % # of measurements of s tatm = A(9)+273.15; % Atmospheric temperature [K] (input units are [C]) Utatm = A(10); % standard Uncertainty in Tatm [K] or [C] nT = A(11); % # of temperature measurements patm = A(12)*1000; % Atmospheric pressure [Pa] (input units are [kPa]) Upatm = A(13)*1000; % standard Uncertainty in Patm [Pa] (input units are [kPa]) nP = A(14); % # of pressure measurements clear A ; switch trig_pulse case 0 % Read pulse data file load pulsedata % Define the cross-spectrums

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246 Gxyo = Cxyo+j*Qxyo; Gxys = Cxys+j*Qxys; case 1 % Read text data file A = dlmread(fname2, '\t' ); f = A(:,1); % Frequency array [Hz] w = 2*pi*f; % Angular frequency array [rad/s] Gxxo = A(:,2); % Mic1 power spectrum [Pa^2] Gyyo = A(:,3); % Mic2 power spectrum [Pa^2] Cxyo = A(:,4); % Co-spectrum [Pa^2] Qxyo = A(:,5); % Quad-spectrum [Pa^2] Gxyo = Cxyo+j*Qxyo; h12o = A(:,6)+j*A(:,7); % Frequency Responce function Original coho = A(:,8); % Coherence Original Gxxs = A(:,9); % Mic1 power spectrum [Pa^2] Gyys = A(:,10); % Mic2 power spectrum [Pa^2] Cxys = A(:,11); % Co-spectrum [Pa^2] Qxys = A(:,12); % Quad-spectrum [Pa^2] Gxys = Cxys+j*Qxys; h12s = A(:,13)+j*A(:,14); % Frequency Responce function switched cohs = A(:,15); % Coherence switched Gmm = A(:,16); % Estimate of the noise spectrum for mic1 [Pa^2] Gnn = A(:,17); % Estimate of the noise spectrum for mic2 [Pa^2] splref = A(:,18); % SPL at reference mic [dBspl] clear A ; end tic %% Constants % Cross-section width, l [m] switch tube case 0 dtube = 0.02544; % 1 in square tube Udtude = 0.00003; % total uncertainty in tube dimension case 1 dtube = 0.0085; % 8.5 x 8.5 mm tube Udtube = 0.001; % total uncertainty in tube dimension case 2 dtube = 4*0.0127; % Mike Jones' PWT end %% Compute the uncertainty of h12o SHo = zeros([2 2 length(f)]); for ii = 1:length(f) SHo(:,:,ii) = H3uncert(Gxxo(ii),Gyyo(ii),Gxyo(ii),Gmm(ii),Gnn(ii),h12o(ii),navg); end

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247 %% Compute the uncertainty of h12s SHs = zeros([2 2 length(f)]); for ii = 1:length(f) SHs(:,:,ii) = H3uncert(Gxxs(ii),Gyys(ii),Gxys(ii),Gmm(ii),Gnn(ii),h12s(ii),navg); end %% Calculate averaged frequency response function h12m = sqrt(abs(h12o)./abs(h12s)); phi = 0.5*(unwrap(angle(h12o))-unwrap(angle(h12s))); h12 = h12m.*exp(i.*phi); % h12 will be passed to the subroutine tmm_sub.m % Calculate the standard uncertainty in the frequency response function SH = zeros([2,2,length(f)]); for ii = 1:length(f) SH(:,:,ii) = have_div(h12o(ii),SHo(:,:,ii),h12s(ii),SHs(:,:,ii)); end %% Call tmm_sub.m to carry out the caclutations [r,z,k,SWR] = tmm_subv6(w,l,s,h12,tatm,patm,dtube,trig); % Save the outputdata save outputdata f k r z %% Uncertainty Computations switch trig_uncert case 0 % Linear multivariate uncertianty analysis % Initialize variables for the loops SR = zeros([2 2 length(f)]); SZ = zeros([2 2 length(f)]); % Call the tmm_rv4 and tmm_zv5 to compute the uncertainties for ii = 1:length(f) SR(:,:,ii) = tmm_rv5(w(ii),k(ii),l,s,h12(ii),r(ii),Ul,Us,Utatm,SH(:,:,ii),trig); SZ(:,:,ii) = tmm_zv5(r(ii),SR(:,:,ii),trig); end % Estimate the coverage factor kcf p = 2; % # of variates % First estimate the effective # of degrees of freedom nu = zeros(length(f),1); nux = [navg nl ns nT]-1; Snew = zeros([p p length(nux)]); for ii = 1:length(f) Snew(:,:,1) = SH(:,:,ii); Snew(:,:,2) = [Ul^2 0; 0 0]; Snew(:,:,3) = [Us^2 0; 0 0]; Snew(:,:,4) = [Utatm^2 0; 0 0]; nu(ii) = nu_eff(Snew,nux); if nu(ii) == 0 disp( 'Try again!' ) break end end kcf = sqrt((nu*p./(nu+1-p)).*finv(0.95,p,nu+1-p));

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248 % Estimate the uncertainty bounds on the real and imaginary parts of % the reflection coefficient and normalized acoustic impedance URr = kcf.*sqrt(squeeze(SR(1,1,:))); URi = kcf.*sqrt(squeeze(SR(2,2,:))); UZr = kcf.*sqrt(squeeze(SZ(1,1,:))); UZi = kcf.*sqrt(squeeze(SZ(2,2,:))); % Convert from rectangular form to polar form SRpolar = rect_to_polar(r,SR); % Estimate the uncertainty bounds on the polar form of % the reflection coefficient URm = kcf.*sqrt(squeeze(SRpolar(1,1,:))); URp = kcf.*sqrt(squeeze(SRpolar(2,2,:))); % Save the multivariate uncertainty data save mv_uncertainty URr URi URm URp UZr UZi SR SRpolar SZ kcf case 1 % Monte Carlo simulation assuming a normal output distribution %iter = 25000; % number of iterations tmm_mcv6 % Performs Monte Carlo simulations and saves the data % compute the magnitude and phase % compute the coverage factor nu = iter-1; p = 2; kcf = sqrt((nu*p./(nu+1-p)).*finv(0.95,p,nu+1-p)); % compute the covariance of the magnitude and phase for each freqnency % compute the covariance of the real and imaginar parts of Z for each % freqnency SR = zeros([2,2,length(f)]); SRpolar = zeros([2,2,length(f)]); SZ = zeros([2,2,length(f)]); for ii = 1:length(f) SR(:,:,ii) = cov(real(r_data(:,ii)),imag(r_data(:,ii))); SRpolar(:,:,ii) = cov(abs(r_data(:,ii)),angle(r_data(:,ii))); SZ(:,:,ii) = cov(real(z_data(:,ii)),imag(z_data(:,ii))); end % Estimate the uncertainty bounds on the real and imaginary parts of % the reflection coefficient and normalized acoustic impedance URr = kcf.*sqrt(squeeze(SR(1,1,:))); URi = kcf.*sqrt(squeeze(SR(2,2,:))); URm = kcf.*sqrt(squeeze(SRpolar(1,1,:)));

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249 URp = kcf.*sqrt(squeeze(SRpolar(2,2,:))); UZr = kcf.*sqrt(squeeze(SZ(1,1,:))); UZi = kcf.*sqrt(squeeze(SZ(2,2,:))); % Save the Monte Carlo uncertainty data save mcnormal_uncertainty URr URi URm URp UZr UZi SR SRpolar SZ kcf case 2 % Monte Carlo simulaton assuming an arbitrary distribution % For this case, the uncertainty is given as a lower and upper % limit range. This is done aviod an assumption regarding the % symmetry of the distribution. %iter = 25000; % number of iterations cilevel = 0.95; % desired level for the CI's bins = 40; % number of bins to use to estimate the pdf cbins = 100; % number of contours to use tmm_mcv6 % Performs Monte Carlo simulations and saves the data % Initialize variables URr = zeros(length(f),2); URi = zeros(length(f),2); URm = zeros(length(f),2); URp = zeros(length(f),2); UZr = zeros(length(f),2); UZi = zeros(length(f),2); probrri = zeros(length(f),1); probrmp = zeros(length(f),1); probz = zeros(length(f),1); xrri = cell(length(f),1); % xrmp = cell(length(f),1); % v6 change xz = cell(length(f),1); for ii = 1:length(f) % compute the Confidence intervals [probrri(ii),x] = numericCI([real(r_data(:,ii)) imag(r_data(:,ii))],cilevel,bins,cbins); xrri{ii} = x; URr(ii,1) = min(x(:,1)); URr(ii,2) = max(x(:,1)); URi(ii,1) = min(x(:,2)); URi(ii,2) = max(x(:,2)); % v6 changes here % Old way of mag/phase uncertainty (not consistant with with % how I quote the nominal value. % [probrmp(ii),x] = numericCI([abs(r_data(:,ii)) angle(r_data(:,ii))],cilevel,bins,cbins); % xrmp{ii} = x; % URm(ii,1) = min(x(:,1)); URm(ii,2) = max(x(:,1)); % URp(ii,1) = min(x(:,2)); URp(ii,2) = max(x(:,2)); % New way

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250 xamp = sqrt(x(:,1).^2+x(:,2).^2); xphase = atan2(x(:,2),x(:,1)); URm(ii,1) = min(xamp); URm(ii,2) = max(xamp); URp(ii,1) = min(xphase); URp(ii,2) = max(xphase); [probz(ii),x] = numericCI([real(z_data(:,ii)) imag(z_data(:,ii))],cilevel,bins,cbins); xz{ii} = x; UZr(ii,1) = min(x(:,1)); UZr(ii,2) = max(x(:,1)); UZi(ii,1) = min(x(:,2)); UZi(ii,2) = max(x(:,2)); end % Save the Monte Carlo uncertainty data save mcarbitrary_uncertainty URr URi URm URp UZr UZi probrri probrmp probz xrri xz end %% Write the data to text file fname_out = [fname3 '.out' ]; fid=fopen(fname_out, 'w' ); switch trig_uncert case {0,1} fprintf(fid, 'F[Hz] \t SPLref \t SWR[dB] \t Rmag \t +/-Rmag \t Rpha[deg] \t +/-Rpha[deg] \t Resist \t +/-Resist \t React \t +/-React \r' ); fprintf(fid, '%7.2f\t %5.1f\t %5.1f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\r' ... [f';splref';abs(SWR)';abs(r)';URm';(180*angle(r)/pi)';(180*URp/pi)';rea l(z)';UZr';imag(z)';UZi']); case 2 fprintf(fid, 'F[Hz] \t SPLref \t SWR[dB] \t Rmag \t lowerRmag \t upperRmag \t Rpha[deg] \t lowerRpha[deg] \t upperRpha[deg] \t Resist \t lowerResist \t upperResist \t React \t lowerReact \t upperReact \r' ); fprintf(fid, '%7.2f\t %5.1f\t %5.1f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\t %9.5f\r' ... [f';splref';abs(SWR)';abs(r)';URm(:,1)';URm(:,2)';(angle(r)*180/pi)';(U Rp(:,1)*180/pi)';(URp(:,2)*180/pi)';real(z)';UZr(:,1)';UZr(:,2)';imag(z )';UZi(:,1)';UZi(:,2)']); end fclose(fid); toc %% Plots % Plot the coherence for the two TF figure (1) set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) plot(f/1000,coho,f/1000,cohs) xlim([f(1) f(end)]/1000)

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251 xlabel( 'Freq [kHz]' ); ylabel( 'Coherence' ); title( 'Coherence of the Two Measured Transfer Functions' ) legend( 'Original Position' 'Switched Position' ) %axis tight %grid on % Plot the two frequency response functions figure(2) set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f/1000,h12m,f/1000,abs(h12o),f/1000,abs(h12s)) subplot(2,1,2); plot(f/1000,180.*phi./pi,f/1000,180.*angle(h12o)./pi,f/1000,180.*angle( h12s)./pi) subplot(h1); xlim([f(1) f(end)]/1000) ylabel( '|H_3|' ); xlabel( 'Freq [kHz]' ); title( 'Frequency Response Function' ) legend( 'Averaged' 'Original' 'Switched' ) %axis tight %grid on subplot(h2); xlim([f(1) f(end)]/1000) ylabel( 'arg(H_3) [deg]' ); xlabel( 'Freq [kHz]' ); %axis tight %grid on % Plot the reflection coefficient figure (3) set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f/1000,abs(r)) subplot(2,1,2); plot(f/1000,180.*angle(r)./pi) subplot(h1); xlim([f(1) f(end)]/1000) ylabel( '|R|' ); xlabel( 'Freq [kHz]' ); title( 'Reflection Coefficient' ) %axis tight %grid on subplot(h2); xlim([f(1) f(end)]/1000) ylabel( '\phi [deg]' ); xlabel( 'Freq [kHz]' ); %axis tight %grid on % Plot the reflection coefficient with uncertainty figure (4)

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252 set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) h1=subplot(2,1,1); h2=subplot(2,1,2); switch trig_uncert case {0,1} subplot(2,1,1); errorbar(f/1000,abs(r),URm) subplot(2,1,2); errorbar(f/1000,180.*angle(r)./pi,180.*URp./pi) case 2 subplot(2,1,1); plot(f/1000,abs(r),f/1000,URm(:,1), 'r' ,f/1000,URm(:,2), 'r' ) subplot(2,1,2); plot(f/1000,180.*angle(r)./pi,f/1000,URp(:,1)*180./pi, 'r' ,f/1000,URp(:, 2)*180/pi, 'r' ) end subplot(h1); xlim([f(1) f(end)]/1000) ylabel( '|R|' ); xlabel( 'Freq [kHz]' ); title( 'Reflection Coefficient' ) %axis tight %grid on subplot(h2); xlim([f(1) f(end)]/1000) ylabel( '\phi [deg]' ); xlabel( 'Freq [kHz]' ); %axis tight %grid on % Plot the normalized impedance figure (5) set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f/1000,real(z)) subplot(2,1,2); plot(f/1000,imag(z)) subplot(h1); xlim([f(1) f(end)]/1000) ylabel( '\theta' ); xlabel( 'Freq [kHz]' ); title( 'Normalized Acoustic Impedance' ) %axis tight %grid on subplot(h2); xlim([f(1) f(end)]/1000) ylabel( '\chi' ); xlabel( 'Freq [kHz]' ); %axis tight %grid on % Plot the normalized impedance with uncertainty figure (6) set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0])

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253 h1=subplot(2,1,1); h2=subplot(2,1,2); switch trig_uncert case {0,1} subplot(2,1,1); errorbar(f/1000,real(z),UZr) subplot(2,1,2); errorbar(f/1000,imag(z),UZi) case 2 subplot(2,1,1); plot(f/1000,real(z),f/1000,UZr(:,1), 'r' ,f/1000,UZr(:,2), 'r' ) subplot(2,1,2); plot(f/1000,imag(z),f/1000,UZi(:,1), 'r' ,f/1000,UZi(:,2), 'r' ) end subplot(h1); xlim([f(1) f(end)]/1000) ylabel( '\theta' ); xlabel( 'Freq [kHz]' ); title( 'Normalized Acoustic Impedance' ) %axis tight %grid on subplot(h2); xlim([f(1) f(end)]/1000) ylabel( '\chi' ); xlabel( 'Freq [kHz]' ); %axis tight %grid on % Plots %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Save the MatLab figures if pic == 0 h=figure(1); saveas(h, 'coherence.fig' ) h=figure(2); saveas(h, 'frf.fig' ) h=figure(3); saveas(h, 'r.fig' ) h=figure(4); saveas(h, 'runcert.fig' ) h=figure(5); saveas(h, 'z.fig' ) h=figure(6); saveas(h, 'zuncert.fig' ) end % end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G.1.4 TMM Subroutine Program function [r,z,k,SWR] = tmm_subv6(w,l,s,H12,tatm,patm,dtube,trig); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % tmm_subv6.m % % Todd Schultz 7/29/02 revised 4/15/2005 %

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254 % This matlab program calculates the acoustical impedance of a sample % based on the Two Microphone Method. A trigger is given to allow the % user to turn on or off the corrections for dispersion and attentuation. % Assumptions: 1. exp(jwt) sign convention % 2. no mean flow % 3. mic 1 is farther from the sample % 4. mic 2 is closer to the sample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Inputs % w = vector of test angular frequencies [rad/s] % s = spacing between microphones [m] % Us = uncertainty in s [m] % l = location of mic closer to the specimen [m] % H12 = vecotr of the frequency response function [Parms] % tatm = atmospheric temperature [K] % patm = atmospheric pressure [Pa] % dtube = tube cross-section dimension [m] % trig = logical trigger for dispersion and attentuation % 0 = off (analytical method used for the uncertainty of r) % 1 = on (attentuation and dispersion are not accounted for % in the uncertainty analysis which still uses the % analytical method) % % Outputs % r = vector of the reflection coefficients % k = wavenumber vector used for the calculation of R %% Variable definitions global gamma Rair c0 %% Constants % Hydraulic diameter for square duct [m] HD = dtube; % Gas Constant for air [J/(kg*K)] Rair = 287; % Ratio of specific heats for air gamma = 1.4; % Specific heat at constant pressure for air [J/(kg*K)] cp = 3.5*Rair; % Atmospheric density [kg/m^3] rhoatm = patm./(Rair.*tatm); % Atmspheric speed of sound [m/s] c0 = sqrt(gamma.*Rair.*tatm); % Fluid properties look up (data take from Fundamentals of Heat % and Mass Transfer, 4th ed.) % mu_alpha=matrix of temps of the table data (c 1) [K], % the absolute viscosity (c 2) [(N*s)/m^2] % and the thermal diffusivity (c 3) [m^2/s] table = [100 71.10e-7 2.54e-6; 150 103.4e-7 5.84e-6;

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255 200 132.5e-7 10.3e-6; 250 159.6e-7 15.9e-6; 300 184.6e-7 22.5e-6; 350 208.2e-7 29.9e-6; 400 230.1e-7 38.7e-6]; % Interpolate the values for mu and alpha from table mu_alpha = interp1(table(:,1),table(:,2:3),tatm, '*cubic' ); % Prandtl Number Pr = mu_alpha(:,1)./(rhoatm.*mu_alpha(:,2)); % Heat conduction coefficient [W/(m*K)] kappa = cp*mu_alpha(:,1)./Pr; %% Calcate the attenuation constants % Viscothermal attenuation [1/m] bv = (sqrt(2)/(HD))*(w./c0).*(sqrt(mu_alpha(:,1)./(rhoatm.*w))).*(1+(gamma1)./sqrt(Pr)); % Tube speed of sound (taken from DTB) ct = c0.*(1-(sqrt(2)/HD).*sqrt(mu_alpha(:,1)./(rhoatm.*w)).*(1+(gamma1)./sqrt(Pr))); %% Wave number [1/m] (use trigger for dispersion and attentuation) switch trig case 0 k = w./c0; case 1 k = w./ct; k = k j*bv; end % Reflection coefficient%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r = exp(j*2*k*(l+s)).*(H12-exp(-j*k*s))./(exp(j*k*s)-H12); % Normalized Acoustic Impedance z = (1.+r)./(1.-r); % Standing Wave Ratio SWR = 20*log10((1+abs(r))./(1-abs(r))); % end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% G.1.5 TMM Subroutine for the An alytical Uncertainty in R function SR = tmm_rv6(w,k,l,s,H12,r,Ul,Us,Utatm,SH,trig); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % tmm_rv6.m % % Todd Schultz 4/15/2005 % % This matlab program calculates the estimate for the uncertainty % using analytical methods. A trigger is given to allow the user to

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256 % turn on or off the corrections for dispersion and attentuation. % Assumptions: 1. exp(jwt) sign convention % 2. no mean flow % 3. mic 1 is farther from the sample % 4. mic 2 is closer to the sample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Inputs % w = vector of test angular frequencies [rad/s] % k = wavenumber vector used for the calculation of R % s = spacing between microphones [m] % Us = uncertainty in s [m] % l = location of mic closer to the specimen [m] % Ul = uncertainty in d [m] % H12 = vecotr of the frequency response function [Parms] % SH = covariance matrix for the FRF [Parms^2] % tatm = atmospheric temperature [K] % Utatm = uncertainty in atmospheric temperature [K] % trig = logical trigger for dispersion and attentuation % 0 = off (analytical method used for the uncertainty of r) % 1 = on (attentuation and dispersion are not accounted for % in the uncertainty analysis which still uses the % analytical method) % r = vector of the reflection coefficients % % Outputs % Sr = covariance matrix for the reflection coefficient % Variable definitions global gamma Rair c0 Hr = real(H12); Hi = imag(H12); % wavenumber derivative dkdT = -k.*gamma.*Rair./(2*c0.^2); % Partial derivatives of the reflection coefficient % common factor in derivatives Dem = 1+Hr^2+Hi^2-2*Hr*cos(k*s)-2*Hi*sin(k*s); dRrdHr = (2*cos(k*(2*l+s))-2*Hr*cos(2*k*(l+s)))/Dem (2*Hr*cos(k*(2*l+s))-cos(2*k*l)-(Hr^2+Hi^2)*cos(2*k*(l+s)))*(2*Hr2*cos(k*s))/Dem^2; dRrdHi = -2*Hi*cos(2*k*(l+s))/Dem (2*Hr*cos(k*(2*l+s))-cos(2*k*l)(Hr^2+Hi^2)*cos(2*k*(l+s)))*(2*Hi-2*sin(k*s))/Dem^2; dRrdl = -2*k*imag(r); dRrds = -2*k*(Hr*sin(k*(2*l+s))(Hr^2+Hi^2)*sin(2*k*(l+s))+real(r)*(Hr*sin(k*s)-Hi*cos(k*s)))/Dem; dRrdk = (2*Hr*(2*l+s)*sin(k*(2*l+s))+2*l*sin(2*k*l)+(Hr^2+Hi^2)*(2*l+2*s)*sin(2* k*(l+s)))/Dem (2*Hr*cos(k*(2*l+s))-cos(2*k*l)(Hr^2+Hi^2)*cos(2*k*(l+s)))*(2*Hr*s*sin(k*s)-2*Hi*s*cos(k*s))/Dem^2;

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257 dRidHr = (2*sin(k*(2*l+s))-2*Hr*sin(2*k*(l+s)))/Dem (2*Hr*sin(k*(2*l+s))-sin(2*k*l)-(Hr^2+Hi^2)*sin(2*k*(l+s)))*(2*Hr2*cos(k*s))/Dem^2; dRidHi = -2*Hi*sin(2*k*(l+s))/Dem (2*Hr*sin(k*(2*l+s))-sin(2*k*l)(Hr^2+Hi^2)*sin(2*k*(l+s)))*(2*Hi-2*sin(k*s))/Dem^2; dRidl = 2*k*real(r); dRids = 2*k*(Hr*cos(k*(2*l+s))-(Hr^2+Hi^2)*cos(2*k*(l+s))imag(r)*(Hr*sin(k*s)-Hi*cos(k*s)))/Dem; dRidk = (2*Hr*(2*l+s)*cos(k*(2*l+s))-2*l*cos(2*k*l)(Hr^2+Hi^2)*(2*l+2*s)*cos(2*k*(l+s)))/Dem (2*Hr*sin(k*(2*l+s))sin(2*k*l)-(Hr^2+Hi^2)*sin(2*k*(l+s)))*(2*Hr*s*sin(k*s)2*Hi*s*cos(k*s))/Dem^2; % Form the Jacobian matrix J = [dRrdHr dRrdHi dRrdl dRrds dRrdk*dkdT; dRidHr dRidHi dRidl dRids dRidk*dkdT]; % Form the input covariance matrix Sx = [SH(1,1) SH(1,2) 0 0 0; SH(2,1) SH(2,2) 0 0 0; 0 0 Ul^2 0 0; 0 0 0 Us^2 0 ; 0 0 0 0 Utatm^2]; % Compute the covariance of the reflection coefficient SR = J*Sx*J'; % end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% G.1.6 TMM Subroutine for the An alytical Uncertainty in Z function SZ = tmm_zv6(r,Sr,trig); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % tmm_zv6.m % % Todd Schultz 5/1/2005 % % This matlab program calculates the estimate for the uncertainty % using analytical methods. A trigger is given to allow the user to % turn on or off the corrections for dispersion and attentuation. % Assumptions: 1. exp(jwt) sign convention % 2. no mean flow % 3. mic 1 is farther from the sample % 4. mic 2 is closer to the sample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Inputs % r = vector of the reflection coefficients % Sr = sample covariance matrix for R % trig = logical trigger for dispersion and attentuation % 0 = off (analytical method used for the uncertainty of r)

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258 % 1 = on (attentuation and dispersion are not accounted for % in the uncertainty analysis which still uses the % analytical method) % % Outputs % SZ = covariance matrix for the normalized acoustic impedance % Variable definitions Rr = real(r); Ri = imag(r); % Partial derivatives of the normalized acoustic impedance % common factor in derivatives Dem = ((1-Rr).^2+Ri.^2).^2; dthetadRr = 2*((1-Rr).^2-Ri.^2)./Dem; dthetadRi = -4*Ri.*(1-Rr)./Dem; dxidRr = 4*Ri.*(1-Rr)./Dem; dxidRi = 2*((1-Rr).^2-Ri.^2)./Dem; % Form the Jacobian matrix J = [dthetadRr dthetadRi; dxidRr dxidRi]; % Compute the covariance of the reflection coefficient SZ = J*Sr*J'; % end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% G.1.7 TMM Subroutine for the Mo nte Carlo Uncertainty Estimates % tmm_mcv6.m % % Todd Schultz 9/2/2005 % % This program takes the data from the tmmv6 program and carries out a % Monte Carlo simulatio. The output is the array of random data that is to % be processed with other file to determine the statistics. % Initialize Monte Carlo variables r_data = zeros(iter,length(f)); z_data = zeros(iter,length(f)); ld = zeros(iter,1); sd = zeros(iter,1); td = zeros(iter,1); hd = zeros(iter,length(f));

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259 if trig == 1 pd = zeros(iter,length(f)); dtube = zeros(iter,length(f)); end % pertub the values ld = l + Ul*randn(iter,1); sd = s + Us*randn(iter,1); td = tatm + Utatm*randn(iter,1); if trig == 1 pd = patm + Upatm*randn(iter,1); dtubed = dtube + Udtube*randn(iter,1); end for ii = 1:length(f) junk = mvgrnd([0;0],SH(:,:,ii),iter); hd(:,ii) = real(h12(ii))+junk(:,1)+j*(imag(h12(ii))+junk(:,2)); end clear junk % Carry out the Monte Carlo simulations switch trig case 0 % no dissipation and dispersion for jj = 1:iter [r_data(jj,:),z_data(jj,:),k,SWR] = tmm_subv6(w,ld(jj),sd(jj),hd(jj,:).',td(jj),patm,dtube,trig); end case 1 % with dissipation and dispersion for jj = 1:iter [r_data(jj,:),z_data(jj,:),k,SWR] = tmm_subv6(w,ld(jj),sd(jj),hd(jj,:).',td(jj),pd(ii),dtube(ii),trig); end end % Save the Monte Carlo simulation data switch trig case 0 save mc_data r_data z_data ld sd td hd case 1 save mc_data r_data z_data ld sd td hd pd dtubed end G.2 Uncertainty Subroutines G.2.1 Frequency Response Function Uncertainty function [varargout] = H3uncert(Gxx,Gyy,Gxy,Gmm,Gnn,H,nrec) % H3uncert.m % % Todd Schultz 4/14/2005 % % This program computes the random uncertainty in the frequency reponse % function estimate. This program is for when the FRF is estimated using % H3. H3 = sqrt(H1*H2) where H1 = Gxy/Gxx and H2 = Gyy/Gyx

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260 % The input signal is assumed to be deterministic and % that the noise on the signal does not pass through the system. Both % noise signals are assumed to be Gaussian and uncorrelated with eachother. % % Usage: [varargout] = H3uncert(Gxx,Gyy,Gxy,Gmm,Gnn,H,nrec) % % Inputs % Gxx = power spectrum of the input signal % Gyy = power spectrum of the output signal % Gxy = cross spectrum between the input and output signals % Gmm = estimate of the input noise power spectrum % Gnn = estimate of the output noise power spectrum % H = estimate of the FRF % nrec = number of spectral averages % % Outputs % SH = sample covariance matrix for H % Relationships Cxy = real(Gxy); Qxy = imag(Gxy); % Sample Covariance matrix for the spectrums Sx = [Gmm.*(2.*Gxx-Gmm) 0 Gmm.*Cxy Gmm.*Qxy; 0 Gnn.*(2.*Gyy-Gnn) Gnn.*Cxy Gnn.*Qxy; Gmm.*Cxy Gnn.*Cxy (Gnn.*Gxx+Gmm.*GyyGmm.*Gnn)/2 0; Gmm.*Qxy Gnn.*Qxy 0 (Gnn.*Gxx+Gmm.*Gyy-Gmm.*Gnn)/2]; Sx = Sx/nrec; % Derivative for the real and imaginary parts of H dHrdGxx = -sqrt(Gyy/Gxx)*Cxy/(2*Gxx*sqrt(Cxy^2+Qxy^2)); dHrdGyy = Cxy/(2*sqrt(Gxx*Gyy*(Cxy^2+Qxy^2))); dHrdCxy = sqrt(Gyy/Gxx)*Qxy^2./((Cxy^2+Qxy^2)^(3/2)); dHrdQxy = -sqrt(Gyy/Gxx)*Cxy*Qxy/((Cxy^2+Qxy^2)^(3/2)); dHidGxx = -sqrt(Gyy/Gxx)*Qxy/(2*Gxx*sqrt(Cxy^2+Qxy^2)); dHidGyy = Qxy/(2*sqrt(Gxx*Gyy*(Cxy^2+Qxy^2))); dHidCxy = -sqrt(Gyy/Gxx)*Cxy*Qxy/((Cxy^2+Qxy^2)^(3/2)); dHidQxy = sqrt(Gyy/Gxx)*Cxy.^2/((Cxy^2+Qxy^2)^(3/2)); % Define the Jacobian matrix J = [dHrdGxx dHrdGyy dHrdCxy dHrdQxy; dHidGxx dHidGyy dHidCxy dHidQxy]; % Compute the sample covariance matrix of H SH = J*Sx*J'; if nargout == 1 varargout = {SH}; elseif nargout == 2

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261 varargout = {SH Sx}; end G.2.2 Averaged FRF Uncertainty function [SHave] = have_div(Ho,SHo,Hs,SHs) % have.m % % Todd Schultz 4/14/2005 % % This program computes the random uncertainty in the averaged frequency % reponse function estimate. This programs works when the FRFs are % averaged by dvision. Have = sqrt(Ho/Hs) Also, this programs is set to % handle only one frequency at a time. % % Usage: [SHave] = have_div(Ho,SHo,Hs,SHs) % % Inputs % Ho = original FRF % SHo = sample covariance matrix for the original estimate of the FRF % Hs = switched FRF % SHs = sample covariance matrix for the switched estimate of the FRF % % Outputs % SHave = sample covariance matrix for Have % Useful variables Hor = real(Ho); Hoi = imag(Ho); Hsr = real(Hs); Hsi = imag(Hs); theta = (atan2(Hoi,Hor)-atan2(Hsi,Hsr))/2; % Derivatives for Have dHrdHor = (Hor*cos(theta)+Hoi*sin(theta))/((Hsr^2+Hsi^2)^(1/4)*2*(Hor^2+Hoi^2)^(3 /4)); dHrdHoi = (Hoi*cos(theta)Hor*sin(theta))/((Hsr^2+Hsi^2)^(1/4)*2*(Hor^2+Hoi^2)^(3/4)); dHrdHsr = (-Hsr*cos(theta)Hsi*sin(theta))*(Hor^2+Hoi^2)^(1/4)/(2*(Hsr^2+Hsi^2)^(5/4)); dHrdHsi = (Hsi*cos(theta)+Hsr*sin(theta))*(Hor^2+Hoi^2)^(1/4)/(2*(Hsr^2+Hsi^2)^(5/ 4)); dHidHor = (Hoi*cos(theta)+Hor*sin(theta))/((Hsr^2+Hsi^2)^(1/4)*2*(Hor^2+Hoi^2)^(3/ 4)); dHidHoi = (Hor*cos(theta)+Hoi*sin(theta))/((Hsr^2+Hsi^2)^(1/4)*2*(Hor^2+Hoi^2)^(3 /4));

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262 dHidHsr = (Hsi*cos(theta)Hsr*sin(theta))*(Hor^2+Hoi^2)^(1/4)/(2*(Hsr^2+Hsi^2)^(5/4)); dHidHsi = (-Hsr*cos(theta)Hsi*sin(theta))*(Hor^2+Hoi^2)^(1/4)/(2*(Hsr^2+Hsi^2)^(5/4)); % Define the Jocabian matrix J = [dHrdHor dHrdHoi dHrdHsr dHrdHsi; dHidHor dHidHoi dHidHsr dHidHsi]; % Define input covariance matrix Sx = [SHo zeros(2); zeros(2) SHs]; % Compute cavariance matrix for the Have SHave = J*Sx*J'; G.2.3 Effective Number of degrees of Freedom function nu = nu_eff(S,nux) % % Todd Schultz 5/2/2005 % % This program is designed to compute the effective number of degrees of % freedom for the multivariate uncertainty propagation to estimate the % coveage factor. This program uses the trace method but constrains the % minimum number of effective degrees of freedom to be at least as much as % the minimum number of the input values. % Reference: Willink and Hall, "A classical method for uncertainty analysis % with multidimensional data." Metrologia, 2002, 39, p361-369. % % Usage nu = nu_eff(S,nux) % % Inputs % S = 3 dimensional array of the covariance matrices for all input % variables S(x,y,z) S(x,y) = covariance matrix for variable z % nux = vector degrees of freedom for each input variable nux(z) % % Output % nu = effective number of degrees of freedom for the output variable % % Determine the number of variates and variables [x,y,z] = size(S); if x == y p = x; % number of variates else disp( 'Sample covariance matrices must be square.' ) return end

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263 m = z; % number of input variables % Compute the numerator num = 0; for jj = 1:p for kk = jj:p num = num + sum(S(jj,jj,:))*sum(S(kk,kk,:))+sum(S(jj,kk,:))^2; end end % Compute the denominator den = 0; for jj = 1:p for kk = jj:p d = 0; for ii = 1:m d = d + (S(jj,jj,ii)*S(kk,kk,ii) + S(jj,kk,ii)^2)/nux(ii); end den = den + d; end end % Compute the effective number of degrees of freedom nu = num/den; nu = floor(nu); % Check that nu is greater than min(nux) nu = max([min(nux) nu]); % Check that nu is less than sum(nux) nu = min([nu sum(nux)]); G.2.4 Numeric Computation of Bivariate Confidence Regions function [prob,x,vp] = numericCI(z,p,bins,cbins,vin) % numericCI.m % Todd Schultz 9/1/2005 % % This program is designed to take a set of data from a bivariate Monte % Carlo simulation and determine the p% confidence region. This is done % purely numerically and makes no assumptions about the underlying % distribution of the data. First, the data is used to generate a % bivariate probability density function and to compute a certain number of % constant probability density contours. The pdf is then integrated over % the contours to determine the total propability. The two contours that % bound the desired probability level are linearly interpolated to % determine the desired confidence interval. The estimate pdf is smoothed % to help "average out" any irregularities due to the limited sample data. %

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264 % Usage: [prob,x] = numericCI(z,p,bins,cbins) % % Inputs: % z = matrix of the Monte Carlo simulation data % (Should be n rows by 2 columns) % p = desired probability level for the confidence region % bins = number of bins to used in estimating the pdf % (20 is too few, 35 seems to work well for 25,000 points) % cbins = number of contour levels used to estimat the pdf contours % % Output: % prob = probability of the region inside the found contour % x = matrix of the vertices that define the confidence interval when % connected with straight lines % (m rows by 2 columns) % Check the size of input data [m,n] = size(z); if n ~= 2 disp( 'Input matrix does not have enough columns!' ) return end % Compute the bivariate pdf % pdf = pdf, c = grid locations of the pdf values [pdf,c] = bivariate_pdf(z,bins,1); dz1 = c{1}(2)-c{1}(1); dz2 = c(ISO-10534-2:1998)(2)-c(ISO-10534-2:1998)(1); [X, Y] = meshgrid(c{1},c(ISO-10534-2:1998)); % Grid locations in matrix form % Smooth the pdf pdf_old = pdf; H = fspecial( 'disk' ,2); pdf = imfilter(pdf,H, 'replicate' ); % ptest = sum(sum(pdf))*dz1*dz2 % max(max(pdf)) % min(min(pdf)) % figure % set(gcf,'paperorientation','landscape') % set(gcf,'paperposition',[0.25 0.25 10.5 8.0]) % [X, Y] = meshgrid(c{1},c(ISO-10534-2:1998)); % contourf(X,Y,pdf,vin) % colorbar % %axis equal % xlabel('x1') % ylabel('x2') % title('Bivariate PDF') % Compute a set of iso-probability density lines % Cmatrix = contour matrix Cmatrix = contours(X,Y,pdf,cbins);

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265 % Pick a contour and intregate the pdf to determine the probability pr = 100; % Initialize the loop variable v = 0; xy = []; ii = 0; % Initialize loop counter while pr >= p ii = ii + 1; % Increase the loop counter % Save the old values v_old = v; xy_old = xy; pr_old = pr; % Select the contour [v,xy] = contourline(Cmatrix,ii); % Find all points of the pdf with a value greater then or equal to the % value of the contour line K = find(pdf >= v); pr = sum(pdf(K))*dz1*dz2; % Compute the probability inside the contour end % Interpolate between v and v_old to estimate contour value that should % represent the p level CI and compute the contour vp = interp1([pr_old pr]',[v_old v]',p, 'linear' ); Cmatrix = contours(X,Y,pdf,[v_old vp v]); [v,xy]=contourline(Cmatrix,2); % Check the probability K = find(pdf >= v); pr = sum(pdf(K))*dz1*dz2; if pr > p*(1+0.02) | pr < p*(1-0.02) disp( 'Proper contour was not found. Try agian with more bins.' ) end % Define the output variables x = xy; % Contour definition prob = pr; % Probability of the contour % End G.2.5 Analytical Propagation of Uncertainty from Rectangular Form to Polar Form function Spolar = rect_to_polar(A,Srect) % rect_to_polar.m % % Todd Schultz 6/6/2005 % edited 3/15/2006 % % This program is designed to convert a bivariate uncertainty estimate for % a complex number from rectangular form to polar form. % % Usage: Spolar = rect_to_polar(A,Srect) % % Inputs

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266 % A = vector of the complex numbers % Srect = sample covariance matrices for the complex numbers in rectangular % form (real,imag,measurement number) % % Outputs % Spolar = sample covariance matrices for the complex numbers in polar form % Constants n = length(A); xr = real(A); xi = imag(A); % Loop to iterate the list of numbers S = zeros(2,2,n); for ii = 1:n % Jacobian matrix J = [xr(ii)/sqrt(xr(ii)^2+xi(ii)^2) xi(ii)/sqrt(xr(ii)^2+xi(ii)^2); -xi(ii)/(xr(ii)^2+xi(ii)^2) xr(ii)/(xr(ii)^2+xi(ii)^2)]; % propagate the sample covariance matrix S(:,:,ii) = J*Srect(:,:,ii)*J'; end Spolar = S; % end function G.3 Multivariate Statistics Subroutines G.3.1 Computation of Bivariate PDF function [pdf,c] = bivariate_pdf(x,k,pl) %% % Todd Schultz 6/11/2005 % % This function is designed to compute the bivariate probability density % function of a bivariate random variabe. The pdf is first computed as a % histogram, then normalized and plotted. % % Usage: [pdf] = bivariate_pdf(x,k,pl) % % Inputs: % x = n times 2 matrix containing the bivariate data, n is the number of % data points % k = number of divisions to break up the two axes in for the pdf % approximation [x y] % pl = logical trigger to disply the plot (0=yes, 1=no) % % Output: % pdf = computed pdf values

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267 % c = grid locations of the pdf values %% Constants [n,p] = size(x); %% Compute the histogram [pdf,c] = hist3(x,[k k]); % Normalized the histogram to compute the pdf dx1 = c{1}(2)-c{1}(1); dx2 = c(ISO-10534-2:1998)(2)-c(ISO-10534-2:1998)(1); pdf = pdf/n/dx1/dx2; pdf = pdf'; % transpose the matrix for plotting and consistency if pl == 0 % Plot the pdf figure set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) [X, Y] = meshgrid(c{1},c(ISO-10534-2:1998)); contourf(X,Y,pdf) colorbar %axis equal xlabel( 'x1' ) ylabel( 'x2' ) title( 'Bivariate PDF' ) end G.3.2 Numerical Computation of Constant PDF Contours function [v,xy] = contourline(C,n) % contourline.m % Todd Schultz 8/31/2005 % % contourline extracts the nth set of vertices from the output of % contourc.m. This set of vertices defines the nth contour line that now % can be used for other purposes. % % Usage: [v,xy] = contourline(C,n) % % Inputs % C = output from contourc % n = nth contour to extract % % Output % v = contour value % xy = m x 2 matrix containg the ordered vertices of the contour % Find out how many contours there are total and where they start ncon = 1; jj = 1; while ncon < size(C,2)

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268 conmark(jj) = ncon; % record where each contour starts value(jj) = C(1,ncon); % record the value of each contour nvert(jj) = C(2,ncon); % record the number of vertices ncon = ncon + nvert(jj) + 1; % compute the start of the next contour jj = jj + 1; % increase the counter end % pick the desired contour line v = value(n); xy = C(:,conmark(n)+1:conmark(n)+nvert(n))'; % end G.3.3 Multivariate Normal Random Number Generator % Multivariate Normal Random Generator function f = mvgrnd(m,sigma,n) % m is the mean column vector % sigma is the variance-covariance matrix % n is the number of iterations U = chol(sigma); % Cholesky decomposition. U is an upper triangular matrix d = length(m); for i = 1:n y(i,1:d) = m' + randn(1,d)*U; end f = y; G.4 Modal Decomposition Programs G.4.1 Pulse to MATLAB Conversion Program %% mdm_pulse4.m % % Todd Schultz v1 5/20/2005 % v4 12/2/2005 % % Version 4: The output of the Pulse system is assumed to be the time % series data for all eight microphones and for all 4 sources and the % background noise measurements. BKFiles is no longer used due to

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269 % the large size of the text files generated. The data is to be % saved as a UFF and loaded into MATLAB using the readuff.m function % found off of MATLAB's community site. The averaged FFTs are % computed in this program. The standard uncertainty in the FFTs are % also computed for each microphone signal. The uncertainty is % computed from an analytical formula that has been verified against % raw statistics of FFT computations. % % This program calls the BKFiles command to read in the data from the Pulse % system and relabels and stores the data for use in the MDM codes. The % data is stored in a mat binary file for use by the mdm.m file for % processing the data. Thus, this program is a stand alone file converter % from the Pulse format to the Matlab binary format. % Currently, this is designed to work with MDM1.pls Pulse file. The % discrete Fourier transform coefficients are label with the names of the % form PG1M1. % G1 represents group 1 which is farther from the specimen and M1 is % microphone 1 in that group. % Mic 1 = x1=a/4,y1=0 % Mic 2 = x2=a,y2=a/4 % Mic 3 = x3=3a/4,y3=a % Mic 4 = x4=0,y4=3a/4 clear all ; close all ; clc; pack; tic %% Input the file name. % fnoise = input('Input the file name for the background noise measurement data.\n','s'); % n_source = input('Input the number of independent sources measured.\n'); % fname = cell(n_source,1); % disp('Do not include the extension on the file names.') % fname{1} = input('Input the file name for the first source UFF file.\n','s'); % for ii = 2:n_source % fname{ii} = input('Input the file name for the next source UFF file.\n','s'); % end fnoise = 'background' ; n_source = 5; fname = { 'noplate' ; 'top' ; 'left' ; 'topleft' ; 'tri' }; % fname = {'noplate'}; %% Data Analysis settings. span = 25600; % Usable span [Hz] fs = 2.56*span; % Effective sampling rate [Hz] dt = 1/fs; % Effective sampling time [s] nfft = 2^12; % Desired number of fft lines

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270 df = 1/(nfft*dt); % Desired frequency resolution [Hz] T = dt*nfft; % Spectrum period npoints = 4096000; % Expected points per signal nave = 1000; % Desired number of averages noverlap = 0; % Desired overlap win = ones(nfft,1); % Desired rectangular window fstart = 300; % Lowest frequency of interest [Hz] fstop = 13500; % Highest frequency of interest [Hz] save( 'fftsettings' 'span' 'fs' 'dt' 'nfft' 'df' 'T' 'npoints' 'nave' 'n overlap' 'win' 'fstart' 'fstop' ) %% Signal number to mic name converter % signal 1 == g1m1 (a/4,0,d1) % signal 2 == g1m2 (a,a/4,d1) % signal 3 == g1m3 (3a/4,a,d1) % signal 4 == g1m4 (0,3a/4,d1) % signal 5 == g2m1 (a/4,0,d2) % signal 6 == g2m2 (a,a/4,d2) % signal 7 == g2m3 (3a/4,a,d2) % signal 8 == g2m4 (0,3a/4,d2) %% Load background noise data %[NoiseTime,Info,errmsg] = readuff([fnoise '.uff']); %clear all [NoiseTime,Info,errmsg] = readuff([fnoise '.uff' ]); if length(NoiseTime) ~= 16 disp([ 'Error reading file fnoise '.uff.' ]) disp( 'Please try again.' ) break end % Extract time series g1m1 = NoiseTime(ISO-10534-2:1998).measData; g1m2 = NoiseTime(ISO-10534-2:1998).measData; g1m3 = NoiseTime{6}.measData; g1m4 = NoiseTime{8}.measData; g2m1 = NoiseTime{10}.measData; g2m2 = NoiseTime{12}.measData; g2m3 = NoiseTime{14}.measData; g2m4 = NoiseTime{16}.measData; % Clear extra data clear NoiseTime Info errmsg % Compute auto-spectrums [Gg1m1,f] = pwelch(g1m1,win,noverlap,nfft,fs); I = find(f >= fstart & f <= fstop); Gg1m1 = Gg1m1(I)*df; f = f(I); [Gg1m2,f] = pwelch(g1m2,win,noverlap,nfft,fs); Gg1m2 = Gg1m2(I)*df; f = f(I);

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271 [Gg1m3,f] = pwelch(g1m3,win,noverlap,nfft,fs); Gg1m3 = Gg1m3(I)*df; f = f(I); [Gg1m4,f] = pwelch(g1m4,win,noverlap,nfft,fs); Gg1m4 = Gg1m4(I)*df; f = f(I); [Gg2m1,f] = pwelch(g2m1,win,noverlap,nfft,fs); Gg2m1 = Gg2m1(I)*df; f = f(I); [Gg2m2,f] = pwelch(g2m2,win,noverlap,nfft,fs); Gg2m2 = Gg2m2(I)*df; f = f(I); [Gg2m3,f] = pwelch(g2m3,win,noverlap,nfft,fs); Gg2m3 = Gg2m3(I)*df; f = f(I); [Gg2m4,f] = pwelch(g2m4,win,noverlap,nfft,fs); Gg2m4 = Gg2m4(I)*df; f = f(I); % Save the data save(fnoise, 'f' 'Gg1m1' 'Gg1m2' 'Gg1m3' 'Gg1m4' 'Gg2m1' 'Gg2m2' 'Gg2m3' 'Gg2m4' '-compress' ) % Clear the time series data clear g1m1 g1m2 g1m3 g1m4 g2m1 g2m2 g2m3 g2m4 %% Compute the standard uncertainty of the FFTs % The uncertainty is for single-sided FFTs only and the real and imaginary % parts are assumed to be uncorrelated. The standard uncertainty in the % real and imaginary parts are identical. uGg1m1 = zeros(2,2,length(f)); uGg2m1 = zeros(2,2,length(f)); uGg1m2 = zeros(2,2,length(f)); uGg2m2 = zeros(2,2,length(f)); uGg1m3 = zeros(2,2,length(f)); uGg2m3 = zeros(2,2,length(f)); uGg1m4 = zeros(2,2,length(f)); uGg2m4 = zeros(2,2,length(f)); for ii = 1:length(f) uGg1m1(:,:,ii) = (T/(4*nave))*Gg1m1(ii)*eye(2); uGg1m2(:,:,ii) = (T/(4*nave))*Gg1m2(ii)*eye(2); uGg1m3(:,:,ii) = (T/(4*nave))*Gg1m3(ii)*eye(2); uGg1m4(:,:,ii) = (T/(4*nave))*Gg1m4(ii)*eye(2); uGg2m1(:,:,ii) = (T/(4*nave))*Gg2m1(ii)*eye(2); uGg2m2(:,:,ii) = (T/(4*nave))*Gg2m2(ii)*eye(2); uGg2m3(:,:,ii) = (T/(4*nave))*Gg2m3(ii)*eye(2); uGg2m4(:,:,ii) = (T/(4*nave))*Gg2m4(ii)*eye(2); end % Save the uncertainty data save([fnoise 'uncertainty' ], 'uGg1m1' 'uGg1m2' 'uGg1m3' 'uGg1m4' 'uGg2m1' 'uGg2m2' 'u Gg2m3' 'uGg2m4' ) % Prepare for the next file

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272 clear Gg1m1 Gg1m2 Gg1m3 Gg1m4 Gg2m1 Gg2m2 Gg2m3 Gg2m4 clear uGg1m1 uGg1m2 uGg1m3 uGg1m4 uGg2m1 uGg2m2 uGg2m3 uGg2m4 pack; %% Read in the remaining files and compute the FFTs for ii = 1:n_source [MeasTime,Info,errmsg] = readuff([fname{ii} '.uff' ]); if length(MeasTime) ~= 16 disp([ 'Error reading file fname{ii} '.uff.' ]) disp( 'Please try again.' ) %break end % Extract time series g1m1 = MeasTime(ISO-10534-2:1998).measData; g1m2 = MeasTime(ISO-10534-2:1998).measData; g1m3 = MeasTime{6}.measData; g1m4 = MeasTime{8}.measData; g2m1 = MeasTime{10}.measData; g2m2 = MeasTime{12}.measData; g2m3 = MeasTime{14}.measData; g2m4 = MeasTime{16}.measData; % Clear unwanted data clear MeasTime Info errmsg % Resize time series data into FFT blocks g1m1 = reshape(g1m1,nfft,nave); PG1M1 = fft(g1m1,nfft,1)/nfft; PG1M1 = mean(PG1M1(I),2); % Convert to single-sided FFT g1m2 = reshape(g1m2,nfft,nave); PG1M2 = fft(g1m2,nfft,1)/nfft; PG1M2 = mean(PG1M2(I),2); % Convert to single-sided FFT g1m3 = reshape(g1m3,nfft,nave); PG1M3 = fft(g1m3,nfft,1)/nfft; PG1M3 = mean(PG1M3(I),2); % Convert to single-sided FFT g1m4 = reshape(g1m4,nfft,nave); PG1M4 = fft(g1m4,nfft,1)/nfft; PG1M4 = mean(PG1M4(I),2); % Convert to single-sided FFT g2m1 = reshape(g2m1,nfft,nave); PG2M1 = fft(g2m1,nfft,1)/nfft; PG2M1 = mean(PG2M1(I),2); % Convert to single-sided FFT g2m2 = reshape(g2m2,nfft,nave); PG2M2 = fft(g2m2,nfft,1)/nfft; PG2M2 = mean(PG2M2(I),2); % Convert to single-sided FFT g2m3 = reshape(g2m3,nfft,nave); PG2M3 = fft(g2m3,nfft,1)/nfft; PG2M3 = mean(PG2M3(I),2); % Convert to single-sided FFT

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273 g2m4 = reshape(g2m4,nfft,nave); PG2M4 = fft(g2m4,nfft,1)/nfft; PG2M4 = mean(PG2M4(I),2); % Convert to single-sided FFT % Clear time series clear g1m1 g1m2 g1m3 g1m4 g2m1 g2m2 g2m3 g2m4 % Save FFTs to mat file save(fname{ii}, 'f' 'PG1M1' 'PG1M2' 'PG1M3' 'PG1M4' 'PG2M1' 'PG2M2' 'PG2 M3' 'PG2M4' '-compress' ) % Prepare for next data set clear PG1M1 PG1M2 PG1M3 PG1M4 PG2M1 PG2M2 PG2M3 PG2M4 pack end toc % End Program G.4.2 MDM Main Program %% mdmv4.m % % Todd Schultz v1 9/12/2005 % v2 10/6/2005 % v3 10/27/2005 % v4 12/2/2005 % % % Version 4 changes: added Monte Carlo simulation option for uncertainty % analysis. Input uncertainties are assumed to be Gaussian. % % Version 3 changes: updated solution method for determining the pressure % amplitudes and the reflection coefficient matrix. I now use all % avialable data and use either Guassian elimination for a deterministic % system or a least-squares fit for an over determined system. This % appears to have smoothed out the data around cut-on frequencies. % % This program is design to carry out the data reduction for the MDM % acoustic impedance testing method. The spectral data is retreived from % the mat files exported by the Pulse system from the measurement of the % four independent sources and the two set up text files % are read in directly. The first set up file contains a 8 by 3 matrix of % the microphone locations with respect to the X-, Y-, and D-axis, an

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274 % 8 by 3 matrix with the uncertainties of the microphone locations. The % second text file contains an array with the information about the % waveguide used and the atmospheric temperature and pressure and % uncertainties. The first element is the waveguide number, second element % is the effective number spectral averages, the third element is % temperature, fourth element is the uncertainty in temperature, fifth % element pressure, sixth element is the is the uncertainty in pressure. % Waveguide number: 0 = large waveguide (1 inch by 1 inch) % 1 = small waveguide (8.5 mm by 8.5 mm) % A +jwt sign convention is used through this program. % clear all ; close all ; clc; pack; tic %% Input set up file names. % fnmic = input('Input the file name for the microphone location set up information file.\n','s'); % fnenv = input('Input the file name for the enviromental set up file.\n','s'); % n_source = input('Input the number of independent sources measured.\n'); % fndata = cell(n_source,1); % fndata{1} = input('Input the file name for the first source mat file.\n','s'); % for ii = 2:n_source % fndata{ii} = input('Input the file name for the next source mat file.\n','s'); % end % fsave = input('Input the file name to save the data in.\n','s'); % fsave_mc = input('Input the file name to save the uncertainty estimates to.\n','s'); % pic = input('Press 0 if you would like the figures saved or press 1 otherwise.\n'); % trig = input('Press 0 for NO uncertainty analysis or press 1 for a Monte Carlo uncertainty analysis.\n','s'); % if trig == 1 % funcert = input('Input the file name for the FFT uncertainty estimates.\n','s'); % end fnmic = 'C:\Documents and Settings\Todd\My Documents\Matlab\mdm\version4\xydmic.txt' ; fnenv = 'shmdm.txt' ; fndata = { 'noplate' ; 'top' ; 'left' ; 'topleft' }; % fndata = {'noplate';'top';'left';'topleft';'tri'}; n_source = length(fndata); % fsave = 'mdmCT65'; fsave = 'junk1' ; % fsave_mc = 'mdmCT65_mc'; fsave_mc = 'junk2' ; pic = 0; trig = 0; % funcert = 'backgrounduncertainty';

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275 funcert = 'junk3' ; %% Data anlaysis constants n_mc = 1000; % # of Monte Carlo iterations to use %% Read in Pulse data and rename variables to allow all data sets to be % loaded. The frequency range is already taken care of in mdm_pulsev4.m. PG1M1_s = cell(n_source,1); PG1M2_s = cell(n_source,1); PG1M3_s = cell(n_source,1); PG1M4_s = cell(n_source,1); PG2M1_s = cell(n_source,1); PG2M2_s = cell(n_source,1); PG2M3_s = cell(n_source,1); PG2M4_s = cell(n_source,1); for ii = 1:n_source load(fndata{ii}) PG1M1_s{ii} = PG1M1; PG1M2_s{ii} = PG1M2; PG1M3_s{ii} = PG1M3; PG1M4_s{ii} = PG1M4; PG2M1_s{ii} = PG2M1; PG2M2_s{ii} = PG2M2; PG2M3_s{ii} = PG2M3; PG2M4_s{ii} = PG2M4; end clear PG1M1 PG1M2 PG1M3 PG1M4 PG2M1 PG2M2 PG2M3 PG2M4 %% Read in microphone locations. A = dlmread(fnmic, '\t' ); % Read in set up file XYD = A(1:8,:); % Microphone locations [m] UXYD = A(9:16,:); % standard Uncertainty in Mic locations [m] nXYD = 35; % # of measurements clear A ; %% Read in experimental setup information A = dlmread(fnenv, '\t' ); % Read in set up file tube = A(1); % Tube number navg = A(2); % Number of averages used l = A(3); % mic location closer to the specimen [m] Ul = A(4); % standard Uncertainty in d [m] nl = A(5); % # of measurements of l s = A(6); % mic spacing [m] Us = A(7); % standard Uncertainty in s [m] ns = A(8); % # of measurements of s tatm = A(9)+273.15; % Atmospheric temperature [K] (input units are [C]) Utatm = A(10); % standard Uncertainty in Tatm [K] or [C] nT = A(11); % # of temperature measurements patm = A(12)*1000; % Atmospheric pressure [Pa] (input units are [kPa]) Upatm = A(13)*1000; % standard Uncertainty in Patm [Pa] (input units are [kPa]) nP = A(14); % # of pressure measurements clear A ; %% Constants

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276 % Cross-section width, l [m] switch tube case 0 dtube = 0.02544; % 1 in square tube Udtube = 1.5e-5; % standard uncertainty in tube dimension ndtube = 30; ab = [dtube dtube]; % ab(1)=a=x-axis, ab(2)=b=y-axis case 1 dtube = 0.0085; % Old 8.5 x 8.5 mm tube Udtude = 3.8e-5; % standard uncertainty in tube dimension ndtube = 30; ab = [dtube dtube]; % ab(1)=a=x-axis, ab(2)=b=y-axis end % Air properties and wavenumber gamma = 1.4; % ratio of specific heats for air Rair = 287; % gas constant for air [J/(kg*K)] c0 = sqrt(gamma*Rair*tatm); % speed of sound [m/s] rho = patm/(Rair*tatm); % density of air [kg/m^3] k = 2*pi*f/c0; % wavenumber [1/m] % Cut-on frequencies for the higher-order modes % x-axis is rows, y-axis is columns M_max = ceil(2*ab(1)*max(f)/c0); N_max = ceil(2*ab(2)*max(f)/c0); [N,M] = meshgrid(0:N_max,0:M_max); fco = (c0/(2*pi))*sqrt((M*pi/ab(1)).^2+(N*pi/ab(2)).^2); fco_sort = reshape(fco,size(fco,1)*size(fco,2),1); fco_sort = sort(fco_sort); I = find(max(f) >= fco_sort); I = find(fco_sort(I(length(I))) == fco); [N_max,M_max] = size(fco(I)); [N,M] = meshgrid(0:N_max,0:M_max); fco = (c0/(2*pi))*sqrt((M*pi/ab(1)).^2+(N*pi/ab(2)).^2); fco_sort = reshape(fco,size(fco,1)*size(fco,2),1); fco_sort = sort(fco_sort); %% In = cell(N_max+1,M_max+1); for nn = 0:N_max for mm = 0:M_max In{nn+1,mm+1} = find(f > fco(nn+1,mm+1)); end end %% Call mdm_sub.m to carry out computations of the wave mode coefficients. % Will have to carry out the computations in a loop, once per frequency. % A,B = {(0,0),(1,0),(0,1),(1,1)} % Initialize variables

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277 A = cell(n_source,length(f)); B = cell(n_source,length(f)); kz = cell(length(f),1); for ii = 1:length(f) floop = f(ii); for jj = 1:n_source GM = [PG1M1_s{jj}(ii);PG1M2_s{jj}(ii);PG1M3_s{jj}(ii);PG1M4_s{jj}(ii);PG2M1_ s{jj}(ii);PG2M2_s{jj}(ii);PG2M3_s{jj}(ii);PG2M4_s{jj}(ii)]; [AA,BB,kkz] = mdm_subv4(f(ii),GM,ab,XYD,c0,fco); A{jj,ii} = AA; B{jj,ii} = BB; kz{ii} = kkz; end end %% Compute the incident and reflected power for each source % W = ((a*b)/(8*rho*c0*k))*sum(kzmn*abs(Amn)^2) Wi = zeros(n_source,length(f)); Wr = zeros(n_source,length(f)); alpha = zeros(n_source,length(f)); for ii = 1:length(f) for jj = 1:n_source Amn = A{jj,ii}; Bmn = B{jj,ii}; kzmn = kz{ii}; Wi(jj,ii) = (ab(1)*ab(2)/(8*rho*c0*k(ii)))*sum(kzmn.*abs(Amn).^2); Wr(jj,ii) = (ab(1)*ab(2)/(8*rho*c0*k(ii)))*sum(kzmn.*abs(Bmn).^2); alpha(jj,ii) = 1-(Wr(jj,ii)/Wi(jj,ii)); end end %% Construct the A and B matrices and compute the reflection coefficient % matrix, acoustic impedance, and specific acoustic impedance % Initialize the variable R (mode order 00 10 01 11) R = zeros([M_max,N_max,length(f)]); Zac = zeros(length(A{1,length(f)}),1); Zspac = zeros(length(A{1,length(f)}),1); A_cond = zeros(length(f),1); for ii = 1:length(f) len = length(A{1,ii}); % # of modes to resolve A_matrix = [A{1,ii}]; B_matrix = [B{1,ii}]; for jj = 2:n_source A_matrix = [A_matrix A{jj,ii}]; B_matrix = [B_matrix B{jj,ii}]; end % check the rank of A (should equal 4) A_rank = rank(A_matrix); A_cond(ii) = cond(A_matrix);

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278 R_temp = B_matrix/A_matrix; % Store results in final variables R(1:len,1:len,ii) = R_temp; Zac(1:len,ii) = (1+diag(R_temp))./(1-diag(R_temp)); % Acoustic impedance ratio Zspac(1:len,ii) = Zac(1:len,ii)./sqrt(1(fco_sort(1:len)/f(ii)).^2); % Specific acoustic impedance ratio end %% Save the results save(fsave, 'f' 'fco' 'kz' 'A' 'B' 'R' 'Zac' 'Zspac' 'A_cond' 'Wi' 'Wr' 'alpha' '-compress' ); %% Plotting functions mdm_plotv4 toc tic %% Monte Carlo simulation for uncertainty estimates if trig == 1 % Prepare workspace close all clear AA A_1 A_2 A_3 A_4 A_cond A_matrix A_rank Amn Atemp clear BB B_1 B_2 B_3 B_4 B_cond B_matrix B_rank Bmn Btemp clear I In In00 In10 In01 In11 M N R_temp Wi Wr clear floop h h1 h2 h3 h4 h5 h6 h7 h8 ii jj mm kz kkz kzmn load(funcert) % Load FFT uncertainties pack pause(0.1) % Initialize variables to save the output in A_mc = cell(n_source,length(f),n_mc); B_mc = cell(n_source,length(f),n_mc); R_mc = zeros([M_max,N_max,length(f),n_mc]); Zac_mc = zeros(length(A{1,length(f)}),n_mc); Zspac_mc = zeros(length(A{1,length(f)}),n_mc); %A_cond_mc = zeros(length(f),n_mc); PG1M1_s_mc = PG1M1_s; PG1M2_s_mc = PG1M2_s; PG1M3_s_mc = PG1M3_s; PG1M4_s_mc = PG1M4_s; PG2M1_s_mc = PG2M1_s; PG2M2_s_mc = PG2M2_s; PG2M3_s_mc = PG2M3_s; PG2M4_s_mc = PG2M4_s; for ii = 1:length(f) for kk = 1:n_mc % Perturb input quantities by a random amount given by their % uncertainty distribution

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279 tatm_mc = tatm + Utatm*randn(1); patm_mc = patm + Upatm*randn(1); ab_mc = ab + Udtube*randn(1,2); XYD_mc = XYD + UXYD.*randn(8,3); % for ii = 1:length(f) for jj = 1:n_source junk = mvgrnd([real(PG1M1_s{jj}(ii));imag(PG1M1_s{jj}(ii))],uGg1m1(:,:,ii),1); PG1M1_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG1M2_s{jj}(ii));imag(PG1M2_s{jj}(ii))],uGg1m2(:,:,ii),1); PG1M2_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG1M3_s{jj}(ii));imag(PG1M3_s{jj}(ii))],uGg1m3(:,:,ii),1); PG1M3_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG1M4_s{jj}(ii));imag(PG1M4_s{jj}(ii))],uGg1m4(:,:,ii),1); PG1M4_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG2M1_s{jj}(ii));imag(PG2M1_s{jj}(ii))],uGg2m1(:,:,ii),1); PG2M1_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG2M2_s{jj}(ii));imag(PG2M2_s{jj}(ii))],uGg2m2(:,:,ii),1); PG2M2_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG2M3_s{jj}(ii));imag(PG2M3_s{jj}(ii))],uGg2m3(:,:,ii),1); PG2M3_s_mc{jj}(ii) = junk(1)+j*junk(2); junk = mvgrnd([real(PG2M4_s{jj}(ii));imag(PG2M4_s{jj}(ii))],uGg2m4(:,:,ii),1); PG2M4_s_mc{jj}(ii) = junk(1)+j*junk(2); end % n_source loop jj % end % frequency loop ii % Perturbed speed of sound and wavenumber c0_mc = sqrt(gamma*Rair*tatm_mc); % speed of sound [m/s] rho_mc = patm_mc/(Rair*tatm_mc); % density of air [kg/m^3] k_mc = 2*pi*f/c0_mc; % wavenumber [1/m] %% Call mdm_sub.m to carry out computations of the wave mode coefficients.

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280 % Will have to carry out the computations in a loop, once per frequency. % A,B = {(0,0),(1,0),(0,1),(1,1)} % for ii = 1:length(f) floop = f(ii); for jj = 1:n_source GM = [PG1M1_s_mc{jj}(ii);PG1M2_s_mc{jj}(ii);PG1M3_s_mc{jj}(ii);PG1M4_s_mc{jj }(ii);PG2M1_s_mc{jj}(ii);PG2M2_s_mc{jj}(ii);PG2M3_s_mc{jj}(ii);PG2M4_s_ mc{jj}(ii)]; [AA,BB,kkz] = mdm_subv4(f(ii),GM,ab_mc,XYD_mc,c0_mc,fco); A_mc{jj,ii,kk} = AA; B_mc{jj,ii,kk} = BB; end % end %% Construct the A and B matrices and compute the reflection coefficient % matrix, acoustic impedance, and specific acoustic impedance % Initialize the variable R (mode order 00 10 01 11) % for ii = 1:length(f) len = length(A{1,ii}); % # of modes to resolve A_matrix = [A_mc{1,ii,kk}]; B_matrix = [B_mc{1,ii,kk}]; for jj = 2:n_source A_matrix = [A_matrix A_mc{jj,ii,kk}]; B_matrix = [B_matrix B_mc{jj,ii,kk}]; end % check the rank of A (should equal 4) %A_rank = rank(A_matrix); %A_cond_mc(ii,kk) = cond(A_matrix); R_temp = B_matrix/A_matrix; % Store results in final variables R_mc(1:len,1:len,ii,kk) = R_temp; Zac_mc(1:len,ii,kk) = (1+diag(R_temp))./(1diag(R_temp)); % Acoustic impedance ratio Zspac_mc(1:len,ii,kk) = Zac_mc(1:len,ii,kk)./sqrt(1(fco_sort(1:len)/f(ii)).^2); % Specific acoustic impedance ratio % end % end for frequency loop ii end % end for Monte Carlo loop kk end % end for frequency loop ii %% Save Monte Carlo data save([fsave '_MCdata' ], 'f' 'A_mc' 'B_mc' 'R_mc' 'Zac_mc' 'Zspac_mc' '-compress' )

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281 end % end for if statement toc % End Program G.4.3 MDM Subroutine to Compute the Decomposition function [A,B,kz] = mdm_subv4(f,GM,ab,XYD,c0,fco) %% mdm_subv4.m % % Todd Schultz v1 5/21/2005 % v2 10/6/2005 % v3 10/27/2005 % v4 112/2/2005 % % This program carries out the data reduction routine for the multimode % decomposition technique for acoustical impedance testing. This code % requires the tube dimensions and microphone locations as an input. This % program assumes a +jwt sign convention and neglects dissipation and % dispersion. This code also uses all avialable data to compute the % desired coefficients. For an over-determined system, the coefficients % are solved for using a least-squares fit. For a deterministic system, % the coefficients are solved for using Gaussian elimination. % % Inputs % f = test frequency [Hz] % GM = complex amplitude vector [Pa] % ab = duct dimensions ab(1)-x-axis ab(2)-y-axis % XYD = 8x3 matrix with the location of the microphones % note: first 4 mics should be d1=axial location group1 (farther from sample) [m] % last 4 mics should be d2=axial location group2 (closer to the sample) [m] % c0 = isentropic speed of sound [m/s] % fco = matrix of the cut-on frequencies [Hz] % % Outputs % A = vector of complex incident wave mode coefficients % B = vector of complex reflected wave mode coefficients %% Constants w = 2*pi*f; % angular frequency [1/s] a = ab(1); % tube dimension in x-axis [m] b = ab(2); % tube dimension in y-axis [m]

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282 %% Find the maximum mode numbers that are needed [M,N] = find(f >= fco); % mode number indices M = M-1; N = N-1; % convert to mode numbers n_modes = length(M); % Number of modes to include %% Define augmented amplitude vector GMaug = [real(GM); imag(GM)]; % Augmented amplitude vector %% Define the required microphone locations for a deterministic system x = XYD(:,1); y = XYD(:,2); d = XYD(:,3); %% Calculate the z-axis wavenumbers kz = sqrt((w./c0).^2-(M*pi/a).^2-(N*pi/b).^2); %% Construct the coefficient matrix L Lr = zeros(length(d),2*n_modes); Li = zeros(length(d),2*n_modes); for ii = 1:n_modes Lr(:,ii) = cos(M(ii)*pi*x/a).*cos(N(ii)*pi*y/b).*cos(kz(ii)*d); Lr(:,ii+n_modes) = cos(M(ii)*pi*x/a).*cos(N(ii)*pi*y/b).*cos(kz(ii)*d); Li(:,ii) = cos(M(ii)*pi*x/a).*cos(N(ii)*pi*y/b).*sin(kz(ii)*d); Li(:,ii+n_modes) = cos(M(ii)*pi*x/a).*cos(N(ii)*pi*y/b).*sin(kz(ii)*d); end Laug = [Lr -Li;Li Lr]; % augmented [L] matrix %% Lrtemp=[cos(kz00*d1) cos(pi*x1/a)*cos(kz10*d1) cos(pi*y1/b)*cos(kz01*d1) cos(pi*x1/a)*cos(pi*y1/b)*cos(kz11*d1) cos(kz00*d1) cos(pi*x1/a)*cos(-kz10*d1) cos(pi*y1/b)*cos(-kz01*d1) cos(pi*x1/a)*cos(pi*y1/b)*cos(-kz11*d1); % cos(kz00*d2) cos(pi*x2/a)*cos(kz10*d2) cos(pi*y2/b)*cos(kz01*d2) cos(pi*x2/a)*cos(pi*y2/b)*cos(kz11*d2) cos(-kz00*d2) cos(pi*x2/a)*cos(kz10*d2) cos(pi*y2/b)*cos(-kz01*d2) cos(pi*x2/a)*cos(pi*y2/b)*cos(kz11*d2); % cos(kz00*d3) cos(pi*x3/a)*cos(kz10*d3) cos(pi*y3/b)*cos(kz01*d3) cos(pi*x3/a)*cos(pi*y3/b)*cos(kz11*d3) cos(-kz00*d3) cos(pi*x3/a)*cos(kz10*d3) cos(pi*y3/b)*cos(-kz01*d3) cos(pi*x3/a)*cos(pi*y3/b)*cos(kz11*d3); % cos(kz00*d4) cos(pi*x4/a)*cos(kz10*d4) cos(pi*y4/b)*cos(kz01*d4) cos(pi*x4/a)*cos(pi*y4/b)*cos(kz11*d4) cos(-kz00*d4) cos(pi*x4/a)*cos(kz10*d4) cos(pi*y4/b)*cos(-kz01*d4) cos(pi*x4/a)*cos(pi*y4/b)*cos(kz11*d4); % cos(kz00*d5) cos(pi*x5/a)*cos(kz10*d5) cos(pi*y5/b)*cos(kz01*d5) cos(pi*x5/a)*cos(pi*y5/b)*cos(kz11*d5) cos(-kz00*d5) cos(pi*x5/a)*cos(kz10*d5) cos(pi*y5/b)*cos(-kz01*d5) cos(pi*x5/a)*cos(pi*y5/b)*cos(kz11*d5); % cos(kz00*d6) cos(pi*x6/a)*cos(kz10*d6) cos(pi*y6/b)*cos(kz01*d6) cos(pi*x6/a)*cos(pi*y6/b)*cos(kz11*d6) cos(-kz00*d6) cos(pi*x6/a)*cos(-

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283 kz10*d6) cos(pi*y6/b)*cos(-kz01*d6) cos(pi*x6/a)*cos(pi*y6/b)*cos(kz11*d6); % cos(kz00*d7) cos(pi*x7/a)*cos(kz10*d7) cos(pi*y7/b)*cos(kz01*d7) cos(pi*x7/a)*cos(pi*y7/b)*cos(kz11*d7) cos(-kz00*d7) cos(pi*x7/a)*cos(kz10*d7) cos(pi*y7/b)*cos(-kz01*d7) cos(pi*x7/a)*cos(pi*y7/b)*cos(kz11*d7); % cos(kz00*d8) cos(pi*x8/a)*cos(kz10*d8) cos(pi*y8/b)*cos(kz01*d8) cos(pi*x8/a)*cos(pi*y8/b)*cos(kz11*d8) cos(-kz00*d8) cos(pi*x8/a)*cos(kz10*d8) cos(pi*y8/b)*cos(-kz01*d8) cos(pi*x8/a)*cos(pi*y8/b)*cos(kz11*d8);]; % % Litemp=[sin(kz00*d1) cos(pi*x1/a)*sin(kz10*d1) cos(pi*y1/b)*sin(kz01*d1) cos(pi*x1/a)*cos(pi*y1/b)*sin(kz11*d1) sin(kz00*d1) cos(pi*x1/a)*sin(-kz10*d1) cos(pi*y1/b)*sin(-kz01*d1) cos(pi*x1/a)*cos(pi*y1/b)*sin(-kz11*d1); % sin(kz00*d2) cos(pi*x2/a)*sin(kz10*d2) cos(pi*y2/b)*sin(kz01*d2) cos(pi*x2/a)*cos(pi*y2/b)*sin(kz11*d2) sin(-kz00*d2) cos(pi*x2/a)*sin(kz10*d2) cos(pi*y2/b)*sin(-kz01*d2) cos(pi*x2/a)*cos(pi*y2/b)*sin(kz11*d2); % sin(kz00*d3) cos(pi*x3/a)*sin(kz10*d3) cos(pi*y3/b)*sin(kz01*d3) cos(pi*x3/a)*cos(pi*y3/b)*sin(kz11*d3) sin(-kz00*d3) cos(pi*x3/a)*sin(kz10*d3) cos(pi*y3/b)*sin(-kz01*d3) cos(pi*x3/a)*cos(pi*y3/b)*sin(kz11*d3); % sin(kz00*d4) cos(pi*x4/a)*sin(kz10*d4) cos(pi*y4/b)*sin(kz01*d4) cos(pi*x4/a)*cos(pi*y4/b)*sin(kz11*d4) sin(-kz00*d4) cos(pi*x4/a)*sin(kz10*d4) cos(pi*y4/b)*sin(-kz01*d4) cos(pi*x4/a)*cos(pi*y4/b)*sin(kz11*d4); % sin(kz00*d5) cos(pi*x5/a)*sin(kz10*d5) cos(pi*y5/b)*sin(kz01*d5) cos(pi*x5/a)*cos(pi*y5/b)*sin(kz11*d5) sin(-kz00*d5) cos(pi*x5/a)*sin(kz10*d5) cos(pi*y5/b)*sin(-kz01*d5) cos(pi*x5/a)*cos(pi*y5/b)*sin(kz11*d5); % sin(kz00*d6) cos(pi*x6/a)*sin(kz10*d6) cos(pi*y6/b)*sin(kz01*d6) cos(pi*x6/a)*cos(pi*y6/b)*sin(kz11*d6) sin(-kz00*d6) cos(pi*x6/a)*sin(kz10*d6) cos(pi*y6/b)*sin(-kz01*d6) cos(pi*x6/a)*cos(pi*y6/b)*sin(kz11*d6); % sin(kz00*d7) cos(pi*x7/a)*sin(kz10*d7) cos(pi*y7/b)*sin(kz01*d7) cos(pi*x7/a)*cos(pi*y7/b)*sin(kz11*d7) sin(-kz00*d7) cos(pi*x7/a)*sin(kz10*d7) cos(pi*y7/b)*sin(-kz01*d7) cos(pi*x7/a)*cos(pi*y7/b)*sin(kz11*d7); % sin(kz00*d8) cos(pi*x8/a)*sin(kz10*d8) cos(pi*y8/b)*sin(kz01*d8) cos(pi*x8/a)*cos(pi*y8/b)*sin(kz11*d8) sin(-kz00*d8) cos(pi*x8/a)*sin(kz10*d8) cos(pi*y8/b)*sin(-kz01*d8) cos(pi*x8/a)*cos(pi*y8/b)*sin(kz11*d8);]; % Laugtemp = [Lrtemp -Litemp;Litemp Lrtemp]; %% Solve for the wave-mode coefficients Waug = Laug\GMaug; WM = Waug(1:2*n_modes)+j*Waug(2*n_modes+1:length(Waug)); A = WM(1:n_modes); B = WM(n_modes+1:length(WM)); % End subroutine

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284 G.4.4 MDM Plotting Subroutine %% mdm_plotv4.m % % Todd Schultz 12/17/2005 % % This programs carries out all the plotting functions for the MDM codes. % The plotting functions where removed from the base code to conserve space % and make it easier to edit the actual computations of the results and % Monte Carlo simulations. %% Plotting Functions %% Define helper variables for plotting In00 = find(f > fco(1,1)); In10 = find(f > fco(2,1)); In01 = find(f > fco(1,2)); In11 = find(f > fco(2,2)); A_1 = zeros(length(f),n_source); B_1 = zeros(length(f),n_source); A_2 = zeros(length(f),n_source); B_2 = zeros(length(f),n_source); A_3 = zeros(length(f),n_source); B_3 = zeros(length(f),n_source); A_4 = zeros(length(f),n_source); B_4 = zeros(length(f),n_source); for ii = 1:length(f) for jj = 1:n_source Atemp = A{jj,ii}; Btemp = B{jj,ii}; A_1(ii,jj) = Atemp(1); B_1(ii,jj) = Btemp(1); if length(Atemp) >= 2 A_2(ii,jj) = Atemp(2); B_2(ii,jj) = Btemp(2); end if length(Atemp) >= 3 A_3(ii,jj) = Atemp(3); B_3(ii,jj) = Btemp(3); end if length(Atemp) >= 4 A_4(ii,jj) = Atemp(4); B_4(ii,jj) = Btemp(4); end end end %% Plots pref = 20e-6; %% Plot incident pressure amplitudes figure(1) h1 = subplot(2,2,1); h2 = subplot(2,2,2);

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285 h3 = subplot(2,2,3); h4 = subplot(2,2,4); subplot(2,2,1); plot(f(In00)/1000,20*log10(abs(A_1(In00,:))/pref)) subplot(2,2,3); plot(f(In10)/1000,20*log10(abs(A_2(In10,:))/pref)) subplot(2,2,2); plot(f(In01)/1000,20*log10(abs(A_3(In01,:))/pref)) subplot(2,2,4); plot(f(In11)/1000,20*log10(abs(A_4(In11,:))/pref)) subplot(h1) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(a) Mode (0,0)' }) ylabel( '|P| [dB]' ) title( 'Incident Sound Pressure Levels' ) grid on subplot(h3) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(b) Mode (1,0)' }) ylabel( '|P| [dB]' ) grid on subplot(h2) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(c) Mode (0,1)' }) ylabel( '|P| [dB]' ) grid on subplot(h4) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(d) Mode (1,1)' }) ylabel( '|P| [dB]' ) grid on legend(fndata, 'Location' 'NorthWest' ) %% Plot reflected pressure amplitudes figure(2) set(gcf, 'paperorientation' 'landscape' ) set(gcf, 'paperposition' ,[0.25 0.25 10.5 8.0]) h1 = subplot(2,2,1); h2 = subplot(2,2,2); h3 = subplot(2,2,3); h4 = subplot(2,2,4); subplot(2,2,1); plot(f(In00)/1000,20*log10(abs(B_1(In00,:))/pref)) subplot(2,2,3); plot(f(In10)/1000,20*log10(abs(B_2(In10,:))/pref)) subplot(2,2,2); plot(f(In01)/1000,20*log10(abs(B_3(In01,:))/pref)) subplot(2,2,4); plot(f(In11)/1000,20*log10(abs(B_4(In11,:))/pref)) subplot(h1) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(a) Mode (0,0)' }) ylabel( '|P| [dB]' ) title( 'Reflected Sound Pressure Levels' ) grid on subplot(h3) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(b) Mode (1,0)' }) ylabel( '|P| [dB]' ) grid on subplot(h2) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(c) Mode (0,1)' }) ylabel( '|P| [dB]' ) grid on

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286 subplot(h4) xlim([f(1) f(length(f))]/1000) xlabel({ 'Freq [kHz]' ; '(d) Mode (1,1)' }) ylabel( '|P| [dB]' ) legend(fndata, 'Location' 'NorthWest' ) grid on %% Plot reflection coefficients figure(3) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f(In00)/1000,squeeze(abs(R(1,1,In00))),f(In10)/1000,squeeze(abs(R( 2,2,In10))),f(In01)/1000,squeeze(abs(R(3,3,In01))),f(In11)/1000,squeeze (abs(R(4,4,In11)))) subplot(2,1,2); plot(f(In00)/1000,squeeze(angle(R(1,1,In00)))*180/pi,f(In10)/1000,squee ze(angle(R(2,2,In10)))*180/pi,f(In01)/1000,squeeze(angle(R(3,3,In01)))* 180/pi,f(In11)/1000,squeeze(angle(R(4,4,In11)))*180/pi) subplot(h1); ylabel( '|R|' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(0,0)' '(1,0)' '(0,1)' '(1,1)' ) title( 'Reflection Coefficients' ) grid on subplot(h2); xlim([f(1) f(length(f))]/1000) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) grid on %% Plot mode scattering coefficients figure(4) h1=subplot(4,2,1); h2=subplot(4,2,2); h3=subplot(4,2,3); h4=subplot(4,2,4); h5=subplot(4,2,5); h6=subplot(4,2,6); h7=subplot(4,2,7); h8=subplot(4,2,8); % From (0,0) subplot(4,2,1); plot(f(In10)/1000,squeeze(abs(R(2,1,In10))),f(In01)/1000,squeeze(abs(R( 3,1,In01))),f(In11)/1000,squeeze(abs(R(4,1,In11)))) subplot(4,2,3); plot(f(In10)/1000,squeeze(angle(R(2,1,In10)))*180/pi,f(In01)/1000,squee ze(angle(R(3,1,In01)))*180/pi,f(In11)/1000,squeeze(angle(R(4,1,In11)))* 180/pi) subplot(h1); ylabel( '|R|' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(0,0) to (1,0)' '(0,0) to (0,1)' '(0,0) to (1,1)' )

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287 title( 'Mode Scattering Coefficients' ) grid on subplot(h3); xlim([f(1) f(length(f))]/1000) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) grid on % From (1,0) subplot(4,2,2); plot(f(In10)/1000,squeeze(abs(R(1,2,In10))),f(In10)/1000,squeeze(abs(R( 3,2,In10))),f(In11)/1000,squeeze(abs(R(4,2,In11)))) subplot(4,2,4); plot(f(In10)/1000,squeeze(angle(R(1,2,In10)))*180/pi,f(In10)/1000,squee ze(angle(R(3,2,In10)))*180/pi,f(In11)/1000,squeeze(angle(R(4,2,In11)))* 180/pi) subplot(h2); ylabel( '|R|' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(1,0) to (0,0)' '(1,0) to (0,1)' '(1,0) to (1,1)' ) title( 'Mode Scattering Coefficients' ) grid on subplot(h4); xlim([f(1) f(length(f))]/1000) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) grid on % From (0,1) subplot(4,2,5); plot(f(In01)/1000,squeeze(abs(R(1,3,In01))),f(In01)/1000,squeeze(abs(R( 2,3,In01))),f(In11)/1000,squeeze(abs(R(4,3,In11)))) subplot(4,2,7); plot(f(In01)/1000,squeeze(angle(R(1,3,In01)))*180/pi,f(In01)/1000,squee ze(angle(R(2,3,In01)))*180/pi,f(In11)/1000,squeeze(angle(R(4,3,In11)))* 180/pi) subplot(h5); ylabel( '|R|' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(0,1) to (0,0)' '(0,1) to (1,0)' '(0,1) to (1,1)' ) title( 'Mode Scattering Coefficients' ) grid on subplot(h7); xlim([f(1) f(length(f))]/1000) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) grid on % From (1,1) subplot(4,2,6); plot(f(In11)/1000,squeeze(abs(R(1,4,In11))),f(In11)/1000,squeeze(abs(R( 2,4,In11))),f(In11)/1000,squeeze(abs(R(3,4,In11))))

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288 subplot(4,2,8); plot(f(In11)/1000,squeeze(angle(R(1,4,In11)))*180/pi,f(In11)/1000,squee ze(angle(R(2,4,In11)))*180/pi,f(In11)/1000,squeeze(angle(R(3,4,In11)))* 180/pi) subplot(h6); ylabel( '|R|' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(1,1) to (0,0)' '(1,1) to (1,0)' '(1,1) to (0,1)' ) title( 'Mode Scattering Coefficients' ) grid on subplot(h8); xlim([f(1) f(length(f))]/1000) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) grid on %% Plot normal incident normalized acoustic impedance figure(5) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f(In00)/1000,real(Zac(1,In00))) subplot(2,1,2); plot(f(In00)/1000,imag(Zac(1,In00))) subplot(h1); ylabel( '\theta' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) %legend('(0,0)','(1,0)','(0,1)','(1,1)') title( 'Normal Incident Acoustic Impedance' ) grid on subplot(h2); xlim([f(1) f(length(f))]/1000) ylabel( '\chi' ) xlabel( 'Freq [kHz]' ) grid on %% Plot normalized acoustic impedance figure(6) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f(In00)/1000,real(Zac(1,In00)),f(In10)/1000,real(Zac(2,In10)),f(In 01)/1000,real(Zac(3,In01)),f(In11)/1000,real(Zac(4,In11))) subplot(2,1,2); plot(f(In00)/1000,imag(Zac(1,In00)),f(In10)/1000,imag(Zac(2,In10)),f(In 01)/1000,imag(Zac(3,In01)),f(In11)/1000,imag(Zac(4,In11))) subplot(h1); ylabel( '\theta_a_c' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(0,0)' '(1,0)' '(0,1)' '(1,1)' ) title( 'Acoustic Impedance Ratio' ) grid on subplot(h2); xlim([f(1) f(length(f))]/1000)

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289 ylabel( '\chi_a_c' ) xlabel( 'Freq [kHz]' ) grid on %% Plot normalized specific acoustic impedance figure(7) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f(In00)/1000,real(Zspac(1,In00)),f(In10)/1000,real(Zspac(2,In10)), f(In01)/1000,real(Zspac(3,In01)),f(In11)/1000,real(Zspac(4,In11))) subplot(2,1,2); plot(f(In00)/1000,imag(Zspac(1,In00)),f(In10)/1000,imag(Zspac(2,In10)), f(In01)/1000,imag(Zspac(3,In01)),f(In11)/1000,imag(Zspac(4,In11))) subplot(h1); ylabel( '\theta_s_p _a_c' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend( '(0,0)' '(1,0)' '(0,1)' '(1,1)' ) title( 'Specific Acoustic Impedance Ratio' ) grid on subplot(h2); xlim([f(1) f(length(f))]/1000) ylabel( '\chi_s_p _a_c' ) xlabel( 'Freq [kHz]' ) grid on %% Plot the ratio of the reflected power to the incident power for each % source (needs to be less then one) figure(8) plot(f/1000,Wr./Wi) ylabel( 'W_r/W_i' ) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ) legend(fndata) title( 'Ratio of Reflected Power to Incident Power' ) grid on %% Plot the absorption coefficient figure(9) plot(f/1000,alpha) xlim([f(1) f(end)]/1000) ylabel( '\alpha' ) xlabel( 'Freq [kHz]' ) legend(fndata) title( 'Power absorption coefficient' ) grid on %% Plot the condition number for A_matrix figure(10) plot(f/1000,A_cond) xlim([f(1) f(length(f))]/1000) xlabel( 'Freq [kHz]' ); ylabel( 'Condition Number' ) title( 'Condition Number of Amatrix' ) grid on

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290 %% Save the MatLab figures if pic == 0 h=figure(1); saveas(h, 'Pincident.fig' ) h=figure(2); saveas(h, 'Preflected.fig' ) h=figure(3); saveas(h, 'r.fig' ) h=figure(4); saveas(h, 'modescattering.fig' ) h=figure(5); saveas(h, 'znormal.fig' ) h=figure(6); saveas(h, 'zacoustic.fig' ) h=figure(7); saveas(h, 'zspac.fig' ) h=figure(8); saveas(h, 'powerratio.fig' ) h=figure(9); saveas(h, 'alpha.fig' ) h=figure(10); saveas(h, 'Rcond.fig' ) end G.4.5 MDM Mode Scattering Coefficients Plotting Subroutine %% mdm_modev4.m % % Todd Schultz 2/20/2006 % % This programs carries out the plotting of the mode scattering % coefficients. %% File to load pic = 0; fname = 'mdmsdof' ; load(fname) %% Plotting Functions %% Define helper variables for plotting In00 = find(f > fco(1,1)); In10 = find(f > fco(2,1)); In01 = find(f > fco(1,2)); In11 = find(f > fco(2,2)); %% Plot mode scattering coefficients %% From (0,0) figure(1) h1=subplot(2,1,1); h2=subplot(2,1,2);

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291 subplot(h1); plot(f(In10)/1000,squeeze(abs(R(2,1,In10))),f(In01)/1000,squeeze(abs(R( 3,1,In01))),f(In11)/1000,squeeze(abs(R(4,1,In11)))) subplot(h2); plot(f(In10)/1000,squeeze(angle(R(2,1,In10)))*180/pi,f(In01)/1000,squee ze(angle(R(3,1,In01)))*180/pi,f(In11)/1000,squeeze(angle(R(4,1,In11)))* 180/pi) subplot(h1); ylabel( '|R|' ) xlim([f(In01(1)) f(end)]/1000) xlabel( 'Freq [kHz]' ) legend( 'from (0,0) to (1,0)' 'from (0,0) to (0,1)' 'from (0,0) to (1,1)' 'Location' 'Best' ) title( 'Mode Scattering Coefficients' ) %grid on subplot(h2); xlim([f(In01(1)) f(end)]/1000) ylim([-180 180]) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) set(gca, 'YTick' ,-180:90:180) set(gca, 'YTickLabel' ,{ '-180' '-90' '0' '90' '180' }) %grid on %% From (1,0) figure(2) h3=subplot(2,1,1); h4=subplot(2,1,2); subplot(h3); plot(f(In10)/1000,squeeze(abs(R(1,2,In10))),f(In10)/1000,squeeze(abs(R( 3,2,In10))),f(In11)/1000,squeeze(abs(R(4,2,In11)))) subplot(h4); plot(f(In10)/1000,squeeze(angle(R(1,2,In10)))*180/pi,f(In10)/1000,squee ze(angle(R(3,2,In10)))*180/pi,f(In11)/1000,squeeze(angle(R(4,2,In11)))* 180/pi) subplot(h3); ylabel( '|R|' ) xlim([f(In01(1)) f(end)]/1000) xlabel( 'Freq [kHz]' ) legend( 'from (1,0) to (0,0)' 'from (1,0) to (0,1)' 'from (1,0) to (1,1)' 'Location' 'Best' ) title( 'Mode Scattering Coefficients' ) %grid on subplot(h4); xlim([f(In01(1)) f(end)]/1000) ylim([-180 180]) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) set(gca, 'YTick' ,-180:90:180) set(gca, 'YTickLabel' ,{ '-180' '-90' '0' '90' '180' }) %grid on %% From (0,1)

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292 figure(3) h5=subplot(2,1,1); h6=subplot(2,1,2); subplot(h5); plot(f(In01)/1000,squeeze(abs(R(1,3,In01))),f(In01)/1000,squeeze(abs(R( 2,3,In01))),f(In11)/1000,squeeze(abs(R(4,3,In11)))) subplot(h6); plot(f(In01)/1000,squeeze(angle(R(1,3,In01)))*180/pi,f(In01)/1000,squee ze(angle(R(2,3,In01)))*180/pi,f(In11)/1000,squeeze(angle(R(4,3,In11)))* 180/pi) subplot(h5); ylabel( '|R|' ) xlim([f(In01(1)) f(end)]/1000) xlabel( 'Freq [kHz]' ) legend( 'from (0,1) to (0,0)' 'from (0,1) to (1,0)' 'from (0,1) to (1,1)' 'Location' 'Best' ) title( 'Mode Scattering Coefficients' ) %grid on subplot(h6); xlim([f(In01(1)) f(end)]/1000) ylim([-180 180]) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' ) set(gca, 'YTick' ,-180:90:180) set(gca, 'YTickLabel' ,{ '-180' '-90' '0' '90' '180' }) %grid on %% From (1,1) figure(4) h7=subplot(2,1,1); h8=subplot(2,1,2); subplot(h7); plot(f(In11)/1000,squeeze(abs(R(1,4,In11))),f(In11)/1000,squeeze(abs(R( 2,4,In11))),f(In11)/1000,squeeze(abs(R(3,4,In11)))) subplot(h8); plot(f(In11)/1000,squeeze(angle(R(1,4,In11)))*180/pi,f(In11)/1000,squee ze(angle(R(2,4,In11)))*180/pi,f(In11)/1000,squeeze(angle(R(3,4,In11)))* 180/pi) subplot(h7); ylabel( '|R|' ) xlim([f(In11(1)) f(end)]/1000) xlabel( 'Freq [kHz]' ) legend( 'from (1,1) to (0,0)' 'from (1,1) to (1,0)' 'from (1,1) to (0,1)' 'Location' 'Best' ) title( 'Mode Scattering Coefficients' ) %grid on subplot(h8); xlim([f(In11(1)) f(end)]/1000) ylim([-180 180]) ylabel( '\phi [deg]' ) xlabel( 'Freq [kHz]' )

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293 set(gca, 'YTick' ,-180:90:180) set(gca, 'YTickLabel' ,{ '-180' '-90' '0' '90' '180' }) %grid on %% Save figures if pic == 0 h=figure(1); saveas(h, 'from00.fig' ) h=figure(2); saveas(h, 'from10.fig' ) h=figure(3); saveas(h, 'from01.fig' ) h=figure(4); saveas(h, 'from11.fig' ) end

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294 LIST OF REFERENCES bom, M. (1989). "Modal decomposition in ducts based on transfer function measurements between microphone pair s." Journal of Sound and Vibration 135(1): 95-114. bom, M. and H. Bodn (1988). "Error analysis of two-microphone measurements in ducts with flow." Journal of th e Acoustical Society of America 83(6): 2429-2438. Akoum, M. and J. M. Ville (1998). "Meas urement of the reflection matrix of a discontinuity in a duct." Journal of the Acoustical Society of America 103(5): 2463-2468. Arnold, D. P., T. Nishida, L. N. Cattafe sta and M. Sheplak (2 003). "A directional acoustic array using silicon micromachi ned piezoresistive microphones." Journal of the Acoustical Society of America 113(1): 289-298. ASTM-E1050-98 (1998). Impedance and Absorp tion of Acoustical Materials Using a Tube, Two Microphones and Digital Fr equency Analysis System, ASTM International. Bendat, J. S. and A. G. Piersol (2000). Random Data New York, John Wiley & Sons, Inc: 316-345,425-431. Berglund, B., T. Lindvall and D. H. Schwel a, Eds. (1999). Guidelines for Community Noise Geneva, World Health Organization. Bielak, G. W., J. W. Premo and A. S. He rsh (1999). "Advanced Tubrofan Duct Liner Concepts." NASA-La RC: NASA/CR-1999-209002 Blackstock, D. T. (2000). Funda mentals of Physical Acoustics New York, John Wiley & Sons, Inc. Bodn, H. and M. bom (1986). "Influence of errors on the two-microphone method for measuring acoustic properties in ducts. Journal of the Acoustical Society of America 79(2): 541-549. Chapra, S. C. and R. P. Canale ( 2002). Numerical Met hods for Engineers New York, McGraw-Hill: 231-261, 440-471.

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295 Chung, J. Y. and D. A. Blaser (1980). "Tra nsfer function method of measuring in-duct acoustic properties, I-Theory, II-Experim ent." Journal of the Acoustical Society of America 68(3): 907-921. Coleman, H. W. and W. G. Steele (1999). E xperimentation and Uncertainty Analysis for Engineers New York, John Wiley & Sons, Inc. Crighton, D. G. (1991). Airframe Noise. Aero acoustics of Flight Vehicles: Theory and Practice, vol. 1 H. H. Hubbard. Hampton, VA, NASA Reference Publication 1258. Volume 1: Noise Sources: 391-447. Dowling, A. P. and J. E. Ffowcs-Williams (1983). Sound and Sound Sources New York, John Wiley & Sons: 157-163. Eversman, W. (1970). "Energy flow criteria for acoustic propag ation in ducts with flow." Journal of the Acoustical Society of America 49(6): 1717-1721. Franzoni, L. P. and C. M. Elliott (1998). "An innovative design of a probe-tube attachment for a (1/2)-in microphone." Journal of th e Acoustical Society of America 104(5): 2903-2910. Gade, S. and H. Herlufsen (1987). Windows to FFT Analysis. Technical Review 3. Brel and Kjr. Golub, R. A., J. J. W. Rawls and J. W. Rusell (2005). Evaluation of the Advanced Subsonic Technology Program Noise Reduction Benefits. TM-2005-212144. Hampton, VA, NASA Langley Research Center. Groeneweg, J. F., T. G. Sofrin, P. R. Glie be and E. J. Rice (1991). Turbomachinery noise. Aeroacoustics of Flight Vehicles : Theory and Practice, vol. 2 H. H. Hubbard. Hampton, VA, NASA Reference Public ation 1258. Volume 1: Noise Sources: 151-209. Hall, B. D. (2003). "Calculating measurement uncertainty for complex-valued quantities." Measurement Science and Technology 14: 368-375. Hall, B. D. (2004). "On the propagation of un certainty in complex-valued quantities." Metrologia 41: 173-177. Herlufsen, H. (1984). Dual Channel FFT Analysis. Technical Review 1. Nrum Denmark, Brel and Kjr. Incropera, F. P. and D. P. DeWitt (2002) Fundamentals of Heat and Mass Transfer New York, John Wiley & Sons, Inc.

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296 Ingard, U. and V. K. Singhal (1974). "Sound a ttenuation in turbulent pipe flow." Journal of the Acoustical Society of America 55(3): 535-538. ISO (International Organization for Sta ndardization)-10534-2:1998 (1998). AcousticsDetermination of Sound Absorption Coefficient and Impedance in Impedance Tubes: Part 2: Transfer-function me thod. International Organization for Standardization. ISO (International Organization for Standardization) (1995). Guide to the Expression of Uncertainty in Measurement. Geneva, International Organization for Standardization. Johnson, R. A. and D. W. Wichern (2002). Applied Multivariate Statistical Analysis Upper Saddle River, NJ, Prentice Hall. Jones, M. G. and T. L. Parrott (1989). "Evaluation of a mu lti-point method for determining acoustic impedance." Mech anical Systems and Signal Processing 3(1): 15-35. Jones, M. G., T. L. Parrott and W. R. Watson (2003). "Comparison of acoustic impedance eduction techniques for lo cally-reacting liners." 9th AIAA/CEAS Aeroacoustics Conference and Exhibit, Hilton Head, South Carolina, AIAA. Jones, M. G. and P. E. Stiede (1997). "Com parison of methods for determining specific acoustic impedance." Journal of th e Acoustical Society of America 101(5): 26942704. Jones, M. G., W. R. Watson and T. L. Parrott (2004). "Design and evaluation of modifications to the NASA Langley flow impedance tube." 10th AIAA/CEAS Aeroacoustics Conference, Manche ster, United Kingdom, AIAA. Katz, B. F. G. (2000). "Method to resolve mi crophone and sample location errors in the two-microphone duct measurement method." J ournal of the Acoustical Society of America 108(5): 2231-2237. Kerschen, E. J. and J. P. Johnston (1981). "A modal separation measurement technique for broadband noise propagating inside circular ducts." Journal of Sound and Vibration 76(4): 499-515. Kinsler, L. E., A. R. Frey, A. B. Coppen s and J. V. Sanders (2000). Fundamentals of Acoustics New York, John Wiley & Sons, Inc. Kirchhoff, G. (1869). "On the influence of thermal conduction in a gas on sound propagation." Ann. Physik Chem. 134: 177-193.

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297 Kline, S. J. (1985). "The purposes of uncertain ty analysis." Journal of Fluids Engineering 107: 153-160. Kline, S. J. and F. A. McClintock (1953) "Describing uncertainties in single-sample experiments." Mechanical Engineering 75(1): 3-8. Kraft, R. E., J. Yu, H. W. Kwan, B. Beer, A. F. Seybert and P. Tathavadekar (2003). Acoustic Treatment Design Scaling Met hods: Phase II Final Report. NASA/CR2003-212428. Hampton, VA, NASA-Langley Research Center. Kraft, R. E., J. Yu, H. W. Kwan, D. K. Echt ernach, A. A. Syed and W. E. Chien (1999). Acoustic Treatment Design Scaling Me thods. NASA/CR-1999-209120. Hampton, VA, NASA-Langley Research Center. Moffat, R. J. (1985). "Using uncertainty anal ysis in the planning of an experiment." Journal of Fluids Engineering 107: 173-178. Moore, C. J. (1972). "In-duct investigation of subsonic fan "rotor alone" noise." Journal of the Acoustical Society of America 51(5): 1471-1482. Moore, C. J. (1979). "Measurement of radial and circumferential modes in annular and circular fan ducts." Jour nal of Sound and Vibration 62(2): 235-256. Morse, P. (1981). Vibration and Sound Sewickley, PA, Acoustical Society of America. Morse, P. M. and K. U. Ingard (1986). Theoretical Acoustics Princeton, NJ, Princeton University Press. Motsinger, R. E. and R. E. Kraft (1991) Design and performance of duct acoustic treatments. Aeroacoustics of Flight Ve hicles: Theory and Practice, vol. 2 H. H. Hubbard. Hampton, VA, NASA Reference Publication 1258. Volume 2: Noise Control: 165-206. Owens, R. E. (1979). Energy Efficient Engi ne Propulsion System-Aircraft Integration Evaluation. NASA/ CR-159488. NASA. Pasqualini, J. P., J. M. Ville and J. F. d. Belleval (1985). "Devel opment of a method of determining the transverse wave struct ure in a rigid wall axisymmetric duct." Journal of the Acoustical Society of America 77(5): 1921-1926. Pickett, G. F., T. G. Sofrin and R. A. We lls (1977). Method of fan sound mode structure determination. NASA-CR-135293. NAS A-Lewis Research Center. Pierce, A. D. (1994). Acoustics: An Intr oduction to its Physic al Principles and Applications Woodbury, NY, Acoustical Society of America.

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298 Pintelon, R., Y. Rolain and W. V. Moer (2002). "Probability density function for frequency response function measuremen ts using periodic signals." IEEE Instrumentation and Measurement T echnology Conference, Anchorage, AK, IEEE. Pintelon, R. and J. Schoukens (2001a). Syst em Indentification: A Frequency Domain Approach New York, IEEE: 33-68. Pintelon, R. and J. Schoukens (2001b). "Measu rement of frequency response functions using periodic excitations, corrupted by correlated input/output errors." IEEE Transactions on Instrumentation and Measurement 50(6): 1753-1760. Randall, R. B. (1987). Frequency Analysis Nrum, Denmark, Brel and Kjr. Rao, S. S. (2002). Applied Numerical Methods for Engineers and Scientists New York, Prentice Hall. Rayleigh, J. W. S. (1945). The Theory of Sound New York, Dover Publications. Ridler, N. M. and M. J. Salter (2002). "An a pproach to the treatment of uncertainty in complex S-parameter measurements." Metrologia 39: 295-302. Rolls-Royce (1996). Noise suppression. The Jet Engine Derby, England, Rolls-Royce plc: 199-205. Salikuddin, M. (1987). "Sound radi ation from single and annular stream nozzles, with modal decomposition of induct acoustic power." Journa l of Sound and Vibration 113(3): 473-501. Salikuddin, M. and R. Ramakrishnan (1987). "Acoustic power measurement for single and annular stream duct-nozzle syst ems utilizing a modal decomposition scheme." Journal of Sound and Vibration 113(3): 441-472. Schmidt, H. (1985). "Resolution bias errors in spectral density, frequency response and coherence function measurements, I-V I." Journal of Sound and Vibration 101(3): 347-427. Schoukens, J., Y. Rolain, G. Simon and R. Pintelon (2003). "Fully automated spectral analysis of periodic si gnals." IEEE Transactions on Instrumentation and Measurement 52(4): 1021-1024. Schultz, T., M. Sheplak and L. Cattafesta ( 2005). "Multivariate uncer tainty analysis and application to the frequency response f unction estimate." Submitted to Journal of Sound and Vibration

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299 Seybert, A. F. and J. F. Hamilton (1978). "Tim e delay bias errors in estimating frequency response and coherence functions." Journal of Sound and Vibration 60(1): 1-9. Seybert, A. F. and D. F. Ross (1977). "Experi mental determination of acoustic properties using a two-microphone random-excitation technique." Journal of the Acoustical Society of America 61(5): 1362-1370. Seybert, A. F. and B. Soenarko (1981). "Erro r analysis of spectral estimates with application to the measurement of acoustic parameters using random sound fields in ducts." Journal of the Acoustical Society of America 69(4): 1190-1199. Smith, M. J. T. (1989). Aircraft Noise Cambridge, Cambridge University Press. Taylor, B. N. and C. E. Kuyatt (1994). Gu idelines for Evaluating and Expressing the Uncertainty of NIST Measurement Result s. Gaithersburg, MD, National Institute of Standards and Technology. Tijdeman, H. (1975). "On the propagation of so und waves in cylindrical tubes." Journal of Sound and Vibration 39(1): 1-33. Underbrink, J. R. (2002). Aeroacoustic Ph ased Array Testing in Low Speed Wind Tunnels. Aeroacoustic Measurements T. J. Mueller. New York, Springer: 165179. Vold, H., J. Crowley and G. T. Rocklin ( 1984). "New ways of estimating frequency response functions." Sound and Vibration : 34-38. Weston, D. E. (1953). "The theory of the propagation of plane sound waves in tubes." Proceeds of the Physical Society of London B66: 695-709. White, F. M. (1991). Viscous Fluid Flow New York, McGraw-Hill. Willink, R. and B. D. Hall (2002). "A classi cal method for uncertainty analysis with multidimensional data." Metrologia 39: 361-369. Yardley, P. D. (1974). Measurement of noi se and turbulence generated by rotating machinery. Southampton, University of Southampton. Ph.D. dissertation. Zinn, B. T., W. A. Bell, B. R. Daniel and A. J. S. Jr ( 1973). "Experimental determination of three-dimensional liquid rocket nozzle admittances." AIAA Journal 11(3): 267272.

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300 BIOGRAPHICAL SKETCH Todd Schultz was born on March 18, 1978, in Winchester, Connecticut. He graduated from Torrington High School in 1996. In the fall of that year, he entered the University of Florida to study aerospace engi neering and earned hi s B.S. degree in May 2001. Todd was award the National Defens e Science and Engineering Graduate Fellowship from the Department of Defens e in the spring of 2001 to attend graduate school at the University of Florida the fo llowing fall. In December 2003, he earned a M.S. degree in mechanical engineering whil e still pursuing his Ph .D. in mechanical engineering with a focus on acoustical measur ements. In the summer of 2004, Todd was award a Graduate Student Researchers Program Fellowship from NASA Langley Research Center to finish his dissertation research.


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ACOUSTIC IMPEDANCE TESTING FOR AEROACOUSTIC APPLICATIONS


By

TODD SCHIULTZ













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006
































Copyright 2006

by

Todd Schultz
















ACKNOWLEDGMENTS

Financial support for the research proj ect was provided by a NASA-Langley

Research Center Grant (Grant # NAG-1-2261). I thank Mr. Michael Jones and Mr. Tony

Parrot at the NASA-Langley Research Center for their guidance and support. I also

thank the University of Florida, Department of Mechanical and Aerospace Engineering,

the NASA Graduate Student Research Program Fellowship, and the National Defense

Science and Engineering Graduate Fellowship administered by the American Society for

Engineering Education for their financial support.

My advisors, Mark Sheplak, Lou Cattafesta, Toshi Nishida, and Tony Schmitz,

deserve special thanks. I thank all of the students in the Interdisciplinary Microsystems

Group, particularly Steve Horowitz, Fei Liu, Ryan Holman, Tai-An Chen and David

Martin, for their assistance and friendship. I thank Paul Hubner for his help with data

acquisition systems and LabVIEW programming. I would also like to thank the staff at

the University of Florida, including Pam Simon, Becky Hoover, Jan Machnik, Mark

Riedy and Teresa Mathia, for helping with the administrative aspects of this project. I

also want to thank Ken Reed at TMR Engineering for machining the equipment needed to

make this proj ect possible.




















TABLE OF CONTENTS


page


ACKNOWLEDGMENT S ............ ...... ._._ .............._ iii..


LI ST OF T ABLE S ........._.._ ......___ .............._ ix...


LIST OF FIGURES .............. ...............x.....


AB STRAC T ......__................ ........_._ ........xvi


CHAPTER


1 INTRODUCTION ................. ...............1.......... ......


1.1 Research Goals .............. ...............10....
1.2 Research Contributions ................. ...............12................
1.3 Dissertation Organization ................ ...............12................


2 ACOUSTIC WAVEGUIDE THEORY ................. ...............13................


2.1 Waveguide Acoustics .............. .... ..... ..............1
2.1.1 Solution to the Wave Equation............... ...............14
2. 1.2 W ave M odes ................. ...............16........... ...
2. 1.3 Phase Speed ................. ...............18........... ...
2. 1.4 Wave Mode Attenuation................ ...............2
2. 1.5 Reflection Coefficient and Acoustic Impedance ................. ................ ..26
2.2 Two-Microphone Method............... ...............28.
2.2.1 Derivation of the TMM ............... ..... ...............29.
2.2.2 Dissipation and Dispersion for Plane Waves .............. ....................3


3 UNCERTAINTY ANALYSIS FOR THE TWO-MICROPHONE METHOD.........36


3.1 Multivariate Form of the TMM Data Reduction Equations ................ ...............37
3.2 TMM Uncertainty Analysis............... ...............38
3.2.1 Multivariate Uncertainty Analysis .............. ...............39....
3.2.2 Monte Carlo Method .............. ...... .... ..............4
3.2.3 Frequency Response Function Estimate............... ...............41
3.2.4 Microphone Locations ................. ...............45...............
3.2.5 Tem perature .................. .... ..... ... ... .......... ............4
3.2.6 Normalized Acoustic Impedance Uncertainty ................. .....................48












3.3 Numerical Simulations .............. ...............48....
3.3.1 Sound-Hard Sample............... ...............50.
3.3.2 Ideal Impedance M odel .............. ...............53....
3.4 Experimental Methodology .............. ...............64....
3.4.1 W ave guides .............. ...............64....
3.4.2 Equipment Description ................. ...............66................
3.4.3 Signal Processing............... ...............6
3.4.4 Procedure ................. ...............66........... ....


4 MODAL DECOMPOSITION METHOD ................. ...............68................


4.1 Data Reduction Al gorithm............... ...............7
4.1.1 Complex Modal Amplitudes .............. ...............71....
4.2.2 Reflection Coefficient Matrix............... ...............73.
4.2.3 Acoustic Impedance .............. ...............74....
4.2.4 Acoustic Power............... ...............75.
4.2 Experimental Methodology .............. ...............77....
4.2. 1W aveguide ................. ...............77........... ....
4.2.2 Equipment Description ................. ...............78........... ....
4.2.3 Signal Processing............... .... ............8
4.2.4 Numerical Study of Uncertainties .............. ...............80....


5 EXPERIMENTAL RESULTS FOR ACOUSTIC IMPEDANCE SPECIMENS ......82


5.1 Ceramic Tubular Honeycomb with 65% Porosity ........__. ........ _.._.............83
5.1.1 TM M Results ................... ........... ...............84....
5.1.2 High Frequency TMM Results ................. ...............87..............
5.1.3 MDM Results and Comparison .................. ...............89..............
5.2 Ceramic Tubular Honeycomb with 73% Porosity ................. .......................97
5.2. 1 TM M Results .................. ........... ...............98.....
5.2.2 High Frequency TMM Results ................. ...............100..............
5.2.3 MDM Results and Comparison ................. ...............101..............
5.3 Rigid Termination............... ..............10
5.3.1 TM M Results .................. ........... ...............109 ...
5.3.2 High Frequency TMM Results ................. ...............111........... ..
5.3.3 MDM Results and Comparison ................. ...............113........... ..
5.4 SDOF Liner .............. ...............120....
5.4. 1 TM M Results ................. .......... ...............121 ....
5.4.2 MDM Results and Comparison ................. ...............123..............
5.5 Mode Scattering Specimen ................. ...............130..............
5.5.1 M DM Results .............. ...............130....


6 CONCLUSIONS AND FUTURE WORK ......._ ......... __ ........._ ......13


6. 1 TMM Uncertainty Analy si s............... .............. 13
6.2 Modal Decomposition Method ................. ...............140...............













APPENDIX


A VISCOTHERMAL LOS SES ................. ...............143........... ...


A. 1 Nondimensionalization and Linearization of the Navier-Stokes Equations......144
A. 1.1 Continuity .............. .... .. .......... ...............147.....
A. 1.2 x -direction Momentum Equation .............. .....................147
A. 1.3 y -direction Momentum Equation ................. .............................148
A. 1.4 Thermal Energy Equation............... ................150
A. 1.5 Equation of State for an Ideal Gas ................. ..... .... ........ ........._ ..151.
A. 1.6 Summary of the Nondimensional, Linearized Equations ......................152
A.2 Boundary Layer Solution............... ...............15
A.2.1 Wall Shear Stress............... ...............157
A.2.2 W all Heat Flux. ............ ...... ...............158...
A.3 M mainstream Flow ................. ...............158._._.. ......
A.3.1 Axial Momentum............... ...............15
A.3.2 Energy Equation ........._..... ... ....._ ... ...............161.
A.3.3 Summary of the Mainstream Flow Equations ................ ........._.._..... 166
A.3.4 Mainstream Flow Wave Equation ..............._ ................ ....._.....167
A.3.5 Dissipation and Dispersion Relations .....__.___ ........___ ................168


B RANDOM UNCERTAINTY ESTIMATES FOR THE FREQUENCY RESPONSE
FUNCTION ................. ...............173......... ......

B. 1 Introduction ................. ...............174........... ...
B.2 Uncertainty Analysis................ ..............17
B.2. 1 Classical Uncertainty Analysis ................ ...............176........... ...
B.2.2 Multivariate Uncertainty Analysis............... ...............17
B.2.2. 1 Fundamentals ................. ........... ...............178...
B.2.2.2 Multivariate uncertainty propagation .............. .. .................18
B.2.2.3 Application: Converting uncertainty from real and imaginary parts
to magnitude and phase............... ...............182.
B.3 Frequency Response Function Estimates ................. .............................185
B.3.1 Output Noise Only System Model ................... ........... ................. .186
B.3.2 Uncorrelated Input/Output Noise System Model .................. ...............191
B.4 Application: Measurement of the FRF Between Two Microphones in a
W aveguide .............. ...............200....
B.5 Conclusions ................. ...............206...............


C FREQUENCY RESPONSE FUNCTION BIAS UNCERTAINTY ESTIMATES..208

C.1 Bias Uncertainty............... ..............20
C.2 Conclusions ................. ...............214.....__ ....


D SOUND POWER FOR WAVES PROPAGATING IN A WAVEGUIDE ..............215













E MODAL DECOMPOSITION METHOD NUMERICAL ERROR STUDY ..........219


E. 1 Signal-to-Noise Ratio ............ ...... ..._. ...............222...
E.2 Microphone Phase Mismatch ......... ........_____ ......... ...........22
E.3 Microphone Locations................ ..............22
E.4 Speed of Sound ................. ...............228...............
E.5 Frequency ........._..... ...._... ...............229....
E.6 Conclusions ............ ........... ...............2 1....


F AUXILIARY GRAPHS .............. ...............233....


F.1 CT65 .............. ...............23 3...
F.2 CT73 ................ ...............235...
F.3 Rigid Termination .............. ...............237....
F.4 SDOF Liner ....._._ ................ ...............239 ....


G COMPUTER CODES .............. ...............241....


G. 1 TMM Program Files .............. .... ...............241..
G. 1.1 TMM Program Readme File............... ...............241.
G. 1.2 Pulse to MATLAB Conversion Program .............. ....................24
G. 1.3 TMM Main Program .............. ...............244....
G. 1.4 TMM Subroutine Program .................. .......... ...............25
G. 1.5 TMM Subroutine for the Analytical Uncertainty in R ............................255
G. 1.6 TMM Subroutine for the Analytical Uncertainty in Z ............................257
G. 1.7 TMM Subroutine for the Monte Carlo Uncertainty Estimates ................258
G.2 Uncertainty Subroutines .............. ..... .... .... ..........5
G. 2.1 Frequency Response Function Uncertainty ................. ........__ .......259
G.2.2 Averaged FRF Uncertainty .....__.....___ ............. ...........26
G.2.3 Effective Number of degrees of Freedom ................. ......._.. .........262
G.2.4 Numeric Computation of Bivariate Confidence Regions.............._._......263
G.2.5 Analytical Propagation of Uncertainty from Rectangular Form to Polar
Form ..........._.... .... ...... ._ .. ...............265...
G.3 Multivariate Statistics Subroutines .............. ...............266....
G.3.1 Computation of Bivariate PDF ................. .... ...............266
G. 3.2 Numerical Computation of Constant PDF Contours .............. ..............267
G.3.3 Multivariate Normal Random Number Generator ................. ...............268
G.4 Modal Decomposition Programs .............. ...............268....
G.4.1 Pulse to MATLAB Conversion Program .............. ....................26
G. 4.2 MDM Main Program ............. ... ....._ .. ...............27
G.4.3 MDM Subroutine to Compute the Decomposition............... ..............8
G.4.4 MDM Plotting Subroutine.............__ .........__ ............._ ........28
G.4.5 MDM Mode Scattering Coefficients Plotting Subroutine ....................290


LIST OF REFERENCES .........____... ...___ ............. ..294.












BIOGRAPHICAL SKETCH .............. ...............300....


















LIST OF TABLES


Table pg

2-1 Cut-on frequencies in k
2-2 Cut-on frequencies in k
2-3 Minimum frequencies to keep effects of dispersion and dissipation <5%...............3 5

3-1 Elemental bias and precision error sources for the TMM. ................... ...............43

3-2 Nominal values for input parameters of numeric simulations. ............. .................50

4-1 Cut-on frequencies in k
4-2 Microphone measurement locations (a = 25.4 mm). ............. .....................7

A-1 Minimum frequency required for series expansion for the two waveguides for air at
200C............... ...............170.

C-1 Simulation results for the second-order system. ............. ...............212....

E-1 Power in Pa2 for all signals from all simulation sources ................. ................. .220


















LIST OF FIGURES


Figure pg

1-1 Illustration of the approach, takeoff, and cutback flight segments and measurement
points. ............. ...............2.....

1-2 Typical noise sources on an aircraft. .............. ...............4.....

1-3 Component noise levels during approach, cutback, and take-off for a Boeing 767-
300 with GEAE CF6-80C2 engines. .............. ...............4.....

1-4 Comparative overall noise levels of various engine types. ........._.... ........_........5

1-5 Engine cutaway showing the acoustic liner locations .................... ...............7

1-6 An example of a SDOF liner showing the atypical honeycomb and the perforate
face sheet. .............. ...............7.....

1-7 An example of a 2DOF liner. .............. ...............8.....

2-1 Illustration of the waveguide coordinate system. ........._. ............ ..............14

2-2 Illustration of the first four mode shapes. .........._.... ........_ ......_..._......1

2-3 Illustration of the wave front and the incidence angle to the waveguide wall and to
the termination................ .............2

2-4 Phase speed versus frequency for the first four modes. ............. .....................2

2-5 Angle of incidence to the sidewall versus frequency for the first four modes.........21

2-6 Angle of incidence to the termination versus frequency for the first four modes....22

2-7 Attenuation of higher-order modes in the large waveguide over a distance of 25.4
m m ............... ...............24...

2-8 Attenuation of higher-order modes in the small waveguide over a distance of 8.5
m m ............... ...............24...

2-9 Attenuation of the first higher-order mode ((1,0) or (0,1)) in the large waveguide at
the microphone locations used for the TMM experiments. .................. ...............26










2-10 Reflection and transmission of a wave off an impedance boundary. ................... ....28

2-11 Experimental setup for the TMM. .............. ...............29....

3-1 Flow chart for the Monte Carlo methods. ............. ...............51.....

3-2 Absolute uncertainty of Roo,oo due to the uncertainties in 1, s, and Tfor

Roo,oo = 0.999 at f~ 5 k

3-3 Absolute uncertainty Ro,oo, due to the SNR for Ro,oo, = 0.999 at f~5 kHz..............53


3-4 Estimated value for the (a) reflection coefficient and (b) total uncertainty for the
sound-hard boundary ...........__......___ ...............54....

3-5 Ideal impedance model and estimated values for (a) reflection coefficient and (b)
normalized specific acoustic impedance. ............. ...............56.....

3-6 Absolute uncertainty of (a) Ro,oo, and (b) ac, due to the uncertainties in 1, s, and T
for the ideal impedance model at f~5 k
3-7 Absolute uncertainty in (a) Ro,oo, (b) ~sac due to the SNR for the ideal impedance
model at/f 5 k
3-8 Total uncertainty in (a) Ro,oo, and (b) ac, as a function of frequency for the ideal
impedance model............... ...............60.

3-9 The confidence region contours for the resistance and reactance for the ideal
impedance model at 5 k
3-10 Confidence region of the ideal impedance model at 5 k uncertainty and 40 dB SNR. ........... _......__ ...............63.

4-1 Schematic of the experimental setup for the MDM. ................ .......................78

4-2 Schematic of the four restrictor plates ................. ...............79......__. ..

5-1 Photograph of the CT65 material ....__. ................ ............... ........ ...83

5-2 Reflection coefficient for CT65 for the TMM. ............. ...............86.....

5-3 Normalized specific acoustic impedance estimates for CT65 via TMM. ................86

5-4 Reflection coefficient for CT65 for the high frequency TMM. ............. ................88

5-5 Normalized specific acoustic impedance estimates for CT65 via the high frequency
TM M ............. ...............89.....











5-6 Incident pressure field for the MDM for CT65 ................. ....._._ ................ 90

5-7 Reflected pressure field for the MDM for CT65................ ...............90.

5-8 Absorption coefficient for CT65 .............. ...............91....

5-9 Comparison of the reflection coefficient estimates for CT65 via all three methods.92

5-10 Mode scattering coefficients for CT65 from the (0,0) mode to the other propagating
m odes. ............. ...............93.....

5-11 Mode scattering coefficients for CT65 from the (1,0) mode to the other propagating
m odes. ............. ...............93.....

5-12 Mode scattering coefficients for CT65 from the (0,1) mode to the other propagating
m odes. ............. ...............94.....

5-13 Mode scattering coefficients for CT65 from the (1,1) mode to the other propagating
m odes. ............. ...............94.....

5-14 Comparison of the acoustic impedance ratio estimates for CT65 via all three
m ethod s. ............. ...............96.....

5-15 Comparison of the normalized specific acoustic impedance estimates for CT65 via
all three methods. ............. ...............97.....

5-16 Photograph of the CT73 material. ............. ...............98.....

5-17 Reflection coefficient for CT73 for the TMM. ............. ...............99.....

5-18 Normalized specific acoustic impedance estimates for CT73 via the TMM. .........99

5-19 Reflection coefficient for CT73 for the high frequency TMM. ............. ................100

5-20 Normalized specific acoustic impedance estimates for CT73 via the high frequency
TM M .......... ................ ...............101......

5-21 Incident pressure field for the MDM for CT73 ................ .......... ...............102

5-22 Reflected pressure field for the MDM for CT73 ......____ ........_ ..............103

5-23 Absorption coefficient for CT73 ................. ....__ ...._ ...............103

5-24 Comparison of the reflection coefficient estimates for CT73 via all three methods.104

5-25 Mode scattering coefficients for CT73 from the (0,0) mode to the other propagating
m odes. ............. ...............105....











5-26 Mode scattering coefficients for CT73 from the (1,0) mode to the other propagating
m odes. ............. ...............105....

5-27 Mode scattering coefficients for CT73 from the (0,1) mode to the other propagating
m odes. ............. ...............106....

5-28 Mode scattering coefficients for CT73 from the (1,1) mode to the other propagating
m odes. ............. ...............106....

5-29 Comparison of the acoustic impedance ratio estimates for CT73 via all three
m ethods. ............. ...............108....

5-30 Comparison of the normalized specific acoustic impedance estimates for CT73 via
all three methods. ............. ...............108....

5-3 1 Photograph of the rigid termination for the large waveguide. ............. ..............109

5-32 Reflection coefficient for the rigid termination for the TMM. ............._ .............110

5-33 Standing wave ratio for the rigid termination measured by the TMM. ..................1 11

5-34 Reflection coefficient for the rigid termination for the high frequency TMM. .....112

5-35 SWR for the rigid termination calculated from the high frequency TMM. ...........112

5-36 Triangle restrictor plate. ................ ...............114......... .....

5-37 Incident pressure field for the MDM for the rigid termination .............. ..... ..........114

5-38 Reflected pressure Hield for the MDM for the rigid termination. ........................115

5-39 Power absorption coeffieient for the rigid termination for the MDM..................115

5-40 Comparison of the reflection coefficient estimates for the rigid termination via all
three methods. ........... _...... __ ...............117..

5-41 SWR for the rigid termination calculated from the MDM. .............. ................11 7

5-42 Mode scattering coefficients for rigid termination from the (0,0) mode to the other
propagating modes. ................ ...............118......... ......

5-43 Mode scattering coefficients for rigid termination from the (1,0) mode to the other
propagating modes. ................ ...............118......... ......

5-44 Mode scattering coefficients for rigid termination from the (0, 1) mode to the other
propagating modes. ................ ...............119......... ......

5-45 Mode scattering coefficients for rigid termination from the (1,1) mode to the other
propagating modes. ................ ...............119......... ......











5-46 SDOF liner showing the irregular honeycomb and perforated face sheet. ............120

5-47 Reflection coefficient for the SDOF specimen for the TMM. ............. ................122

5-48 Normalized specific acoustic impedance estimates for the SDOF specimen via
TM M .......... ................ ...............122......

5-49 Incident pressure field for the MDM for the SDOF specimen. .............. .............124

5-50 Reflected pressure field for the MDM for the SDOF specimen. ...........................124

5-51 Power absorption coefficient for the SDOF specimen for the MDM. ................126

5-52 Comparison of the reflection coefficient estimates for the SDOF specimen via the
TMM and MDM ................. ...............126................

5-53 Mode scattering coefficients for SDOF specimen from the (0,0) mode to the other
propagating modes. ............. ...............127....

5-54 Mode scattering coefficients for SDOF specimen from the (1,0) mode to the other
propagating modes. ............. ...............127....

5-55 Mode scattering coefficients for SDOF specimen from the (0, 1) mode to the other
propagating modes. ............. ...............128....

5-56 Mode scattering coefficients for SDOF specimen from the (1,1) mode to the other
propagating modes. ............. ...............128....

5-57 Comparison of the acoustic impedance ratio estimates for the SDOF specimen via
the TMM and MDM ................. ...............129...............

5-58 Comparison of the normalized specific acoustic impedance estimates for the SDOF
specimen via the TMM and MDM ................. ...............129..............

5-59 Photograph of the mode scattering specimen ................. ................. ........ 130

5-60 Incident pressure field for the MDM for the mode scattering specimen................132

5-61 Reflected pressure field for the MDM for the mode scattering specimen. ............132

5-62 Power absorption coefficient for the mode scattering specimen for the MDM.....133

5-63 Comparison of the reflection coefficient estimates for the mode scattering specimen
via the M DM ............. ...............134....

5-64 Mode scattering coefficients for the mode scattering specimen from the (0,0) mode
to the other propagating modes. ............. ...............135....











5-65 Mode scattering coeffieients for the mode scattering specimen from the (1,0) mode
to the other propagating modes. ............. ...............135....

5-66 Mode scattering coefficients for the mode scattering specimen from the (0, 1) mode
to the other propagating modes. ............. ...............136....

5-67 Mode scattering coefficients for the mode scattering specimen from the (1,1) mode
to the other propagating modes. ............. ...............136....

5-68 Comparison of the acoustic impedance ratio estimates for the mode scattering
specimen via the MDM. .............. ...............137....

5-69 Comparison of the normalized specific acoustic impedance estimates for the mode
scattering specimen via the MDM. ............. ...............137....

A-1 Oscillating flow over a flat plate. .............. ...............145....

A-2 Control volume showing the external forces and flows crossing the boundaries. .161

A-3 Control volume showing the external heat fluxes and flows crossing the
boundaries. ............. ...............166....

B-1 A plot of the raw data and estimates for a randomly generated complex variable..183

B-2 A plot of the raw data and estimates in polar form for a randomly generated
complex variable. ............. ...............185....

B-3 System model with output noise only. ............. ...............187....

B-4 System model with uncorrelated input/output noise. .............. ...................19

B-5 Bode plot of the true FRF and the experimental estimate. .............. ..............199

B-6 Magnitude and phase plot of the uncertainty estimates. ................ ................199

B-7 The experimentally measured FRF between the two microphones. ......................205

B-8 Comparison for the uncertainty estimated by the multivariate method and by the
direct statistics. ............. ...............205....

C-1 FRF for the simulation with random noise............... ...............213.

E-1 The rms normalized error for the modal coefficients versus noise power added to
the si gnal s. .............. ...............223....

E-2 The rms normalized error for the reflection coefficient matrix versus noise power.224

E-3 The rms normalized error versus the number of averages for a noise power of 0.01
Pa2.............. ...............224..










E-4 The rms normalized error for the reflection coefficient versus the number of
averages for a noise power of 0.01 Pa2. ............ ...............225.....

E-5 The rms normalized error for the modal coefficients versus a phase error applied to
microphone 4 in group 2 for each source. .............. ...............225....

E-6 The rms normalized error for the reflection coefficient matrix versus a phase error
applied to microphone 4 in group 2 for all sources ................. ............ .........226

E-7 The rms normalized error for the modal coefficients versus a microphone location
error applied to microphone 1 in group 1 for each source. ............. ...................227

E-8 The rms normalized error for the reflection coefficient matrix versus a microphone
location error applied to microphone 1 in group 1 for all sources. ........................227

E-9 The rms normalized error for the modal coefficients versus a temperature error. 228

E-10 The rms normalized error for the reflection coefficient matrix versus a temperature
error. .............. ...............229....

E-11 The rms normalized error for the modal coefficients versus frequency. .............230

E-12 The rms normalized error for the reflection coefficient matrix versus frequency. 230

F-1 Ordinary coherence function for the TMM measurement of CT65 .....................233

F-2 The measured FRF for CT65 for the TMM. ............. ...............234....

F-3 Ordinary coherence function for the high frequency TMM measurement of CT65.234

F-4 The measured FRF for CT65 for the high frequency TMM. ............. .................235

F-5 Ordinary coherence function for the TMM measurement of CT73 .................. .....235

F-6 The measured FRF for CT73 for the TMM. ............. ...............236....

F-7 Ordinary coherence function for the high frequency TMM measurement of CT73.236

F-8 The measured FRF for CT73 for the high frequency TMM. ............. .................237

F-9 Ordinary coherence function between the two microphones for the TMM
measurement of the rigid termination. ................ ...............237........... ...

F-10 The measured FRF for the rigid termination for the TMM ................. ................238

F-11 Ordinary coherence function for the high frequency TMM measurement of the rigid
term nation. ............. ...............238....

F-12 The measured FRF for the rigid termination for the high frequency TMM...........239










F-13 Ordinary coherence function for the TMM measurement of the SDOF specimen.239

F-14 The measured FRF for the SDOF specimen for the TMM. ............. ..................240
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ACOUSTIC IMPEDANCE TESTING FOR AEROACOUSTIC APPLICATIONS

By

Todd Schultz

August 2006

Chair: Mark Sheplak
Cochair: Louis N. Cattafesta
Major Department: Mechanical and Aerospace Engineering

Accurate acoustic propagation models are required to characterize and

subsequently reduce aircraft engine noise. These models ultimately rely on acoustic

impedance measurements of candidate materials used in sound-absorbing liners. The

standard two-microphone method (TMM) is widely used to estimate acoustic impedance

but is limited in frequency range and does not provide uncertainty estimates, which are

essential for data quality assessment and model validation. This dissertation presents a

systematic framework to estimate uncertainty and extend the frequency range of acoustic

impedance testing.

Uncertainty estimation for acoustic impedance data using the TMM is made via

two methods. The first employs a standard analytical technique based on linear

perturbations and provides useful scaling information. The second uses a Monte Carlo

technique that permits the propagation of arbitrarily large uncertainties. Both methods

are applied to the TMM for simulated data representative of sound-hard and sound-soft


XV111









acoustic materials. The results indicate that the analytical technique can lead to false

conclusions about the magnitude and importance of specific error sources. Furthermore,

the uncertainty in acoustic impedance is strongly dependent on the frequency and the

uncertainty in the microphone locations.

Next, an increased frequency range of acoustic impedance testing is investigated

via two methods. The first method reduces the size of the test specimen (from 25.4 mm

square to 8.5 mm square) and uses the standard TMM. This method has issues

concerning specimen nonuniformity because the small specimens may not be

representative of the material. The second method increases the duct cross section and,

hence, the required complexity of the sound Hield propagation model. A comparison

among all three methods is conducted for each of the three specimens: two different

ceramic tubular specimens and a single degree-of-freedom liner. The results show good

agreement between the TMM and the modal decomposition method for the larger

specimens, but the methods disagree for the smaller specimen size. The results for the

two ceramic tubular materials show a repeating resonant pattern with a monotonic

decrease in the resonant peaks of the acoustic resistance with increasing frequency. Also,

significant mode scattering is evident in most of the specimens tested.















CHAPTER 1
INTTRODUCTION

Modern society has increasingly demanded a safer, more pleasant living

environment. Studies have also shown that exposure to noise pollution has adverse

health effects such as hearing impairment, reduced speech perception, sleep deprivation,

increased stress levels, and general annoyance (Berglund et al. 1999). These results have

led to increased noise restrictions on industrial factories, automobiles, and aircraft. The

common element of these sources is that they all produce a complex noise spectrum and

broad bandwidth. Yet increased noise restrictions have been readily met because of an

increased research effort in acoustics, involving sound generation, propagation, and

suppression (Golub et al. 2005).

The aerospace industry has been a maj or focus for increased noise regulations due

to community noise concerns around commercial airports (Motsinger and Kraft 1991;

Berglund, Lindvall and Schwela 1999). Takeoff, landing and cutback are the flight

segments of greatest relevance to community noise concerns because of the relative

proximity of the aircraft to the community. These flight segments and reference

measurement points are shown in Figure 1-1. During these flight segments, the

configuration of the aircraft is altered from the clean cruise configuration via the

deployment of high-lift devices and landing gear. Also, during take-off the engines

operate at full power, further increasing the noise levels. In order to reduce the fly-over

noise, the power to the engines is reduced after take-off, during climb or cutback when

the aircraft is still relatively close to the ground near populated areas. The overall noise









level from the aircraft has contributions from many sources as shown in Figure 1-2.

These contributions can be separated into two broad categories: airframe noise and

engine noise. Examples of airframe noise include noise generated from flaps, slats,

landing gear, and vertical and horizontal tails. Engine noise consists of jet noise from the

exhaust, combustion noise, turbomachinery noise, and the noise due to the integration of

the engines with the airframe. The effective perceived noise levels (EPNL) (Smith 1989)

of the component noise sources for the three flight segments are shown in Figure 1-3 for

a Boeing 767-300 with GEAE CF6-80C2 engines. The figure shows that the major

contributors to the total aircraft noise are jet and fan noise for takeoff, jet noise for

cutback, and inlet, fan and airframe noise for approach. Thus, to reduce aircraft noise for

takeoff and cutback, engine noise should be reduced, whereas airframe noise must be

considered for the approach flight segment.

















Figure 1-1: Illustration of the approach, takeoff, and cutback flight segments and
measurement points (adapted from Smith 1989).

Early commercial aircraft used turbojet engines, and the resulting noise was

dominated by the jet noise component. With the advent of ultra high bypass turbofan

engines, the dominant noise sources for modern commercial jet aircraft are now engine









noise (during takeoff) and airframe noise (during approach). Because it was the

dominant noise source in earlier j et aircraft, j et noise has been studied for many decades.

Lighthill's analogy can be used to understand the scaling issues in the evolution of the j et

engine noise from the first turbojet engines to modern high-bypass-ratio turbofan engines.

Lighthill's analogy relates the mean square value of the radiated density perturbations

(p'2 ) fTOm a subsonic turbulent j et to the velocity and diameter of the j et as (Dowling

and Ffowes-Williams 1983)

r2 2 D2
pr2 p2M' 2


where po is the mean or atmospheric density, M~ is the exit Mach number of the jet, D

is the diameter of the jet and r is the distance from the jet. Equation (1.1) is only valid

for subsonic flows and shows that the magnitude of the sound from a j et is more

dependent on the velocity of the j et than the size of the j et. In particular, the mean square

density perturbations are proportional to the eighth power of the Mach number but only

to the second power of the diameter. A comparative chart of the perceived noise levels

from various engine types is given in Figure 1-4. The first generation of jet aircraft relied

on propulsion from a single, high velocity j et from the aft of the engine (Rolls-Royce

1996). This generated a tremendous amount of noise, as seen from Lighthill's analogy.

Subsequently, noise suppression devices for these engines have been developed to

reduce the noise generated from the jets. The devices included suppressor nozzles that

promote rapid mixing of the exhaust j et with the ambient fluid (Owens 1979). The

overall effect is to quickly reduce the velocity of the j et and thus reduce the noise levels.

The next development was the low-bypass-ratio turbofan engine (Rolls-Royce 1996).













OII, JVertical tail
eness Slat Tip vortices


Nose landing/ o a
gear Horizontal tail


Engines lni Flap/side-edge
\/ Man ladingvortices
gear
Figure 1-2: Typical noise sources on an aircraft (adapted from Crighton 1991).




Inlet

ARl fan

Contbustor




Total Airfranse

Total Aircraft


60 65 70 75 80 85 90 95 100
EPF\L [dB]

Figure 1-3: Component noise levels during approach, cutback, and take-off for a Boeing
767-300 with GEAE CF6-80C2 engines (adapted from Golub et al. 2005).

For this engine, the maj ority of the propulsion force is generated by the bypass flow with


an increased area, and at the exit the bypass air is mixed with the j et, significantly


lowering the exit Mach number. The diameter of the engine and j et was enlarged, but


noise levels were significantly reduced because of the lower-velocity j et from the fan and


the mixing of the two j ets. Modern turbofan engines use high-bypass-ratio inlets, with a


bypass ratio of approximately three or greater (Rolls-Royce 1996). The diameter of the

fans on these engines can thus be 2.5 m or larger. The diameter of the exiting air flow is


increased, the mixing is increased, and the jet velocity is decreased, thus lowering the










propagated noise. The growth of the diameter from these large high-bypass turbofans is

now restricted by problems related to the large weight and large frontal area, such as drag

(Rolls-Royce 1996).



120-
Turbojets without noise suppressors

Turbojets with noise suppressors
110-
I Low bypass ratio turbofans
I with noise suppresors

w High bypass ratio turbofans
100-
with noise suppressors
Overall trend



00 years
Figure 1-4: Comparative overall noise levels of various engine types (adapted from Rolls-
Royce 1996).

Since the use of high-bypass-ratio turbofans has reduced the perceived noise levels

from the jet by approximately 20 dB, other noise sources have become important

contributors to the overall noise level of the aircraft (Smith 1989). Two such sources are

engine noise (other than j et noise) and airframe noise. To reduce the engine noise,

designs have focused on acoustic treatments to the interior of the engine nacelles to alter

the propagation of the sound and reduce the radiation of noise from the engine into the

far-field (Motsinger and Kraft 1991). These nacelle liners are placed at various locations

throughout the engine to suppress noise from a particular region, as shown in Figure 1-5.

The liners minimize the radiation of sound by altering the acoustic impedance boundary

condition along the walls of the nacelle. The acoustic impedance, which is the complex









ratio of the acoustic pressure to the acoustic volume velocity, is a property of the liner

configuration and materials. As shown in Figure 1-6, typical single degree-of-freedom

(SDOF) liners are a composite structure of a layer of honeycomb support sandwiched

between a solid backing sheet and a perforated face sheet. These liners act as Helmholtz

resonators and are used to attenuate the noise spectrum (Motsinger and Kraft 1991; Rolls-

Royce 1996). The bandwidth over which a SDOF liner is effective is about one octave,

centered around its resonant frequency (Motsinger and Kraft 1991). If the liner has two

layers of honeycomb separated by a second perforate face sheet, the liner is called a two

degree-of-freedom (2DOF) liner as shown in Figure 1-7. The 2DOF liner has two

resonant frequencies and a larger bandwidth, about two octaves, of effectiveness relative

to the SDOF liner, but weigh more than SDOF liner (Motsinger and Kraft 1991; Bielak et

al. 1999). Another type of liner uses a bulk absorber, which is designed to attenuate

sound over a broad bandwidth. These liners are less effective at reducing the propagation

of engine at a given frequency as compared to the SDOF or 2DOF liners, and usually are

not able to provide structural support (Motsinger and Kraft 1991; Bielak, Premo and

Hersh 1999). Typical materials used for bulk absorbers include woven wire mesh,

ceramic tubular materials, and acoustic foams and fibers such as polyurethane, melamine,

fiberglass, etc.

When designing an engine nacelle for noise suppression, semi-empirical analytical

models can be used to Eind the optimal acoustic impedance for acoustic treatment

(Motsinger and Kraft 1991). Potential liner candidates must be experimentally tested to

determine their acoustic impedance. The experimentally measured values can then be

used in new models to predict the noise levels from the engine for that particular










configuration. Scale model and full-size engine testing can be done to verify the noise

level predictions and to certify the engine.


Stators


Exhaust


Inlet


Figure 1-5: Engine cutaway showing the acoustic liner locations (adapted from
Groeneweg et al. 1991; Rolls-Royce 1996).


Honeycomb layer


Perforate face
.-' '"-' sheet

Solid back sheet I--:




Figure 1-6: An example of a SDOF liner showing the atypical honeycomb and the
perforate face sheet (courtesy of Pratt and Whitney Aircraft).

One of the limiting factors for the computational models is the experimental

database for the acoustic properties of any material used for noise control (Kraft et al.

1999; Kraft et al. 2003). Current applications require extending the frequency range of









acoustic impedance testing out to 20 k
(Kraft, Yu, Kwan, Echternach, Syed and Chien 1999; Kraft, Yu, Kwan, Beer, Seybert

and Tathavadekar 2003). Existing methods for measuring normal-incident acoustic

impedance have their limitations. Of these, the most noticeable limitation is the

frequency range within which the methods are valid. Therefore, existing sound

propagation models must extrapolate the acoustic impedance to the frequency range of

interest for applications. This introduces a potentially large source of error in the models.

Better results could be realized if the actual acoustic impedance of the materials could be

measured in the frequency range of interest.













Honeycomb layers

Perforate face
Solid back sheet set


Figure 1-7: An example of a 2DOF liner (adapted from Rolls-Royce 1996).

The Two-Microphone Method (TMM) (Seybert and Ross 1977; Chung and Blaser

1980; ASTM-E1050-98 1998; ISO-10534-2:1998 1998) and the Multi-Point Method

(MPM) (Jones and Parrott 1989; Jones and Stiede 1997) are two techniques to determine

the normal-incidence acoustic impedance of materials. For the TMM, a compression

driver is mounted at one end of a rigid-walled waveguide and the test specimen is

mounted at the other end (Figure 2-11). Two microphones are flush-mounted in the duct









wall at two locations along the tube near the specimen to measure the incident and

reflected waves with respect to the sample. The data are used to estimate the complex

reflection coefficient and the corresponding acoustic impedance of the test specimen.

More detailed information on the TMM is presented later in Chapter 2. The test

procedure for the MPM is similar to the TMM except the number of microphones is

increased and the computations rely on a least-squares approach. However, the MPM

still assumes that only plane waves exist in the waveguide. Since the TMM is supported

by ISO and ASTM standards, it is the method used in this dissertation.

Both methods have produced results for materials up to a frequency of

approximately 12 k
frequencies. However, in order to do, the specimen size of the material would need to be

reduced to maintain the plane wave assumption, since the upper frequency limit of the

method is inversely proportional to the specimen size or waveguide dimensions (ASTM-

E1050-98 1998; ISO-10534-2:1998 1998). The size is limited in order to prevent the

propagation of higher-order modes and thus maintain the plane wave assumption. For

square cross-section of length a, the maximum frequency for plane waves, fanewave, is

(Blackstock 2000)


f < o0 (1.2)

where co is the isentropic speed of sound inside the waveguide. A specimen of 25.4 mm

by 25.4 mm is limited to a frequency range up to approximately 6.7 k
using the TMM, but a specimen of 8.5 mm by 8.5 mm has a frequency range up to 20

k
and in local material variations that can cause changes in the measured acoustic










impedance. The installation and fabrication issues arise from having to cut a finite

specimen, often resulting in damage to its edges. Furthermore, the smaller the specimen

size, the larger the percentage of the total area composed of the damaged edges. For the

local material variations, testing a large number of specimens can quantify these

statistical variations. However, this approach is time consuming and costly.

Another method to increase the frequency range is to permit the propagation of

higher-order modes (A+bom 1989; Kraft et al. 2003). This allows large specimens but

increases the complexity of the measurement setup and data reduction routine. For a 25.4

mm-square duct, the bandwidth is increased to 13.5 k
or to 20 k
decomposition method (MDM) is that the higher-order modes can also be modeled as

plane waves at oblique angles of incidence; thus this method can yield information

regarding the effects of angle of incidence. The oblique-incidence information can used

to verify the local reactivity assumption, which states that the acoustic impedance is

independent of the angle of incidence (Dowling and Ffowes-Williams 1983).

Before the frequency range can be extended, the accuracy of the existing methods

must be understood. Accurate uncertainty estimates give insight into how errors scale

versus frequency and will aid in the design of new measurement techniques and

improved liners. Without understanding uncertainty, there is no way to ensure that such

measurements will meet the needs of the aeroacoustic application.

1.1 Research Goals

The focus of this dissertation is to increase the frequency range of acoustic

impedance measurement technology to the range of interest in aeroacoustic applications

and to supply experimental uncertainty estimates with the data. These data will help to









evaluate potential liner candidates and improve the accuracy of models of the sound Hield.

Also, design procedures and codes that predict the acoustic impedance of typical liners

can be validated using the acoustic impedance data with corresponding uncertainty

estimates. The implementation of improved experimental techniques and corresponding

uncertainty analyses with existing design and computational tools will assist in the

reduction of time and cost required to meet community noise restrictions.

To meet the goal of extending the frequency range of acoustic impedance

measuring technology, two different approaches are used. The first seeks to reduce the

size of the cross-section of the waveguide and test specimen to increase the cut-on

frequency for the first higher-order mode to 20 k
will limit the specimen size to 8.5 mm by 8.5mm and thus the above mentioned

specimen size issues may affect the results. The second approach is to keep the specimen

at 25.4mm by 25.4mm and allow for the propagation of higher order modes. A direct

Modal Decomposition Method (MDM) is used that allows for and computes the

amplitudes of the incident and reflected waves for the higher-order modes. This allows

the frequency range to increase. This method will provide a comparison for the data

measured with the small specimen to elucidate any issues associated with the specimen

size. Also, acoustic impedance data at angles of incidence other than perpendicular to the

specimen surface are measured, because higher-order modes can also be thought of as

plane waves traveling at an angle with respect to the axis of the duct.

Before either path is pursued, two techniques are first developed to estimate the

uncertainty for the complex reflection coefficient. One method is an analytical approach

that provides scaling information, and the other is a Monte Carlo method that is not










restricted to small perturbations. The two methods are compared to each other to help

determine their strengths and weaknesses.

1.2 Research Contributions

The contributions of this dissertation to the aeroacoustic community are as follows.

* Development of an analytical and a numerical method for the propagation of
experimental uncertainty in data reduction routines with complex variables.

* Application of the uncertainty analysis methods to the Two-Microphone Method.

* Application of a Modal Decomposition Method for measuring normal-incident
acoustic impedance in the presence of higher-order modes in the waveguide.

* Comparison of experimental data with uncertainty estimates for acoustic
impedance from the TMM and the MDM. The specimens compared are a rigid
termination and two ceramic tubular materials.

1.3 Dissertation Organization

This dissertation is organized into six chapters. This chapter introduced and

discussed the motivation for the research present in this dissertation. The next chapter

reviews the theory of acoustic waveguides. The derivation of the TMM is presented

there as well. Chapter 3 presents the derivation and application of the uncertainty

methods for the TMM and includes a discussion of the issues present when increasing the

bandwidth of the TMM up to 20 k
accounts for the propagation of higher-order modes through the waveguide. This chapter

presents the derivation of this method and a discussion of the requirements for the data

acquisition hardware to ensure reasonable accuracy. Chapter 5 presents detailed

experimental results for different acoustic impedance specimens. The final chapter offers

concluding remarks and future directions.















CHAPTER 2
ACOUSTIC WAVEGUIDE THEORY

This chapter introduces the basic analytical analysis for rectangular duct acoustic

waveguides. First, the acoustic wave equation is presented, and its solution is given.

Next, a discussion of the solution properties is presented. Then this chapter concludes

with a derivation of the TIVM.

2.1 Waveguide Acoustics

A waveguide is a device that is used to contain and direct the propagation of a

wave. A simple example of an acoustic waveguide is a plastic tube with a sound source

at one end. For simple geometries of the internal cross section, the exact sound field in

the waveguide can be solved from the linear lossless acoustic wave equation, as long as

the wave equation assumptions are not violated (Pierce 1994; Blackstock 2000). The

lossy wave equation can also be solved for some simple cases but an ad hoc method will

be introduced in a later section in this chapter to account for attenuation. The linear

lossless acoustic wave equation assumes that an acoustic wave is isentropic, the pressure

perturbations are small compared to the medium' s bulk modulus (p,c2 ), and that there is


no mean flow of the medium. Under these conditions, the wave equation for pressure

fluctuations is

1 8 p'
SV p' = 0, (2.1)
c,2 dt









where p' is the acoustic pressure, co is the isentropic propagation speed given by


co = iyR, 7 is the ratio of specific heats, Reas is the ideal gas constant, T is the

absolute temperature, t is time and V2 is the Laplacian operator. First, let the coordinate

system for the waveguide be defined as a Cartesian coordinate system with the z-axis

aligned with the axis of the tube, and the origin located at a corner of the tube as shown in

Figure 2-1. The d-axis as shown is a useful auxiliary coordinate axis. Note that the d-

direction is the direction along the axis of the tube and is the propagation direction for

reflected waves.











Figure 2-1: Illustration of the waveguide coordinate system.

2.1.1 Solution to the Wave Equation

Solutions to the wave equation are present in many sources such as (Rayleigh

1945), (Dowling and Ffowes-Williams 1983), (Pierce 1994), (Morse and Ingard 1986),

(Kinsler et al. 2000) and (Blackstock 2000). For the solution presented in this section,

the acoustic pressure signal is assumed to be time-harmonic which states

p' =Re Pe'"cot (.2)

where P is the complex acoustic pressure amplitude and co, is the angular frequency.

After substituting Equation (2.2) into the acoustic wave equation, it reduces to

V2P+ k2 P = 0, (2.3)









which is known as Helmholtz's Equation (Pierce 1994; Blackstock 2000). The constant

k is the wavenumber, which is defined as


k =Z (2.4)


The general solution to the Helmholtz equation, assuming propagation in the d' -direction,

is a summation of normal modes given as


P = CC Cv,,, (x, y) A4,, e y k + B,,,, e- y d) (2.5)


where ~j= J-l, A_,,, and B_,,, are the complex modal amplitudes of the incident and

reflected wave, respectively, m and n are the mode numbers, kZ is the propagation

constant, and ry(x, y) is the transverse factor. The transverse factor is a product of two

eigenfunctions determined by the boundary conditions. For the waveguides used in this

dissertation, the tube walls are assumed to be sound-hard or rigid and therefore do not

vibrate or transmit sound. Practically, the sound-hard boundary condition can be realized

for a gaseous medium by utilizing tube walls made of a thick, rigid material, such as steel

or aluminum. The only boundary condition provided by a sound-hard boundary for an

inviscid flow is that the particle velocity normal to the surface is zero at the walls. From

the conservation of momentum (i.e. Euler' s Equation), this is represented as the normal

gradient component of the acoustic pressure being equal to zero. Hence, the transverse

factor for a rectangular duct with rigid walls is


.,,,(x,y)= cos xcos -yZi (2.6)


where a and b are the side lengths of the waveguide shown in Figure 2-1. The

remaining constants, the complex modal amplitudes, A,,,, and B,,,,, are determined by









two boundary conditions. The first boundary condition is a given acoustic impedance at

d' = 0 The second boundary condition would be a known pressure or velocity source at

the other end of the waveguide at d' = Lw where Lw is the length of the waveguide.

Applying the boundary conditions, Equation (2.5) can be solved for each A,, and Bmn

The A,, eykzd terms represent waves traveling from the source to the other end of the

tube. The Bmn e- ykzd terms represent the reflected waves returning to the source after

bouncing off the sample.

The dispersion relation for the rectangular waveguide comes for the separation

constant from applying a separation of variables solution to Equation (2.3) and is


co mx nzi
k = :I I I(2.7)
zco a b


and, for a normal mode to propagate, kZ must contain a real-valued component.

If kZ has an imaginary component, there will be two solutions to Equation (2.7) that will

be complex conjugates. For a waveguide, only the solution that causes the amplitude to

exponentially decay is physically valid from conservation of energy. This term will force

the acoustic pressure amplitude to zero as the axial distance increases in the direction of

propagation, and the wave is deemed an evanescent wave.

2.1.2 Wave Modes

The indices m and n represent the mode numbers and are denoted by (m, n) .

Physically, the indices m and n represent the number of half-wavelengths in the x-

direction and y-direction, respectively. The frequency at which a mode makes the

transition from evanescent to propagating is known as the cut-on frequency. Below the










cut-on frequency, the mode is evanescent. Above the cut-on frequency, the mode is

propagating and present along the entire length of the waveguide. The cut-on frequencies

are calculated from the dispersion relation in Equation (2.7) when kZ = 0,



I = 2 2 (2.8)


The experiments for this dissertation will use two different waveguides. Both of

them have a square cross-section. The length of the sides of the first waveguide is 8.5

mm. For air at 298 K and 101.3 kPa, co = 343 m/s and the cut-on frequencies for the

different modes are given in Table 2-1. Notice that the cut-on frequency for the first

higher-order mode is approximately 20 kHz. This implies that only plane waves are

present below this frequency and that the TMM can be used.

Table 2-1: Cut-on frequencies in k m O 1 2 3
0 0 20.2 40.4 60.5
1 20.2 28.5 45.1 63.8
2 40.4 45.1 57.1 72.8
3 60.5 63.8 72.8 85.6

The second waveguide that will be used for this experiment has a square cross-

section measuring 25.4 mm on each side. For the same conditions as above, the cut-on

frequencies for the different modes are given in Table 2-2. Note that the plane wave

mode, mode (0,0), is present for all frequencies. Also note that only the (0,0), (1,0), (0,1)

and (1,1) modes are present at frequencies less than 13.5 k
have one half wavelength in the x-direction and y-direction, respectively. The (1,1) mode

has a half wavelength in both the x-direction and the y-direction. The mode shapes are










given in Figure 2-2 as observed from the sound source. The node lines indicate where

the acoustic pressure is zero. Also note that the wavenumbers are a function of the mode.

Table 2-2: Cut-on frequencies in k m 0 1 2 3
0 0 6.75 13.5 20.3
1 6.75 9.55 15.1 21.4
2 13.5 15.1 19.1 24.4
3 20.3 21.4 24.4 28.7


(1,0) mode


(0,0) mode


(0,1) mode


(1,1) mode


---- Node Lines

Figure 2-2: Illustration of the first four mode shapes.

2.1.3 Phase Speed

The speed at which a wave front travels down the axis of the waveguide is known

as the phase speed. The phase speed, cph iS defined for a rectangular waveguide as

(Blackstock 2000)










c"= (2.9)


and can be found for each mode:


c'" (2.10)

co : mia ); nib );

which can be rewritten as


c, = "O (2.11)




The concept of the phase speed allows for higher-order modes to be considered as plane

waves traveling at an angle inside the waveguide, as shown in Figure 2-3. From

Equation (2. 11), as the frequency is increased the phase speed approaches the isentropic

speed of sound but the phase speed at the cut-on frequencies for each mode tends to

infinity. The incidence angle with respect to the waveguide wall normal as seen in Figure

2-3, 0,,,,, is found from the geometric relationship between the phase speed for that mode

and the speed of sound. Thus the incidence angle is found from



0,,, = sin cp = sin 1 (2.12)


Another useful angle that is developed from the concept of phase speed is the angle that

the wave makes with the normal to a flat termination at the end of the waveguide (d = 0 ),

denoted by 4,, From the geometry given in Figure 2-3 and from specular reflection,

,,, is complementary to 0,,,, by











~,= 90o = cos cc= cos 1- 2 (2.13)



V ve front













Figure 2-3: Illustration of the wave front and the incidence angle to the waveguide wall,
8, and to the termination, m .

The expressions given above for the phase speed, Equation (2. 11) and the two

angles of incidence, Equations (2. 12) and (2. 13), are shown only to depend on the

waveguide geometry, the bandwidth of interest, and the mode number. For the

waveguide with the 8.5 mm by 8.5 mm cross-section and a bandwidth of 20 k
phase speed is simply the isentropic speed of sound and the wave is normally incident to

the termination. Continuing with the example of a waveguide with a square cross-section

of 25.4 mm by 25.4 mm, at room temperature and pressure the phase speed and the two

angles of incidence are graphed in Figure 2-4, Figure 2-5 and Figure 2-6, respectively. A

bandwidth of 13.5 k
(0,1) and (1,1). Figure 2-4 shows the phase speed, Figure 2-5 shows the angle of

incidence the mode makes to the sidewall of the waveguide, and Figure 2-6 shows the

angle of incidence the mode makes to the termination. The plane wave mode is present

for all frequencies and is normally incident to the termination for all frequencies as well.











The properties of the other modes vary as a function of frequency. The phase speed

approaches infinity asymptotically at the cut-on frequencies, where the angle of incidence

to the sidewall approaches zero and the angle of incidence to the termination approaches

90 degrees. This shows that the MDM offers the potential to test the impedance of


specimens with oblique incident-waves.


1000







700-

600

500-

400

300
0


Figure


(O, 1) }
(1, O)


(1, 1)


2000 4000 6000 8000 10000 12000


2-4: Phase speed versus frequency for the first four modes.


(0,1)
(1,0)


25 50

E 40


01
0 2000 4000 6000 8000 10000 12000
f[Hz]
Figure 2-5: Angle of incidence to the sidewall versus frequency for the first four modes.











00
(0,1) (11
80 -(1,0)

70

60-

S50-

~40-

30-

20-

10 (0,0)

0 2000 4000 6000 8000 10000 12000
f[Hz]
Figure 2-6: Angle of incidence to the termination versus frequency for the first four
modes.

2.1.4 Wave Mode Attenuation

The energy in the evanescent wave modes exponentially decays as the wave

propagates down the waveguide. The TMM assumes that only the plane wave mode is

present at the microphone locations and that all other modes have decayed and can be

neglected. To ensure that the evanescent waves have decayed sufficiently, the amplitude

of a wave should be measured at two different axial locations in the waveguide.

This analysis of the decay of the amplitude of the evanescent waves assumes only

an incident or right-running wave. This allows Equation (2.5) to be simplified for a

single mode to



g,,,,- = A,, cos xZi~ coye yk d (2. 14)


To measure the loss in amplitude of the evanescent wave, the ratio of Equation (2. 14) is

taken for two locations separated by a distance de to give










nasnz k (~de
P,,,, (d+e = e kde.i (2.15)
P, (d) m ( mia nzCO YZ k-de"'


Recall that for an evanescent wave, the wavenumber is imaginary and thus the amplitude

of the evanescent wave will exponentially decay. The loss in amplitude can be defined

on a decibel scale by

l= 201og,de k e.). (2.16)

The decay of the evanescent waves for the two waveguides introduced in Section 2. 1.2

can be plotted. The distance traveled by the wave is assumed to be de = 25.4 nan for the

large waveguide which is equal to the length of one of the sides of the cross-section.

Figure 2-7 shows the attenuation of the higher-order modes in the large waveguide for

modes (0,1), (1,0), up to (3,3) up to their cut-on frequency. The mode can be determined

by comparing the cut-on frequency in the figure to those listed in Table 2-2. For the

small waveguide, the distance traveled is assumed to be de = 8.5nan, which is again the

length of one of the sides of the cross section. Figure 2-8 shows the attenuation of the

higher-order modes in the small waveguide for modes (0,1), (1,0), up to (3,3) up to their

cut-on frequency. The mode can be determined by comparing the cut-on frequency in the

figure to those listed in Table 2-1. The amplitude of the first evanescent wave ((0,1) and

(1,0)) is reduced by -3.8 dB with the frequency lowered from the cut-on frequency by

only 16 Hz for the large waveguide and only 25 Hz for the small waveguide. The

attenuation of other higher-order modes is larger.

Figure 2-9 shows the attenuation of the first higher-order mode in the large

waveguide for two different distances. The two distances chosen are the distances from















-20


-40









-100
Increasing wavne mode


120
0 5 10 15 20 25 30
Freq [kHz]

Figure 2-7: Attenuation of higher-order modes in the large waveguide over a distance of
25.4 mm.


Increasing wave mode


-120 '
0 10 20 30 40 50 60 70 80 90
Freq [kHz]

Figure 2-8: Attenuation of higher-order modes in the small waveguide over a distance of
8.5 mm.


the specimen test surface to the two microphones used in the TMM for the large


waveguide. The attenuation shown in this figure represents the worst case in terms of


contamination of the microphone signals with unmodeled deterministic signals that will










bias the estimates from the TMM. The figure shows that the attenuation approaches

-34 dB asymptotically for the closer microphone location and -57 dB for the farther

microphone location as the frequency approaches zero, but that the attenuation tends to

zero near the cut-on frequencies. The -20 dB point for the closer microphone location is

at approximately 5.47 kHz. Above this frequency, the signal measured by this

microphone could be affected by the non-negligible amplitude of the higher-order modes

propagating from the specimen to the microphone. The absolute amplitude of the first

higher-order mode may still be negligible when compared to the absolute amplitude of

the plane wave mode, because the overall length of the waveguide provides sufficient

attenuation such that only plane waves are incident on the specimen and such that the

specimen may not strongly scatter incident energy from the plane wave mode into the

higher-order modes upon reflection.

The data shown in the Eigures in this section demonstrates that the attenuation of

the higher-order modes is not an instantaneous effect. At the cut-on frequency, the

higher-order modes have an infinite speed and are felt throughout the entire duct. As the

frequency decreases away from cut-on, the amplitude of the evanescent mode is

decreased, but only by a Einite amount. If the initial amplitude of the evanescent mode is

sufficiently high, then the attenuation may not be strong enough to reduce the amplitude

below the noise floor of the measurement microphones. This may introduce a significant

bias error source into the estimates for the TMM or any other method that assumes no

amplitude in the evanescent modes. Numerical simulation of the sound field at the

microphone locations, including an amplitude component for the higher-order modes,

would be required to characterize the impact of the unmodeled evanescent modes. The










relative amplitudes between the microphone locations are known, as shown in Equation

(2.15), and the absolute amplitude at one location could be inferred from experimental

data. The simulated signals could then be processed as experimental data to gauge the

amount of bias error that is introduced into the estimates for the reflection coefficient and

acoustic impedance.



Sd -32 1 mm
Sd -527rmm
-10e










-40





Freq [IRb]


Figure 2-9: Attenuation of the first higher-order mode ((1,0) or (0,1)) in the large
waveguide at the microphone locations used for the TMM experiments.

2.1.5 Reflection Coefficient and Acoustic Impedance

For the remainder of this chapter, only plane waves are assumed to propagate, thus

restricting the bandwidth for a given waveguide. For this case, the reflection coefficient

is the ratio of the acoustic pressure amplitudes of the reflected wave to the incident wave

and is a single complex quantity. The plane wave reflection coefficient is defined as


40,0 Bo (2.17)
Aoo










where R,z,, is the plane wave reflection coefficient, and A~, and B,, are the complex

modal amplitudes for the incident and reflected wave, respectively. The reflection

coefficient indicates the degree to which a material reflects sound. However, the

reflection coefficient can also be used to calculate the normalized specific acoustic

impedance, fspac, of a material. The normalized specific acoustic impedance is defined

by the ratio of the acoustic impedance of the material to that of the medium used during

the test. For most cases, the medium is air. The acoustic impedance is defined as the

complex ratio of the acoustic pressure to the acoustic volume velocity. The specific

acoustic impedance is the complex ratio of the acoustic pressure to the acoustic particle

velocity. The characteristic impedance is the specific acoustic impedance of that

particular medium.

For the purpose of finding the acoustic impedance ratio, consider an incident wave

reflecting off the termination of the waveguide as shown in Figure 2-10. The two

boundary conditions are applied to the interface (Blackstock 2000):

1. The pressure must be continuous across the interface.

2. The normal component of the particle velocity must be continuous across the
interface.

The first boundary condition leads to the following expression

1 + Ro0 oo = To000, (2.18)

where Towan, is the plane wave transmission coefficient defined as the ratio of the

amplitude of the transmitted pressure wave to the amplitude of the incident pressure

wave. The second boundary condition leads to the following expression

1 R~,, T
"""co(#,)= on,"il~ cos(#,), (2. 19)
Z, Z,









where Z, and Z, are the specific acoustic impedances for medium 0 and medium 1,


respectively. The terms Z,/cos(#,) and Z,/cos(#,,) represent the acoustic impedance

for medium 0 and medium 1, respectively, but under the plane wave assumption the

incidence angle and the transmission angle are Oo with respect to the specimen surface

normal and the acoustic impedance becomes identical to the specific acoustic impedance.

Thus, Equation (2.19) simplifies to

1 Ro,o g T
00,00(2.20)


Equations (2.18) and (2.20) can be combined, and then the resulting expression can be

solved for the plane wave specific acoustic impedance ratio, given by


~,, (2.21)


From Equation (2.21), the task of finding the normalized specific acoustic impedance

reduces to finding the reflection coefficient of the incident and reflection plane waves.












Figure 2-10: Reflection and transmission of a wave off an impedance boundary.

2.2 Two-Microphone Method

The TMM (Seybert and Ross 1977; Chung and Blaser 1980; ASTM-E1050-98

1998; ISO-10534-2:1998 1998) is a standardized technique for determining the normal

incident acoustic impedance. A schematic of the test setup for the TMM is given in









Figure 2-11. The notation used here follows the ASTM E1050-98 standard (ASTM-

E1050-98 1998). The advantage of the TMM is the simplicity offered by assuming the

sound field is only comprised of plane waves. Therefore, only two unknown coefficients

are determined and only two microphones are used. The data reduction equation for the

TMM is derived in this section, starting from the basic assumptions and the general

solution of the wave equation given in Equation (2.5). Afterwards, the effects of

dispersion and dissipation are addressed briefly.

Specimen
Compression Rigid Back
Driver Reference Mic Plate



Waveguide



Mic 1 Mic 2
Power
Amplifier -Spectrum Analyzer -Mic Power
& Supply
Signal Generator
Figure 2-11: Experimental setup for the TMM.

2.2.1 Derivation of the TMM

The TMM assumes that the sound field inside the waveguide is composed solely of

plane waves. This simplifies the solution to the wave equation from Equation (2.5) to

P = Aoo0e~lkd Bo e-y~d (2.22)

This equation can then be recast by using the definition of the reflection coefficient given

in Equation (2. 17) to


P = Aoo(ejh~d 0~,00' -y").


(2.23)









Now, the two primary unknowns are Roo,oo and Aoo The two unknowns are solved for by

taking measurements of the complex pressure amplitude at two different locations along

the waveguide. Let I denote the distance between the test specimen and the closest

microphone, P_2, and s denote the distance between the two microphones. The system

of equations is

PI = Aoo e e kz (lrs +%. Roooo y kz(ljs (2.24)


P2 a 0L'1 ]kz 0,00o -Jkzl) (2.25)

The complex pressure amplitude of the incident wave is eliminated from the system of

equations by taking the ratio of P_2 to 21 to get

Pt eykz 000 ykzl e "
H -2 (2.26)
12 P e ykz (l~ s) 0,00 y kz (+si )

where H12 is the frequency response function between microphone 1 and microphone 2.

Then this new expression is solved for the reflection coefficient and simplified as

Hj~' ]ks "
Roo,oo = 2 1ke~ ) (2.27)
eii H12


where H12 = E G12 d11l IS the estimate of the frequency response function between the


two microphones, E [ ] is the expectation operator, G12 is the estimated cross spectrum

and G,, is the estimated autospectrum (Bendat and Piersol 2000). The frequency

response function is switched from the exact H12 to the estimate H12, in Eqluation (2.27)

because Hj12 iS an unbiased estimate of H12 and reduces to H12 in the case of no

measurement noise.









The form of the data reduction equation in Equation (2.27) is the same as the form

presented in the ASTM E1050-98 standard (ASTM-E1050-98 1998). The only

difference between this form and the form presented in the ISO 10534-2:1998 standard

(ISO-10534-2: 1998 1998) is the definition of the reference length, 1 The ISO standard

defines 1 to be the distance from the surface of the specimen to the microphone farther

away (ISO-10534-2:1998 1998). The remainder of this document will use the definition

of I used in the derivation in this section that is consistent with the ASTM E1050-98

standard, which is the distance from the surface of the specimen to the closest

microphone (ASTM-E1050-98 1998). After the reflection coefficient is found from

Equation (2.27), the normalized specific acoustic impedance is computed from Equation

(2.21).

2.2.2 Dissipation and Dispersion for Plane Waves

A dispersion relation is an expression that shows how the wave speed depends on

frequency. An example of a dispersion relation was given in Equation (2. 11) for the

phase speed of the higher-order modes in the waveguide. Dissipation is the removal of

energy from the propagating wave. The main mechanisms for the dissipation of wave

propagation in ducts are viscous losses and thermal conduction in the boundary layer

(Ingard and Singhal 1974; Blackstock 2000). At high frequencies, molecular relaxation

can also be another source of attenuation, but this is neglected in this analysis. The

boundary layer is a thin region near the boundary where the effects of viscosity and heat

transfer are important. The no-slip boundary condition and viscosity produce a transfer

of momentum from the flow to the wall and retards the flow in the boundary layer region.

The no-slip boundary condition states that the velocity of the flow must match the

velocity of the solid boundaries, which for the cases presented in this dissertation are not









moving. The viscous boundary layer thickness for an oscillatory flow over a stationary

plate is


3(m) 6.5 (2.28)

where v is the kinematic viscosity, and 3 is defined as the distance from the boundary to

the point in the flow where the velocity only differs by 1% from the free stream value

(White 1991). As the frequency increases, the viscous boundary layer thickness

decreases and the region where the no-slip boundary condition influences the flow is

reduced. The thermal boundary layer is the region where heat is transferred from the

flow to the boundary. The thermal boundary layer thickness is related to the viscous

boundary thickness and the Prandtl number by (White 1991)


St 2 ,(2.29)
PrZ

where Pr = v/a is the Prandtl number and a is the thermal diffusivity. Both the transfer

of momentum and the transfer of thermal energy from the flow to the wall work to reduce

the amplitude of the pressure wave.

To account for dispersion and dissipation in viscothermal flows, the wavenumber is

allowed to be complex and is given by


k = jp (2.30)


where c is the speed of sound inside the waveguide adjusted for dispersion and P is the

dissipation coefficient. The speed of sound corrected for viscothermal effects is

(Blackstock 2000)










c = co 1 Z 1+ J~: (2.31)


where S is the Stokes number given by


S = l, (2.32)


and L = 4A/1,,ere, is the hydraulic diameter of the waveguide, lymmere, is the wetted

perimeter of the cross section and A is the cross-sectional area. The dissipation

coefficient for viscothermal effects for plane waves is (Ingard and Singhal 1974;

Blackstock 2000)


J= 1+ J~~ ., (2.33)


Both Equations (2.31) and (2.33) contain the Stokes number, which is a nondimensional

number that relates a characteristic length, in this case the hydraulic diameter, to the

viscous boundary layer thickness for oscillating flows. In the limit of thin acoustic

boundary layers (at high frequency), the ratio of the viscous boundary layer thickness to

the hydraulic diameter goes to zero, and the Stokes number approaches infinity. Thus as

c 4 co and p 4 0, the lossless wavenumber is recovered. Physically, as the boundary

layer becomes smaller, the effects of viscosity and heat transfer become less important

and flow should approach the lossless case as shown. Also, Equations (2.30) through

(2.33) show that the wavenumber corrected for dispersion and dissipation is a function of

the angular frequency, the thermodynamic state, and the geometry of the waveguide.

This is in contrast to the wavenumber given in Equation (2.4) for linear lossless acoustic









motion, which was just a function of the angular frequency and the thermodynamic state.

The derivations of both Equations (2.31) and (2.33) are given in Appendix A.

To consider the relative importance of the effects of dispersion and dissipation, the

propagation constants, kd are compared for the lossless case and for the case with

dispersion and dissipation. The propagation constant for the lossless case is

(kd) = d. (2.34)


The propagation constant for the case dispersion and dissipation is


(kd)ih~n~s~ = -j7 d do -p co (2.35)


Simplifying the ratio of (kd)ther~monscous, to (kd);osszes yields



(kd),,, S

In order to neglect the effects of dispersion and dissipation, the ratio in Equation (2.36)

must be close to unity. This requires that the last term in the equation is much less than

unity and as seen in Equation (2.36), this occurs at high frequencies. For air at standard

temperature and pressure with = -1.4, v = 15.7 x10-6 m2 S, Pr = 0.708 (Incropera and

DeWitt 2002), Table 2-3 shows the minimum frequency necessary to keep the last term

in Equation (2.36) under a value of 0.05 for the two waveguides given in this chapter and

for the two ceramic tubular specimens, CT73 and CT65, described in Chapter 5. Notice

that dispersion and dissipation are important for the two ceramic tubular materials in the

frequency range of interest for acoustic impedance testing.






35


Table 2-3: Minimum frequencies to keep effects of dispersion and dissipation <5%.
Waveguide cross-section Frequency [k
25.4 mm x 25.4 mm 0.0067

8.5 mm x 8.5 mm 0.060

CT73 (hydraulic diameter
= 1.10 mm) 4.35

CT65 (hydraulic diameter
= 0.443 mm) 22.2















CHAPTER 3
UNCERTAINTY ANALYSIS FOR THE TWO-MICROPHONE METHOD

Previous studies on the uncertainty of the TMM have discussed in detail specific

error sources due to uncertainties in spectral estimates (Seybert and Soenarko 1981;

Boden and A~bom 1986; A~bom and Boden 1988) and the microphone spacing and

locations (Boden and A~bom 1986; A~bom and Boden 1988; Katz 2000) and have provided

recommendations to minimize the respective error components. However, these efforts

did not provide a method to propagate the estimated uncertainties to the overall

uncertainty in the acoustic impedance and reflection factor. The purpose of this chapter

is to provide a systematic framework to accomplish this task. In particular, a frequency-

dependent 95% confidence interval is estimated using both multivariate uncertainty

analysis and Monte Carlo methods.

The multivariate uncertainty analysis is an analytical method that assumes small

uncertainties which cause only linear variations in the output quantities, but differs from

classical uncertainty methods by allowing multiple, possibly correlated, components to be

tracked. As long as the data reduction equation can be cast into a multivariate equation

and the derivatives can be found, the multivariate uncertainty method provides a

convenient way to propagate the experimental uncertainty. The multivariate technique is

required because the measured data and the final output of the TMM are complex

variables that are treated as bivariate variables. The input covariance matrix and Jacobian

are computed and propagated through the data reduction equation (as shown in Appendix

B). The multivariate method thus provides analytical expressions that are used to extract










important scaling information, while the Monte Carlo simulations are used to account for

the nonlinear perturbations of the input uncertainties observed in practice.

The remainder of this chapter is organized as follows. First, the TMM data

reduction equations are presented in a multivariate form. Next, a general procedure to

estimate the complex uncertainty using the multivariate method is outlined, and a brief

discussion of the major error sources and their respective frequency scaling follows. The

results of numerical simulations to illustrate the relative advantages and disadvantages of

the TMM and the multivariate method follow. Specifically, two impedance cases are

presented, a sound-hard boundary that is representative of a high-impedance sample, and

an "ideal" impedance sample that is representative of an optimum impedance for a ducted

turbofan. Monte Carlo simulations are compared with the results of the multivariate

method.

3.1 Multivariate Form of the TMM Data Reduction Equations

From Equations (2.27) and (2.21), Ro,oo, and 400 are complex quantities that are


functions of another complex variable H~i, the multivariate uncertainty analysis method is

used to propagate the uncertainty (Ridler and Salter 2002; Hall 2003; Hall 2004). To

employ the multivariate method, the data reduction equations for the plane wave

reflection coefficient and the normalized acoustic impedance given in Equations (2.27)

and (2.21), respectively, must be separated into the real and imaginary parts denoted by

the subscripts R and I, respectively. For Ro,oo,,









2HR C~lbsl k(21+s)~l/ir -cs2k) j+HJcos 2k(l+s))

R;I R 1+Hj +Hz -2HR, COs(ks)-2H:, sin (ks)
RI~ ""\ 2H in( 21 ) sin (2krlll)x i+H ) sin2k (l+ s))
1+Hj+H 21iIR COs(ksr)-21i sin(ks)

In this form, the two variates of the reflection are functions of five input variates, HRi,

HIj, 1, s, and k, where HR, and Hz, are the real and imaginary parts of Hj,,,

respectively. The FRF is also treated as two variates instead of a single quantity. The

corresponding form for the normalized specific acoustic impedance is

1 R2 R2
Ts (1-RR)" +R,'
4,spac 1 (3.2)
Zx 2R,
(1- RR)2 + Rf

3.2 TMM Uncertainty Analysis

Previous studies of the error sources for the TMM have focused on determining

general scaling of the error and an experimental design that minimizes such errors with

the use of a Gaussian input signal. Seybert and Soenarko found that the bias error in the

FRF due to spectral leakage can be minimized by using a small value for the bin width of

the spectral analysis (Seybert and Soenarko 1981). Spectral leakage can be eliminated

using a periodic input signal. They also found that locating the microphones too close to

the specimen introduced bias and random errors that are a function of the measured

coherence. To increase the coherence, the microphones should be placed close together

relative to the wavelength, but the coherence will always be low when one of the

microphone locations coincides with a node in the standing wave pattern. One of the









most important findings was that when the value of s approaches an integer number of

half wavelengths, the error increases dramatically.

Boden and A~bom expanded on these results and found that the bias error of the

FRF was impacted by the overall length of the waveguide, the value of the specific

acoustic impedance of the specimen, and the location of the microphones relative to the

specimen (Boden and A~bom 1986). The random error was a function of the coherence

and was influenced by the value of the reflection coefficient, outside noise sources, and

the value of ks They suggest satisfying 0.1x~i < ks < 0.8xi to keep the overall error low.

In combination with their second study (A+bom and Boden 1988), they concluded that

errors in the microphone locations dominated over (1) spatial averaging effects, (2) any

offset the acoustic center has from its assumed location at the geometric center, and (3)

any effects from the finite impedance of the microphones themselves.

3.2.1 Multivariate Uncertainty Analysis

The results from the previous studies provide the necessary guidance to quantify

and minimize component error sources that, together with the multivariate uncertainty

and the Monte Carlo methods, can be used to provide 95% confidence intervals. The

multivariate method propagates the uncertainty estimates through any data reduction

equation (Ridler and Salter 2002; Willink and Hall 2002; Hall 2003; Hall 2004; Schultz

et al. 2005) using

s1' = JsJT (3.3)

where s, is the sample covariance matrix of the output variable, sx is the sample

covariance matrix of the input variates, J is the Jacobian matrix for the data reduction

equation, and the superscript T indicates the transpose. With the sample covariance









matrix of the variable, the 95% confidence region is found from the probability statement

(Johnson and Wichern 2002)


Prob!,, (y-ys (yy) F,, 1-, =1-a, (3.4)

where y is a vector representing the multivariate variable, y is the sample mean vector,

s,, is the sample covariance matrix of the mean, FP( n is the statistic of the F

distribution with p variates (two for a complex variable), and v,f +1- p degrees of

freedom for a probability 1- a, and v,f is the effective number of degrees of freedom

from the measurements (Willink and Hall 2002). If the entire confidence region is not

desired, the confidence level estimates of the uncertainty for each variate can be

computed from the equation

U, = kg un, (3.5)

where un is the estimate of the sample standard deviation for the nth Output variate (i.e.,

the square root of the diagonal elements of s, ), and kc is the coverage factor given by


V,fp
k =F .(3.6)


The Jacobian matrix for the reflection coefficient is



J 1R I3i (3.7)
Roo~oo 8RI 8RI 8RI 8RI 8RI
8H1;R 8H:~i 81 Bs 1 (3T

where, in this model, the wavenumber is treated solely as a function of temperature and

thus, the uncertainty in the wavenumber is solely due to the uncertainty in the










temperature measurement. The Jacobian matrix for the normalized specific acoustic

impedance is



J. (3.8)

8RR aR,

3.2.2 Monte Carlo Method

A Monte Carlo method is also used to compute the uncertainties of the reflection

coefficient and the acoustic impedance ratio. The Monte Carlo method involves

assuming distributions for all of the input uncertainties and then randomly perturbing

each input variable with a perturbation drawn from its uncertainty distribution (Coleman

and Steele 1999). The assumed distributions will be multivariate distributions if the input

variates are correlated. Now, the perturbed input variates are used to compute the

outputs, in this case R,g,, and 4spa. This is repeated until the statistical distribution of

the output variable has converged, and then the output distribution is used to estimate the

95% confidence regions. A summary of the uncertainty sources is given in Table 3-1.

3.2.3 Frequency Response Function Estimate

Estimates of the uncertainty and error sources in the FRF are documented in the

literature (Seybert and Hamilton 1978; Seybert and Soenarko 1981; Schmidt 1985;

Bendat and Piersol 2000; Pintelon and Schoukens 2001b; Pintelon et al. 2002). For this

paper, two uncorrelated noise sources are assumed to affect a single-input/single-output

system with a periodic and deterministic input signal, as described in Appendix B. Also

from Appendix B, the FRF estimate is












r~R G~, zCz 1 2
L t~ ~c 2 cz(3.9)




where G,, and G22 are the estimated autospectral densities of the signals from

microphones 1 and 2, respectively, and C12 and Q12 are the co- and quad-spectral density

functions (i.e., G12 =1 1; +2). Eqluation (3.9) is commonly called the H3, eStimate.

Any phase bias can be eliminated using a switching technique, described in Appendix B.

The final estimate of the FRF is computed from the geometric average of the two

interchanged measurements as


H= ,i" (3.10)


where Ho and Hs are the FRF between the microphones in their original and their

interchanged locations, respectively.

The details on computing the estimate of the FRF for this system model are given

in Appendix B. The sample covariance matrix for H3i and the Jacobian matrix needed to

propagate the uncertainty to the averaged FRF are also given in Appendix B. The

uncertainty estimation requires an additional measurement with the pseudo-random

source turned off to estimate the noise power spectrum.









Table 3-1: E mental bias and precision error sources for the TMM.
Variable or Error Source Error Estimator

T RTD accuracy Manufacturer' s specifications or
calibration accuracy
Ambient temporal Minimize by conducting the test in
variations limited amount of time
Spatial variations Estimate by measuring the
temperature at different locations
along. the waveguide
Random Statistical methods
variations
s,l Caliper accuracy Manufacturer' s specifications or
calibration accuracy
Acoustic centers Calibration or estimate as half
microphone diameter
Random variation Statistical methods
Microphones Spatial averaging Minimize by using microphones
with a diameter much smaller than
the wavelength
Impedance Minimize by using microphones
change of with a diameter much smaller than
waveguide wall the wavelength
t;r Phase mismatch Correct for by using microphone
switching
Magnitude Correct for by calibrating each
mismatch microphone and microphone
switching
A/D limitations Minimize by maximizing the
significant bits
Finite frequency Not present for a periodic random
resolution input signal
Random error Sample covariance matrix given in
(Schultz, Sheplak and Cattafesta
2005)


The reflection coefficient' s sensitivity to uncertainty in the FRF is described by


8RR COs (k (21+ s)) -j HRCOs(2kl 1+ s)) + RR COs (ks) -H
= 2
8H1;R 1+ H~ + Hf -? 21iIR COs(ks4) -2II sin (ksE)


(3.11)



(3.12)


-1, COs 2k (+ s)) + RR (Sin(k~s)-II
2 ,
1+1+H -21iIR COs(ks)- 21i I sin(ks4)


8RR









dR sin (k (21+ s)) -i HRSin (2k (l+ s)) +l RIcos (ksr)-H
rIl = 2 2 2 (3.13)
83HR +H +H 2H;R COs (ksE) -2H: sin (ksr)

and


8R, -H, sin (2k (l+ s))+ R, sin (ks)-H
= 2 (3.14)
i3H;7 1+ Hj + Hf 2HRk COs(ks4) -2H:i sin (ks)

Consider the case when ks4= nzi, which leads to H =(-1)". As a result, the common

denominator in Equations (3.11)-(3.14) equals zero, resulting in a singularity so that any

uncertainty in the FRF will result in a large uncertainty in the reflection coefficient. This

result agrees with previous studies (Seybert and Soenarko 1981; Boden and A~bom 1986;

A~bom and Boden 1988).

Eqluations (3.11)-(3.14) indicate that the sensitivity to the uncertainty in Hi is

dependent on the value of H~ and R ~, As H~ approaches the limiting values of zero or

infinity (i.e., when one of the microphones is located at a node), or as the magnitude of

%,o,, approaches the limit of unity, the sensitivity will increase. This implies that the

accurate measurement of the two extremes, sound-hard R i,;,, = 1) and pressure release


Io,=-) boundaries, which possess cusps in the standing wave patterns, will show

the largest sensitivities to uncertainty. The equations also show a periodic element to the

uncertainty estimates that is dependent on the wavenumber and the locations of the

microphones. Thus for a fixed set of microphone locations, the uncertainty estimates

may vary versus frequency. The actual periodicity is complex to analyze because of the

combinations of trigonometric functions present in the partial derivatives.









3.2.4 Microphone Locations

This section addresses the effects of the uncertainty of the microphone locations on

the reflection coefficient. The respective sensitivity coefficients for the distance between

the specimen and the closest microphone I and for the microphone spacing s are

8RR
= -2kR,, (3.15)


8RI
S= 2kRR (3.16)


8RR HR, Sin (k (21+ s)) -H +j Hf si, 2k (l+s)) + RR HR, Sin (ks) Hz cos (ks))
S- "k ,(3.17)
,s, 1+ H~r + Hf 2HRk COs(ks4) -2H:i sin (ks)

and


dR, HR3 COs(k (21+s)) H~ +~ Hf )cos 2k (1+ s))- R, HRi Sin (ks) Hzi cos (ks))
S= 2k .(3.18)
,s, 1+ Hj + HF 2HR, COs(ks) -2H:, sin (ks)

The sensitivity coefficients for I and s are both directly proportional to the frequency via

the wavenumber, emphasizing the difficulty of making accurate measurements at high

frequency. Equations (3.17) and (3.18) have the same denominator as Equations (3.11)-

(3.14), again showing that half-wavelength spacing ks = nzi should be avoided. Again,

the equations also show a periodic element to the uncertainty estimates that is dependent

on the wavenumber and the locations of the microphones as shown with the frequency

response function derivatives and that analyzing the periodicity would be even more

involved because of the increased number of trigonometric terms.

3.2.5 Temperature

The random uncertainty in the temperature measurement can be handled using

standard statistical procedures. The effects of temporal variations in the atmospheric









conditions can be minimized by limiting the duration of the test. The spatial variation in

the temperature of the waveguide can be characterized by measuring the temperature at

various locations and computing the standard deviation of the measurements, but this will

be a crude estimate since the entire temperature is not measured. The temperature sensor

for this study is mounted on the exterior wall to avoid undesired reflections and scattering

of the sound field inside the waveguide, and is found to give reliable estimate of the gas

temperature if the wall is highly conductive. This was confirmed by comparing the

measured temperature inside the waveguide to the outside surface metal temperature

while the sound source was on for one experimental run. The total uncertainty in

temperature is estimated from the root-sum-square of the individual uncertainties.

The sensitivity coefficients of the reflection coefficient with respect to

temperature are computed using the chain rule


R (3.19)
dT 8k dT'

and


I ,(3.20)
dT 8k dT

where


8RR, -Asin (k(21+s))+1sin(2kl)+BHsin 2k(l+ s))-C
= 2s s (3.21)
Sk 1+ H)~ + Hf~ 2EiR COs (ks) 2H:, sin (ks)


8RI A cos (k (21+ s)) 1cos (2kl) -B cos 2k (1+ s))- C
'= 2.s s (3.22)
Sk 1+I +H 2HRi COs(ksr) -2H:i sin (ks4)


A =(, 1+ I\ 2 1R (3.23)










B = 1s+-i + ), (3.24)


and


C = RR Ri Sin (ks) cos (ksv). (3.25)

Equations (3.19)-(3.25) reveal that the uncertainty in R~,,, ,,is approximately proportional

to the microphone spacing. Reducing the spacing between the microphones will reduce

the sensitivity of the uncertainty in the reflection coefficient with respect to the

wavenumber and temperature. Also, Equations (3.21) and (3.22) possess the same

singularity as the other derivatives at ks = nzi Again, the equations also show a periodic

element to the uncertainty estimates that is dependent on the wavenumber and the

locations of the microphones as shown before with the same difficulties.

For the case with dispersion and dissipation, the complex wavenumber is a function

of the thermodynamic state (ambient temperature and pressure), the frequency, and the

waveguide geometry (Morse and Ingard 1986; Blackstock 2000). The scaling of the

uncertainty in R,, ,, accounting for these effects is difficult to examine analytically. If

dissipation and dispersion are neglected and an ideal gas is assumed, the wavenumber is

given by Equation (2.4) and is only a function of temperature. Thus, the derivative of the

wavenumber with respect to temperature is


= -k g (3.26)
dT 2c2

Equation (3.26) shows that the uncertainty will increase with frequency via the

wavenumber and that the uncertainty is inversely proportional to the square of the speed

of sound.









3.2.6 Normalized Acoustic Impedance Uncertainty

For the uncertainty analysis, the normalized specific acoustic impedance is treated

as solely a function of the reflection coefficient. The Jacobian matrix is


2 (- e -R,$ -4R, (1-,)

((1-R )+R~) (1-R ) +Ri)
J = 4,( 2( (3.27)




Notice that each term has the same denominator and a singularity exists (i.e. when

(1-R )2 + R = 0) for a sound-hard boundary, R ~, = 1 This situation will be studied

further in the section below.

3.3 Numerical Simulations

Much of the observations in Section 3.2 have been previously reported in the

literature (Seybert and Soenarko 1981; Boden and A~bom 1986; A~bom and Boden 1988).

The main contribution of this chapter is to demonstrate how these uncertainty sources

propagate and contribute to the overall uncertainty in R ,~, if they remain linear. But for

typical experimental situations, the uncertainties cause nonlinear perturbations in the

reflection coefficient and acoustic impedance. In order to demonstrate the uncertainty

propagation, numerical experiments on a sound-hard boundary and an "ideal" impedance

sample are carried out using the analytical method outlined in Section 3.2 for the overall

uncertainty estimate.

Time-series data are simulated using Equations (2.2) and (2.23) by choosing a

desired value of R ~,, and the resulting data are processed using the algorithms









described in Section 3.2. The nominal values for the input parameters are given in Table

3-2. The test frequency of 5 k
bounded for this set of microphone locations. A parametric study of the effects of sensor

signal-to-noise ratio (SNR) and uncertainties in temperature, microphone location, and

spacing is completed in isolation, assuming the perturbations remain linear. The relative

uncertainties in the temperature, microphone location, and spacing are independently

varied from 0.1% to 10% at a single frequency, while the other uncertainties are set to

zero and the input signal is noise-free. The effect of the SNR is studied by varying the

SNR from 30 dB to 70 dB while holding the other uncertainties to zero. The SNR for the

numerical simulations is based on the power in the incident wave only at that frequency

compared to the power in the noise signal at that frequency and is kept constant across

the entire bandwidth. Next, the total uncertainty in Roo,oo as a function of frequency is

estimated from the case with the relative input uncertainties of 0.01% and 1% for a SNR

of 40 dB. The estimated 95% confidence intervals are then compared to the results of the

Monte Carlo simulation using 25,000 iterations. All the variables are assumed to be

normally distributed for the Monte Carlo simulation outlined in Figure 3-1 and the real

and imaginary parts of the FRF are assumed to be correlated, as shown in Appendix B.

The simulations used either a zero-mean periodic random signal for a broadband

periodic source or a sinusoid for single-frequency excitation. The bandwidth chosen for

the broadband simulations is 0 to 20 k
sampling frequency of 51.2 k
a frequency resolution of 50 Hz. In these simulations, the microphone spacing is not

designed to avoid the situation where ks = nzi or to maintain the inequality ks < nzi










(A+bom and Boden 1988). This is acceptable since the goal of the simulations is to

demonstrate that the uncertainty analysis methods presented earlier capture the correct

behavior. In an actual experiment, multiple microphone spacings can be used to avoid

the regions where ks nzi .

Table 3-2: Nominal values for input parameters of numeric simulations.
Parameter Value
1 32.1 mm
s 20.6 mm
T 23.8 oC
3.3.1 Sound-Hard Sample

The first specimen studied is a sound-hard boundary. To avoid the singularity

present in the data reduction and uncertainty expressions, the assumed value of the

reflection coefficient is 4,0,00 = 0.999, which gives a standing wave ratio (the ratio of the

maximum to the minimum pressure amplitude along the axis of the waveguide) of greater

than 60 dB. Figure 3-2 shows the absolute uncertainty in the reflection coefficient as a

function of the uncertainty in 1, s' and T at 5 kHz. Figure 3-3 shows the absolute

uncertainty in the reflection coefficient as a function of the SNR. The results in these

Eigures suggest that the dominant source of uncertainty in the magnitude of the reflection

coefficient is the random uncertainty in the FRF measurement for signal-to-noise ratios of

50 dB or lower. The dominant source of uncertainty in the phase of the reflection

coefficient is in the measurement of the distance between the specimen and the nearest

microphone. Improvements in the measurement of the reflection coefficient could be

obtained from improvements in the accuracy of the FRF measurements (reducing the

noise in the system, increasing the number of averages) and the measurement of the

distance between the specimen and the nearest microphone.

































fo r


, s, T


Figure 3-1: Flow chart for the Monte Carlo methods.


1- I-







52


The estimated value of the reflection coefficient for the relative uncertainty in the

measurement of the microphone location, the microphone spacing, and the temperature

each set to 1% and with a SNR of 40 dB is given in Figure 3-4(a) as a function of

frequency. The uncertainty results for the multivariate method and the Monte Carlo

simulation are shown in Figure 3-4(b). Note that the peaks in the uncertainty are at

frequencies 8.4 and 16.7 k
microphones is at a node in the standing wave pattern are 1.6, 2.7, 4.9, 8.1, 8.2, 11.5,

13.5, 14.7, 18.0, and 18.8 k
simulation within 5% for all frequencies except those corresponding to a node in the

standing wave at a microphone location or the singularity where ks nzi, validating the

multivariate method for very small component errors. The true value only fell outside the

estimated 95% confidence region for both the multivariate method and the Monte Carlo


104







108
100 10'

102

10'



101


100 10'
Faelatile Lkicertairty [%|o

Figure 3-2: Absolute uncertainty of Ro,oo, due to the uncertainties in 1, s, and Tfor
Roo,oo = 0.999 at f-5 k









simulation five times for the magnitude and zero times for the phase out of the total 400

frequency bins. The two methods also match at lower values of the input uncertainty, but

such agreement is not universal for all acoustic materials, which is shown in the next

section. Figure 3-4(a) shows that the estimate of the reflection coefficient becomes non-


physical, i.e. %,~, > 1, at the two frequencies where the singularity occurs. The

uncertainty in the estimate also increases to account for the singularity and the confidence


interval for %,,, does include physical values for the estimate.


10 2

10"

104

10-

105
10 2 5 4 5 5 6 6 5 7


101




104
30 35 40 45 50 56 60 65 70
Sigal-to-Noise Ratio [dB]

Figure 3-3: Absolute uncertainty E,, ,, due to the SNR for E,, ,, = 0.999 at f-5 k
3.3.2 Ideal Impedance Model

The second simulation corresponds to the ideal impedance model given in Figure

37 of the NASA CR-1999-209002 (Bielak, Premo and Hersh 1999), designed using

Boeing's Multi-Element Lining Optimization (MELO) program. The data provided in

the NASA CR is limited to a frequency range of 500 Hz to 10 k









1.02

1.01

a~1

0.99


0.98L
0


2 4 6 8 10 12 14 16 18 20


-2


1I
0


0 2 4 6 8 10 12 14 16 18 20
Freq [kHz]

(a)

10
100
101


10"

0 2 4 6 8 10 12 14 16 18 20


40
10"

103"

10"
101


0 2 4 6 8 10 12 14 16 18 20
Freq [k6i|


Figure 3-4: Estimated value for the (a) reflection coefficient and (b) total uncertainty for
the sound-hard boundary with 1% relative uncertainty for 1, s, and T and 40
dB SNR. Multivariate Method, -- -- Monte Carlo simulation. The
two lines are indistinguishable at most frequencies.










frequency range needed for this simulation by assuming that the first and last values are

constant for the ranges of 0 to 500 Hz and 10 to 20 k
reflection coefficient and normalized impedance data are given in Figure 3-5. This

specimen is chosen to determine the extent to which the uncertainties in a typical liner

specimen scale in a manner similar to that of a sound-hard boundary. The primary

distinction between the two cases is that there are no nodes in the standing wave pattern

for this impedance sample. As a result, the coherence between the two microphone

signals is expected to be near unity for all frequencies assuming a reasonable SNR.

Figure 3-6(a) shows the absolute uncertainty in the reflection coefficient as a

function of the uncertainty in 1, s, and T at 5 k
uncertainty in the normalized specific acoustic impedance. Figure 3-7(a) shows the

absolute uncertainty in the reflection coefficient as a function of the SNR, and Figure

3-7(b) shows the absolute uncertainty in the normalized specific acoustic impedance.

The results in these figures suggest that the dominant sources of uncertainty in the

magnitude and the phase of the reflection coefficient are the microphone location and

spacing. In contrast to the sound-hard boundary, there is no dominating uncertainty

source for the total uncertainty in the ideal impedance model data.

The estimated value of the reflection coefficient for the case with a SNR of 40 dB

is included in Figure 3-5(a). The estimates for the normalized specific acoustic

impedance are included in Figure 3-5(b). The uncertainty results for the multivariate

method and the Monte Carlo simulation are shown in Figure 3-8(a) for the reflection

coefficient and in Figure 3-8(b) for the normalized specific acoustic impedance. The

peaks in the uncertainty are at frequencies 8.4 and 16.7 k


















'i


O 24 6 8


10 12 14 16 18 20


50

0



-100


-150~
0


2 4 6 8 10 12 14 16 18 20
Freq [k~b|


2.5
2-
1.5-


0.5

O
O


r
I


2 4 6 8 10 12 14 16 18 20











2 4 6 8 10 12 14 16 18 20
Freq [k~b|


-1.5
O


(b)
Figure 3-5: Ideal impedance model and estimated values, adapted from (Bielak, Premo
and Hersh 1999), for (a) reflection coefficient and (b) normalized specific
acoustic impedance. Model value, Estimated value (40 dB
SNR). The two lines are indistinguishable.
















o


10"


100
Raelatime Lhcertairty
(a)


101

0 10-


>X10


100 10'
Raelatime Lhcertairty
(b)
Figure 3-6: Absolute uncertainty of (a) Ro,oo, and (b) ~,a due to the uncertainties in 1, s,
and T for the ideal impedance model at f-5 k






58



10(


104





1066
30 35 4 45 5 56 60 65 70



10"


101


10-33
30 35 4 45 5 56 60 65 70
Sigal-to-Noise Ratio [dB]

(a)

10-2



10


10

30 35 40 45 50 56 60 65 70


10-2


10-3


104


105
30 35 40 45 50 56 60 65 70
Sigal-to-Noise Ratio [dB|

(b)
Figure 3-7: Absolute uncertainty in (a) Ro,oo, and in (b) ac, due to the SNR for the ideal
impedance model at f 5 k








average percent difference between the two methods is 5% for both the magnitude and

phase for the reflection coefficient for the case with only 0.01% relative uncertainty in

1, s, and T and the average percent difference is 2% for the normalized resistance and

reactance. For the case with 1% relative uncertainty in 1, s, and T large differences can

be seen in the estimate of the uncertainty in the magnitude of the reflection coefficient at

frequencies below 6 k
that the Monte Carlo simulations reveal, but the multivariate method estimates are

conservative for this case. The average percent difference between the two methods

increases to 75% for the magnitude of the reflection coefficient, 14% for the phase of the

reflection coefficient, 13% for the normalized resistance, and 16% for the normalized

reactance. These increases demonstrate that uncertainties in 1, s, and T are causing

nonlinear perturbations in both the reflection coefficient and the normalized acoustic

impedance for the case with only 1% relative uncertainty. Thus, the multivariate method

fails to give accurate values of the true uncertainty estimates. To increase the accuracy of

the multivariate method, the multivariate Taylor series used in the derivations could be

expanded to include as many terms as needed for the desired accuracy. The best option is

to use numerical techniques such as the Monte Carlo simulations used in this dissertation

to propagate the uncertainty.

The probability density function is plotted to further investigate the differences

between the multivariate method and the Monte Carlo simulations for large uncertainties.

This is done for the normalized specific acoustic impedance data and for a frequency of

5 kHz since there is a large difference between the two methods and it avoids

complications due to the microphone spacing (see Figure 3-8). Figure 3-9(a) shows the







60




102



-4"
10

103



102

10"
0 2 4 6 8 10 12 14 16 18 20



102
10'

B 10'0"


101
102
0 2 4 6 8 10 12 14 16 18 20



102



-4'
10"


0 2 4 6 8 10 12 14 16 18 20

Fre [bh
1(b)







indstiguihabeq atmotreuecis









confidence region contours for the case with only 0.01% relative uncertainties in 1, s, and

T whereas Figure 3-9(b) is for the case with 1% relative uncertainties. The figures

show that as the uncertainties become larger and cause nonlinear perturbations in the data

reduction equation, the confidence region contours change from a normal distribution to

an irregular "boomerang-shaped" distribution. Thus, the nonlinear effect invalidates the

normal distribution assumption and the uncertainty must be found from the actual

computed distribution resulting from the Monte Carlo simulation. In general, the

uncertainty cannot be summarized by the sample mean vector and the sample covariance

matrix. The contour line in the joint probability density function (pdf) that represents a

probability of 0.95 should be found and used as the 95% confidence region estimate for

the uncertainty. To find the uncertainty in the resistance and reactance due to 1% relative

uncertainty in each input variable, 25,000 iterations from the Monte Carlo simulation are

used to estimate the joint pdf. The joint pdf is approximated by discretizing the range of

the resistance and reactance into 40 bins each, for a total of 1,600 bins, and is smoothed

using a 2 bin x 2 bin kernel. Next, 100 contours of constant joint probability density are

found, and the j oint pdf is integrated within each contour to find the total probability

within that contour. Next, the contour corresponding to 95% coverage is found via

interpolation. The quoted uncertainty is then taken as the maximum and minimum values

of the contour for each component, such as the real and imaginary parts of the reflection

coefficient or the resistance and reactance. The uncertainty estimates of the magnitude

and phase of the reflection coefficient are found from the maximum and minimum values

of the magnitude and phase for the contour computed from the real and imaginary parts

of the reflection coefficients. For the case of the ideal impedance model with 1% relative














0. 106



0. 104



0. 102



X 0.1



0.098


85


95
- -
2.202


0.096


0.094
2.197


2. 198


2.199


2.2 2.201


0.5


0.4


0.3


0.2


X 0.1


0






-0.2


- 85


- 95


2.1 2.15 2.2 2.25



(b)
Figure 3-9: The confidence region contours for the resistance and reactance for the ideal
impedance model at 5 k 1% relative uncertainties in 1, s, and T.










uncertainty and a SNR of 40 dB for a frequency of 5 k
given in Figure 3-10, along with the estimated 95% confidence region from the

multivariate method and estimated and true values of the normalized impedance. This

figure illustrates the difference in the predicted uncertainty regions between the two

methods and how much larger the Monte Carlo region is. The quoted uncertainty for this

case is best given as a range since it is asymmetrical about the estimate. The estimate of


the normalized resistance is 2.20 with a 9)5% confidence interval of [2.11, 2.23] and the


estimate of the reactance is 0. 1 with a 95% confidence interval of [-0.3, 0.4] For


comparison, the uncertainty estimates from the multivariate method are f0.03 for the

resistance and f0.4 for the reactance.


O.5






0.2




-o0.1




-0.2

-0.3t
2.1 2.15 2.2 2.25


Figure 3-10: Confidence region of the ideal impedance model at 5 k input uncertainty and 40 dB SNR. Monte Carlo confidence region,

Multivariate confidence region, O estimated impedance, ~true
impedance, Monte Carlo method simultaneous confidence interval
estimates, Multivariate method simultaneous confidence interval
estimates.









3.4 Experimental Methodology

To demonstrate the multivariate method and the Monte Carlo method on

experimental data, two experimental setups are developed, using two different size

waveguides. The larger waveguide has a plane wave operating bandwidth up to 6.7 k
whereas the small waveguide has a plane wave operating bandwidth up to 20 k
schematic of the experimental setup is shown in Figure 2-11i. Each component of the

experimental setup and the data acquisition and analysis routine will be discussed in turn.

3.4.1 Waveguides

The larger waveguide is approximately 96 cm long and has a square cross-section

measuring 25.4 mm on a side. The walls of the waveguide are constructed of 22.9 mm-

thick aluminum (type 6061-T6). The cut-on frequencies for the higher-order modes,

given in Table 2-2, show that the limiting bandwidth for the TMM is 6.7 k
waveguide. The location of the microphone, 1, and the microphone spacing, s, is

measured before the experiment using digital calipers (with an accuracy of f0.05 mm ) .

The measurement is repeated 45 times and the data are used to compute the best

estimates of the microphone location and spacing and the random uncertainty of the

geometric center of the microphones. A bias uncertainty due to the difference between

the geometric center and the acoustic center is neglected since over the entire operational

frequency range of the large waveguide, the microphone diameter is assumed to be small

compared to the wavelength and thus the microphones represent point measurements.

The total uncertainty in the locations of the microphones is taken as the root-sum-square

of the random uncertainty and the accuracy of the calipers. The microphone located

closest to the specimen with a 95% confidence interval estimate is located 32.0f 0.8 mm










from the specimen. The spacing between the two microphones with a 95% confidence

interval estimate is 20.7 f1.1 mm .

The smaller waveguide is approximately 87 cm long and has a square cross-section

measuring 8.5 mm on a side. The walls of the waveguide are constructed out of at least

12.7-mm thick aluminum (type 6061-T6). The cut-on frequencies for the higher-order

modes, given in Table 2-1, show that the limiting bandwidth for the TMM is 20 k
this waveguide. The location of the microphone, 1, and the microphone spacing, s, is

measured before the experiment using digital calipers. The measurement is repeated 45

times and the data are used to compute the best estimates of the microphone location and

spacing and the random uncertainty. The uncertainty in the acoustic centers of the

microphones is estimated ad hoc to be fl.5 mm, which is considered here because of the

increased frequency range as compared to the other waveguide. For this waveguide, the

diameter of the microphones can no longer be considered small compared to the

wavelength and the microphone measurements no longer represent a point measurement.

The Helmholtz number (kd) is on the order of unity at approximately 10kHz The total

uncertainty is taken as the root-sum-square of the random uncertainty, the accuracy of the

calipers, and the bias due to the acoustic centers. The microphone located closest to the

specimen with a 95% confidence interval estimate is located 38.11f2.0 mm from the

specimen. The spacing between the two microphones with a 95% confidence interval

estimate is 12.7 f 2.0 mm The maj ority of the uncertainty in the microphone locations is

due to the uncertainty in the acoustic centers of the microphones.









3.4.2 Equipment Description

The compression driver is a BMS 4590P, with an operating frequency range of 0.2

to 22 k
a Briiel and Kj ur Pulse Analyzer System, which also acquired and digitized the two

microphone signals with a 16-bit digitizer. The measurement microphones are Briiel and

Kj ur Type 413 8 microphones (3.18 mm diameter) and are installed into the waveguide

with their protective grids attached to the microphone. The microphones are calibrated

only for magnitude before mounting in the waveguide using a Briiel and Kj ur Type 4228

Pistonphone. Atmospheric temperature is measured using a surface-mounted, 100-02

platinum resistive thermal device (Omega SRTD-1) with an accuracy of +2 K .

3.4.3 Signal Processing

For the large waveguide, the two microphone signals are sampled at a rate of 16.4

k
resolution is 16 Hz. For the small waveguide, the two microphone signals are sampled at

a rate of 65.5 k
The frequency resolution is 32 Hz. A periodic pseudo-random signal is used as the

excitation signal is to the compression driver.

3.4.4 Procedure

The microphones are first calibrated. The excitation signal is applied, and the

amplifier gain is adjusted such that the sound pressure level at the reference microphone

is approximately 100-120 dB (ref. 20 CIPa) for all frequency bins. Then the full-scale

voltage on the two measurement channels of the Pulse Analyzer System is adjusted to

maximize the dynamic range of the data system. The excitation signal is turned off and









the microphone signals are measured to estimate the noise spectra (see Appendix B). The

input and output signals for FRF estimation are assumed to contain uncorrelated noise

and there the real and imaginary parts of the FRF may be correlated as shown in

Appendix B. Next, the excitation signal is turned on and the two microphone signals are

recorded with the microphones in their original positions and switched positions. The

time-series data are used to compute the required spectra and ultimately H, 4, ,, ,,and


Cspa Via Equations (3.10), (2.27), and (2.21), respectively.

For the temperature measurement, the random uncertainty is estimated from the

standard deviation of at least 100 measurements, and the bias uncertainty is estimated by

the accuracy of the RTD (2 K). The total uncertainty in temperature is computed from

the root-sum-square of the random and bias uncertainties.

The uncertainties in the reflection coefficient and the measured normalized acoustic

impedance are estimated using both the multivariate method and a Monte Carlo

simulation (see Figure 3-1). The input distributions for 1, s, and T are assumed to be

independent Gaussian distributions and the input distribution for Hj is assumed to be a

bivariate normal distribution computed from Appendix B. A specific form for the output

distribution of the Monte Carlo simulation is not assumed as described previously at the

end of Section 3.3.2. This approach is chosen because of its ability to handle the large

perturbations that the uncertainties in the temperature and the microphone locations

represent. Results for four specimens are presented in Chapter 5.















CHAPTER 4
MODAL DECOMPO SITION METHOD

Modal decomposition methods presented in the literature can be separated into two

different schemes: correlation and direct methods (A+bom 1989). Correlation approaches

determine the modal amplitudes by measuring the temporal and spatial correlation of

acoustic pressure inside the waveguide. Direct methods, however, use point

measurements to compute the modal amplitudes from a system of equations derived from

an analytical propagation model. Accurate propagation models exist for rectangular,

square, or cylindrical ducts with rigid walls. However, multiple independent sources are

required to resolve the acoustic properties of the test specimen, such as the reflection

coefficients, mode scattering coeffieients, and acoustic impedances. This dissertation

uses the latter approach and computes the modal amplitudes by solving a system of linear

equations. This method is also amenable to a least-squares solution for added robustness.

Focusing now on prior research on direct methods, early work by Eversman (1970)

investigated the energy flow of acoustic waves in rectangular ducts but did not consider

the decomposition of modal components. Moore (1972) was one of the first to

investigate direct methods to determine the source distribution for ducted fans but limited

his results to estimates of the sound pressure levels for each circumferential mode and

neglected radial modes. Following this, Zinn et al. (1973) investigated measuring

acoustic impedance for higher-order modes by adapting the standing-wave method.

Yardley (1974) then added the effects of mean flow and reflected waves to determine the

source distribution of a fan but did not expand the method to compute the reflection









coefficient matrix. Yardley also suggested that the microphones should all be mounted

flush to the waveguide or duct. Pickett et al. (1977) continued to improve the direct

method by adding a discussion of optimum microphone locations but limited their

algorithms to a deterministic system of equations. Only results at the fan blade passage

frequency were reported. Moore (1979) continued the analysis evolution by comparing

integral algorithms for the solution of the deterministic set of equations to the least-

squares approach. He concluded that the deterministic system was susceptible to

measurement noise, and the least-squares solution provided robustness and approached

the integral method solution in the limit of infinite measurement points. Again, his

results were limited to estimates of the modal amplitudes.

Subsequently, Kerschen and Johnston (1981) developed a direct technique for

random signals, but restricted the method to only incident waves. Pasqualini et al. (1985)

concentrated their efforts on a transform scheme for a direct method for circular or

annular ducts only. A method for use with transient signals was then developed by

Salikuddin and Ramakrishnan (Salikuddin 1987; Salikuddin and Ramakrishnan 1987).

Continuing this line of work, A~bom (1989) extended the direct method to any type of

signal by measuring the frequency response function between microphone pairs. A~bom

noted difficulties associated with generating the necessary independent sources to

calculate the reflection coefficient matrix. Akoum and Ville (1998) then developed and

applied a direct method based on a Fourier-Lommel transform to the measurement of the

reflection coefficient matrix at the baffled end of a pipe. They developed an apparatus

for generating the necessary independent sources by mounting a compression driver to

the side of a circular waveguide on a rotating ring. Their results were in good agreement









with theoretical predictions for the normal mode, but they stated that discrepancies

existed for the higher-order modes since all of the data were near the cut-on frequency.

Most recently, Kraft et al. (2003) discussed the development of a modal decomposition

experiment using four microphones but did not provide any results.

The contribution of this chapter is to adapt a direct MDM based on a least-squares

scheme to a square duct and to use simple sources to acquire the data necessary to

estimate the entire reflection coefficient matrix and the acoustic impedance at frequencies

beyond the cut-on frequency of higher-order modes. The outline of the chapter is as

follows. The next section derives the data reduction procedure for estimating the

complex modal amplitudes, the reflection coefficient matrix, and the acoustic impedance

values from the measured data. Section 4.2 outlines the experimental procedure and

analysis parameters. This section concludes with a brief discussion concerning the

sources of error.

4.1 Data Reduction Algorithm

The MDM developed here is restricted to time-harmonic, linear, lossless acoustics

without mean flow governed by the Helmholtz equation. The solution is given in

Equation (2.5) in Chapter 2, but is repeated here for convenience as


P = f 7,,,, (x, y) (A,,,, e y d + B,,, e- 'kd ) (4.1)


where


,,,(x~y)= cos co -y ,i (4.2)


for a rigid-walled square duct. The dispersion relation and an expression for the cut-on

frequencies are given in Section 2. 1.1 in Equation (2.8).









4.1.1 Complex Modal Amplitudes

The experimental procedure flush-mounts a number of microphones in the sides of

the waveguide, as in the TMM. The number and locations of the microphones are

selected to observe the desired modes. A test frequency is selected and the total number

of propagating modes, a is found from the equation for the cut-on frequency, Equation

(2.8). The minimum number of microphone measurements required to uniquely

determine the acoustic pressure is for this test frequency is 20 (A+bom 1989). Next,

Equation (4.1) is written for each microphone measurement, summing only over the

propagating modes for that frequency, to form a system of equations


P, = sn~ (x ~y) Am e y kzd + Ben e- y kz d


P_ n22 mekd Bn-kd (4.3)



P, = n x mneyz'+eey


where the subscript on P represents the microphone location and r is the number of

microphone measurements, which must be equal to or larger than 20 To decompose the

sound field, the microphones should be located with some transverse separation and some

axial separation. A simple way to configure the microphone locations is to group the

microphones into two groups and locate each group at a separate axial location. The

system of equations can be compactly expressed in matrix form as

SP) = L W) (4.4)

where (P_} is the x 1 vector of measured complex acoustic pressure amplitudes, { W) is


the 20 x 1 vector of the complex modal amplitudes given by










(W} = d,(4.5)


and L is the r x 20 matrix of the coefficients from Equation (4.3), composed of the

transverse function and the propagation exponential.

The coefficient matrix has a special form; it is composed of two sub-matrices that

are complex conjugates. As a result of this structure, the determinant of the L matrix has

an imaginary part that is identically equal to zero. To avoid this problem, the matrix

equation is transformed into a system of two real-valued matrix equations, each with a

coefficient matrix that has a non-zero determinant

(R I )P ,= (LR, + jLI,)({WR) + j {W )), (4.6)

where the sub scripts R and I denote the real and imaginary parts, respectively (Rao

2002). The expression is rearranged by carrying out the multiplication and collecting the

real and imaginary parts


r P)(~~[~ -,L t .~(~ (4.7)

The solution to Equation (4.7) is found, for example, via Gaussian elimination for the

deterministic case in which r = 20 For the overdetermined case in which r > 20 a

least-squares solution to Equation (4.7) is desired for a robust solution, and this can be

found by solving the normal equations (Chapra and Canale 2002)

[LR -L,('RfI [R)LR -LI R-L W
L, L 1=L -LL, ifWR>WI)(4.8)

where the superscript T represents the transpose of the matrix (Chapra and Canale

2002).









4.2.2 Reflection Coefficient Matrix

With the existence of higher-order propagating modes, an incident acoustic mode

now may reflect as the same mode and scatter into different modes. This increases the

complexity of characterizing the specimen, as a single reflection coefficient no longer

describes the acoustic interaction. Instead, a reflection coefficient matrix is defined as

{ B) = R {A) (4.9)

where the size of R is ox e and the vectors { A) and {B) are ox 1 (A~bom 1989;

Akoum and Ville 1998). The elements of R are represented by R,,~,,, ,,where the first

index, mn is the mode number for the reflected mode and the second index, qr is the

mode number for the incident mode. The diagonal elements, R,,,,,,,,, represent same-

mode reflection coefficients, while the off-diagonal elements, R,,~,,, ,represent the mode

scattering coefficients. To determine the unknown reflection coefficient matrix, a

minimum of a linearly independent source conditions must be measured (A+bom 1989;

Akoum and Ville 1998). The additional vectors of the incident and reflected complex

modal amplitudes are combined together to form matrices such that

(CB)I (B)Z L {B) ]= R A), {A), L { A) ,l (4.10)

which can then be solved for the reflection coefficient matrix. The approach in this work,

described below, to generate multiple independent sources is to place various restrictor

plates between the waveguide and the compression driver, the purpose of which is to

emphasize one of the modes. Previous researchers placed the compression drivers

perpendicular to the waveguide on a rotating ring and varied the location of the










compression drivers relative to the microphones (Pasqualini, Ville and Belleval 1985;

Akoum and Ville 1998; Blackstock 2000).

4.2.3 Acoustic Impedance

The acoustic impedance ratio is defined only for same-mode reflections as

(Blackstock 2000)

specimen
cos ) 1R,nm
{,,,, = "" '" ,(4. 11)
Z, 1-Rmnm
cos ( ,,,

where Zspeclmne; and Z, are the characteristic impedances of the specimen and medium,


respectively, and 4(; is the angle of transmission for the m, n mode. The acoustic

impedance ratio is also called the ratio of oblique incidence wave impedance by

(Dowling and Ffowes-Williams 1983). The normalized specific acoustic impedance or

the normalized characteristic impedance is obtained from Equation (4. 11) as

Z cos +
<-pa specnlmen H nin1ni
Cs c Z, cos (4,,,) 1- R,, ,,,,'


Without further information concerning ((,, only the acoustic impedance ratio can be

computed from the results of the MDM. However, locally reactive materials are desired

for aeroacoustic applications as engine nacelle liners (Motsinger and Kraft 1991), and

therefore are commonly tested (Jones et al. 2003; Jones et al. 2004). A locally reactive

material is a material whose impedance is independent of the angle of incidence and

therefore is assumed to have a transmission angle of approximately zero (Morse 1981).

In this case, Equation (4.12) simplifies to










Zspe~e 1 1 +R,,~,
Z, cos (95,,,) 1- Rin,,,

which represents the normalized surface response impedance (Dowling and Ffowes-

Williams 1983) and can be estimated from the MDM. To check the validity of the

locally reactive assumption, the normalized specific acoustic impedances, from Equation

(4.13), for all modes at a given frequency should be equal.

For the TMM, only plane waves are present and, hence, only the normal-incident

specific acoustic impedance is determined. The angle of incidence for the (0,0) mode

acoustic impedance is seen from Equation (2.13) have normal incidence. Thus, the (0,0)

mode acoustic impedances from Equations (4. 11)-(4. 13) are identical to the estimate

from the TMM and the two estimates can be compared. The higher-order modes

assumed in the MDM can be thought of as plane waves at an oblique angle of incidence,

as discussed in Section 2. 1.3. The effect of angle of incidence causes the acoustic

impedance value to differ from the specific acoustic impedance value, and thus both

estimates of impedance must be considered to fully characterize the specimen.

4.2.4 Acoustic Power

In addition to the acoustic impedance ratio, the absorption coefficient is an

important parameter to characterize acoustic materials. The absorption coefficient, a, is

defined as the amount of acoustic power absorbed by the specimen normalized by the

incident power (ISO-10534-2:1998 1998) and is given as


a -1 (4.14)


where W and W, represent the power in the incident and reflected acoustic fields,

respectively. In the case of the TMM, the absorption coefficient only considers the









power contained in the plane wave mode, but in the MDM, the absorption coefficient will

encompass the total power absorbed in all the propagating modes. Equation (4.14)

assumes that no acoustic power is transmitted through the waveguide into the

surrounding environment, hence demonstrating the need for terminating the specimen

with a rigid back plate and for insuring proper sealing of the waveguide. In this case, all

the acoustic energy that is not absorbed by the specimen and dissipated as heat is send

back down the waveguide. Expressions for the incident and reflected powers are derived

from integrating the acoustic intensity in the d -direction over the cross-section of the

waveguide to obtain the total power, W, given by


W (f )= IdS dRe PU ~ldxdy (4.15)

where Ud is the acoustic velocity perturbation along the d -axis in the frequency domain

and is found from Euler's equation (Blackstock 2000) as

j dP
L1d= (4.16)
-d pcok Sdd

where k = co/lc, The orthogonal properties of the normal modes in the acoustic pressure

solution given in Equation (4.1), and in the acoustic velocity perturbation solution, given

in Equation (4. 16), allow for the expression of the total power to be simplified and

ultimately separated into two parts. Each part only contains the modal amplitudes for

either the incident waves or the reflected waves. The resulting expressions for the

incident and reflected powers are


W = a fk,,,A,,, (4 .1 7)
8 pc ,k ,,=, ,,










WA N
8pc,k,, ,,, ,

The full derivation is given in Appendix D. The absorption coefficient not only provides

an estimate of the sound absorption capabilities of a material, but also provides a check

on the measurement. The absorption coefficient is bounded between zero and unity, and

values outside this range indicate a problem with the experimental setup and procedure.

4.2 Experimental Methodology

To verify the data reduction routine outlined above and obtain acoustic impedance

data beyond the cut-on frequency, an experimental apparatus is developed. The actual

results are presented in the next chapter along with the results for the TMM. The

experimental procedure to acquire and reduce the data is similar to the TMM. A

compression driver is mounted at one end of a waveguide, and the test specimen is

mounted at the other end. For the MDM, eight microphones are flush-mounted in the

duct wall at two axial locations near the specimen to resolve the incident and reflected

waves. Fourier transforms of the phase-locked, digitized pressure signals at each location

are used to estimate the complex acoustic pressure and thus the modal coefficients and

reflection coefficient matrix. A schematic of the experimental setup is shown in Figure

4-1, with eight microphones flush-mounted in the waveguide. Each component of the

experimental setup and the data acquisition and analysis routine will be discussed in turn.

4.2.1 Waveguide

The waveguide used in the measurements presented is approximately 96 cm long

and has a square cross-section measuring 25.4 mm on a side. The walls of the waveguide

are constructed of 22.9 mm-thick aluminum (type 6061-T6). The cut-on frequencies for

the higher-order modes, given in Table 4-1, show that the limiting bandwidth for the










TMM is 6.7 k
the first four modes. To resolve these four modes, eight microphones are placed in two

groups of four microphones at two axial locations. The placement is chosen such that

each microphone is not located at the node line of any of the modes of interest and to

achieve a sufficient signal-to-noise ratio. The locations of the eight microphones are

provided in Table 4-2. The independent sources for the MDM are generated via the four

different restrictor plates shown in Figure 4-2, each one designed to emphasize one or

more of the first four modes.



















Figure 4-1: Schematic of the experimental setup for the MDM (some microphone
connections are left out for clarity).

Table 4-1: Cut-on fre uencies in k m \n 0 1 2 3
0 0 6.75 13.5 20.3
1 6.75 9.55 15.1 21.4
2 13.5 15.1 19.1 24.4
3 20.3 21.4 24.4 28.7

4.2.2 Equipment Description

The compression driver is a BMS 4590P, with an operating frequency range of 0.2

to 22 k








Table 4-2: Microphone measurement locations (a = 25.4 mm)
Microphone x,v,d Location Microphone x,v,d Location [mm]
Imm1
1 0.25a, 0,1.6a 5 0.25a, 0,1.1a
2 a, 0.25a,1.6a 6 a, 0.25a,1.1a
3 0.75a,a,1.6a 7 0.75a,a,1.1a
4 0, 0.75a, 1.6a 8 0, 0.75a,1i.1a


(0,1i) restrictor plate


(0,0) restrictor plate


(1,0) restrictor plate (1,1) restrictor plate
Figure 4-2: Schematic of the four restrictor plates. (The dotted line represents the
waveguide duct cross-section.)

a Briiel and Kj ur Pulse Analyzer System, which also acquired and digitized the eight

microphone signals with a 16-bit digitizer. The measurement microphones are Briiel and

Kj ur Type 413 8 microphones (3.18 mm diameter) and are installed into the waveguide

with their protective grids attached to the microphone. The microphones are calibrated

only for magnitude before mounting in the waveguide. The phase mismatch between the

eight microphones was measured in previous experiments, with each microphone flush

mounted at the end of the large waveguide with a reference microphone, up to 6.7 k
and was found to be no greater than +So. This error is found to be acceptable (less than

10 % uncertainty for the modal amplitudes and reflection coefficients) from the results of


ID


I I
L___J









the numerical uncertainty studies in Section 4.2.4. Atmospheric temperature is measured

using a 100-02 platinum resistive thermal device with an accuracy of +2 K .

4.2.3 Signal Processing

All eight microphone signals are measured and subsequently processed with a fast

Fourier transform algorithm. The frequency resolution is 16 Hz with a frequency span

from 0.3 to 13.5 k
with no overlap. Leakage is eliminated by the use of a pseudo-random periodic signal to

excite the compression driver. To ensure synchronous data acquisition, the sampling is

triggered by the start of the generator signal in a phase locked acquisition mode. The

data are then processed using the MDM described above.

4.2.4 Numerical Study of Uncertainties

The main sources of error for the MDM are the signal-to-noise ratio, microphone

phase mismatch, uncertainties in the measurements of the microphone locations, and the

temperature. The frequency scaling of the uncertainty in the computed values from the

MDM is also important, as the goal of the MDM is to extend the frequency range of

acoustic impedance testing. Numerical studies have been conducted concerning the

effects of the individual error sources and the frequency scaling of the total error, and are

only summarized here for brevity; the results are given in Appendix E. These studies

were performed for an approximate sound-hard termination, with four different vectors of

incident complex modal amplitudes. The reflected modal amplitudes are computed from

Equation (4.9) and the data are then used to calculate time-series data. The time-series

data are then processed using the MDM described above. The root-mean-square (rms)

normalized error between the elements of the calculated reflection coefficient matrix and









the modeled reflection coefficient matrix is used to gauge the uncertainty of the MDM.

The numerical studies are performed at a frequency of 12 k
the microphone locations listed in Table 4-2. The simulations varied the error introduced

into the simulated input signals to the MDM and computed the perturbed output

reflection coefficient matrix for each of the error sources individually. These results

showed that the MDM gives reliable and accurate estimates (with ~10% uncertainty) for

the complex modal coefficients and the reflection coefficient matrix. The influence of

evanescent modes can be simulated to determine the magnitude of the bias error they

cause if the amplitudes of the incident and reflected evanescent waver are known.