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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 NASALangley Research Center Grant (Grant # NAG12261). I thank Mr. Michael Jones and Mr. Tony Parrot at the NASALangley 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, TaiAn 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 TwoMicrophone 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 TWOMICROPHONE 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 SoundHard 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 NavierStokes 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 SignaltoNoise 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 21 Cuton frequencies in k 22 Cuton frequencies in k 23 Minimum frequencies to keep effects of dispersion and dissipation <5%...............3 5 31 Elemental bias and precision error sources for the TMM. ................... ...............43 32 Nominal values for input parameters of numeric simulations. ............. .................50 41 Cuton frequencies in k 42 Microphone measurement locations (a = 25.4 mm). ............. .....................7 A1 Minimum frequency required for series expansion for the two waveguides for air at 200C............... ...............170. C1 Simulation results for the secondorder system. ............. ...............212.... E1 Power in Pa2 for all signals from all simulation sources ................. ................. .220 LIST OF FIGURES Figure pg 11 Illustration of the approach, takeoff, and cutback flight segments and measurement points. ............. ...............2..... 12 Typical noise sources on an aircraft. .............. ...............4..... 13 Component noise levels during approach, cutback, and takeoff for a Boeing 767 300 with GEAE CF680C2 engines. .............. ...............4..... 14 Comparative overall noise levels of various engine types. ........._.... ........_........5 15 Engine cutaway showing the acoustic liner locations .................... ...............7 16 An example of a SDOF liner showing the atypical honeycomb and the perforate face sheet. .............. ...............7..... 17 An example of a 2DOF liner. .............. ...............8..... 21 Illustration of the waveguide coordinate system. ........._. ............ ..............14 22 Illustration of the first four mode shapes. .........._.... ........_ ......_..._......1 23 Illustration of the wave front and the incidence angle to the waveguide wall and to the termination................ .............2 24 Phase speed versus frequency for the first four modes. ............. .....................2 25 Angle of incidence to the sidewall versus frequency for the first four modes.........21 26 Angle of incidence to the termination versus frequency for the first four modes....22 27 Attenuation of higherorder modes in the large waveguide over a distance of 25.4 m m ............... ...............24... 28 Attenuation of higherorder modes in the small waveguide over a distance of 8.5 m m ............... ...............24... 29 Attenuation of the first higherorder mode ((1,0) or (0,1)) in the large waveguide at the microphone locations used for the TMM experiments. .................. ...............26 210 Reflection and transmission of a wave off an impedance boundary. ................... ....28 211 Experimental setup for the TMM. .............. ...............29.... 31 Flow chart for the Monte Carlo methods. ............. ...............51..... 32 Absolute uncertainty of Roo,oo due to the uncertainties in 1, s, and Tfor Roo,oo = 0.999 at f~ 5 k 33 Absolute uncertainty Ro,oo, due to the SNR for Ro,oo, = 0.999 at f~5 kHz..............53 34 Estimated value for the (a) reflection coefficient and (b) total uncertainty for the soundhard boundary ...........__......___ ...............54.... 35 Ideal impedance model and estimated values for (a) reflection coefficient and (b) normalized specific acoustic impedance. ............. ...............56..... 36 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 37 Absolute uncertainty in (a) Ro,oo, (b) ~sac due to the SNR for the ideal impedance model at/f 5 k 38 Total uncertainty in (a) Ro,oo, and (b) ac, as a function of frequency for the ideal impedance model............... ...............60. 39 The confidence region contours for the resistance and reactance for the ideal impedance model at 5 k 310 Confidence region of the ideal impedance model at 5 k 41 Schematic of the experimental setup for the MDM. ................ .......................78 42 Schematic of the four restrictor plates ................. ...............79......__. .. 51 Photograph of the CT65 material ....__. ................ ............... ........ ...83 52 Reflection coefficient for CT65 for the TMM. ............. ...............86..... 53 Normalized specific acoustic impedance estimates for CT65 via TMM. ................86 54 Reflection coefficient for CT65 for the high frequency TMM. ............. ................88 55 Normalized specific acoustic impedance estimates for CT65 via the high frequency TM M ............. ...............89..... 56 Incident pressure field for the MDM for CT65 ................. ....._._ ................ 90 57 Reflected pressure field for the MDM for CT65................ ...............90. 58 Absorption coefficient for CT65 .............. ...............91.... 59 Comparison of the reflection coefficient estimates for CT65 via all three methods.92 510 Mode scattering coefficients for CT65 from the (0,0) mode to the other propagating m odes. ............. ...............93..... 511 Mode scattering coefficients for CT65 from the (1,0) mode to the other propagating m odes. ............. ...............93..... 512 Mode scattering coefficients for CT65 from the (0,1) mode to the other propagating m odes. ............. ...............94..... 513 Mode scattering coefficients for CT65 from the (1,1) mode to the other propagating m odes. ............. ...............94..... 514 Comparison of the acoustic impedance ratio estimates for CT65 via all three m ethod s. ............. ...............96..... 515 Comparison of the normalized specific acoustic impedance estimates for CT65 via all three methods. ............. ...............97..... 516 Photograph of the CT73 material. ............. ...............98..... 517 Reflection coefficient for CT73 for the TMM. ............. ...............99..... 518 Normalized specific acoustic impedance estimates for CT73 via the TMM. .........99 519 Reflection coefficient for CT73 for the high frequency TMM. ............. ................100 520 Normalized specific acoustic impedance estimates for CT73 via the high frequency TM M .......... ................ ...............101...... 521 Incident pressure field for the MDM for CT73 ................ .......... ...............102 522 Reflected pressure field for the MDM for CT73 ......____ ........_ ..............103 523 Absorption coefficient for CT73 ................. ....__ ...._ ...............103 524 Comparison of the reflection coefficient estimates for CT73 via all three methods.104 525 Mode scattering coefficients for CT73 from the (0,0) mode to the other propagating m odes. ............. ...............105.... 526 Mode scattering coefficients for CT73 from the (1,0) mode to the other propagating m odes. ............. ...............105.... 527 Mode scattering coefficients for CT73 from the (0,1) mode to the other propagating m odes. ............. ...............106.... 528 Mode scattering coefficients for CT73 from the (1,1) mode to the other propagating m odes. ............. ...............106.... 529 Comparison of the acoustic impedance ratio estimates for CT73 via all three m ethods. ............. ...............108.... 530 Comparison of the normalized specific acoustic impedance estimates for CT73 via all three methods. ............. ...............108.... 53 1 Photograph of the rigid termination for the large waveguide. ............. ..............109 532 Reflection coefficient for the rigid termination for the TMM. ............._ .............110 533 Standing wave ratio for the rigid termination measured by the TMM. ..................1 11 534 Reflection coefficient for the rigid termination for the high frequency TMM. .....112 535 SWR for the rigid termination calculated from the high frequency TMM. ...........112 536 Triangle restrictor plate. ................ ...............114......... ..... 537 Incident pressure field for the MDM for the rigid termination .............. ..... ..........114 538 Reflected pressure Hield for the MDM for the rigid termination. ........................115 539 Power absorption coeffieient for the rigid termination for the MDM..................115 540 Comparison of the reflection coefficient estimates for the rigid termination via all three methods. ........... _...... __ ...............117.. 541 SWR for the rigid termination calculated from the MDM. .............. ................11 7 542 Mode scattering coefficients for rigid termination from the (0,0) mode to the other propagating modes. ................ ...............118......... ...... 543 Mode scattering coefficients for rigid termination from the (1,0) mode to the other propagating modes. ................ ...............118......... ...... 544 Mode scattering coefficients for rigid termination from the (0, 1) mode to the other propagating modes. ................ ...............119......... ...... 545 Mode scattering coefficients for rigid termination from the (1,1) mode to the other propagating modes. ................ ...............119......... ...... 546 SDOF liner showing the irregular honeycomb and perforated face sheet. ............120 547 Reflection coefficient for the SDOF specimen for the TMM. ............. ................122 548 Normalized specific acoustic impedance estimates for the SDOF specimen via TM M .......... ................ ...............122...... 549 Incident pressure field for the MDM for the SDOF specimen. .............. .............124 550 Reflected pressure field for the MDM for the SDOF specimen. ...........................124 551 Power absorption coefficient for the SDOF specimen for the MDM. ................126 552 Comparison of the reflection coefficient estimates for the SDOF specimen via the TMM and MDM ................. ...............126................ 553 Mode scattering coefficients for SDOF specimen from the (0,0) mode to the other propagating modes. ............. ...............127.... 554 Mode scattering coefficients for SDOF specimen from the (1,0) mode to the other propagating modes. ............. ...............127.... 555 Mode scattering coefficients for SDOF specimen from the (0, 1) mode to the other propagating modes. ............. ...............128.... 556 Mode scattering coefficients for SDOF specimen from the (1,1) mode to the other propagating modes. ............. ...............128.... 557 Comparison of the acoustic impedance ratio estimates for the SDOF specimen via the TMM and MDM ................. ...............129............... 558 Comparison of the normalized specific acoustic impedance estimates for the SDOF specimen via the TMM and MDM ................. ...............129.............. 559 Photograph of the mode scattering specimen ................. ................. ........ 130 560 Incident pressure field for the MDM for the mode scattering specimen................132 561 Reflected pressure field for the MDM for the mode scattering specimen. ............132 562 Power absorption coefficient for the mode scattering specimen for the MDM.....133 563 Comparison of the reflection coefficient estimates for the mode scattering specimen via the M DM ............. ...............134.... 564 Mode scattering coefficients for the mode scattering specimen from the (0,0) mode to the other propagating modes. ............. ...............135.... 565 Mode scattering coeffieients for the mode scattering specimen from the (1,0) mode to the other propagating modes. ............. ...............135.... 566 Mode scattering coefficients for the mode scattering specimen from the (0, 1) mode to the other propagating modes. ............. ...............136.... 567 Mode scattering coefficients for the mode scattering specimen from the (1,1) mode to the other propagating modes. ............. ...............136.... 568 Comparison of the acoustic impedance ratio estimates for the mode scattering specimen via the MDM. .............. ...............137.... 569 Comparison of the normalized specific acoustic impedance estimates for the mode scattering specimen via the MDM. ............. ...............137.... A1 Oscillating flow over a flat plate. .............. ...............145.... A2 Control volume showing the external forces and flows crossing the boundaries. .161 A3 Control volume showing the external heat fluxes and flows crossing the boundaries. ............. ...............166.... B1 A plot of the raw data and estimates for a randomly generated complex variable..183 B2 A plot of the raw data and estimates in polar form for a randomly generated complex variable. ............. ...............185.... B3 System model with output noise only. ............. ...............187.... B4 System model with uncorrelated input/output noise. .............. ...................19 B5 Bode plot of the true FRF and the experimental estimate. .............. ..............199 B6 Magnitude and phase plot of the uncertainty estimates. ................ ................199 B7 The experimentally measured FRF between the two microphones. ......................205 B8 Comparison for the uncertainty estimated by the multivariate method and by the direct statistics. ............. ...............205.... C1 FRF for the simulation with random noise............... ...............213. E1 The rms normalized error for the modal coefficients versus noise power added to the si gnal s. .............. ...............223.... E2 The rms normalized error for the reflection coefficient matrix versus noise power.224 E3 The rms normalized error versus the number of averages for a noise power of 0.01 Pa2.............. ...............224.. E4 The rms normalized error for the reflection coefficient versus the number of averages for a noise power of 0.01 Pa2. ............ ...............225..... E5 The rms normalized error for the modal coefficients versus a phase error applied to microphone 4 in group 2 for each source. .............. ...............225.... E6 The rms normalized error for the reflection coefficient matrix versus a phase error applied to microphone 4 in group 2 for all sources ................. ............ .........226 E7 The rms normalized error for the modal coefficients versus a microphone location error applied to microphone 1 in group 1 for each source. ............. ...................227 E8 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 E9 The rms normalized error for the modal coefficients versus a temperature error. 228 E10 The rms normalized error for the reflection coefficient matrix versus a temperature error. .............. ...............229.... E11 The rms normalized error for the modal coefficients versus frequency. .............230 E12 The rms normalized error for the reflection coefficient matrix versus frequency. 230 F1 Ordinary coherence function for the TMM measurement of CT65 .....................233 F2 The measured FRF for CT65 for the TMM. ............. ...............234.... F3 Ordinary coherence function for the high frequency TMM measurement of CT65.234 F4 The measured FRF for CT65 for the high frequency TMM. ............. .................235 F5 Ordinary coherence function for the TMM measurement of CT73 .................. .....235 F6 The measured FRF for CT73 for the TMM. ............. ...............236.... F7 Ordinary coherence function for the high frequency TMM measurement of CT73.236 F8 The measured FRF for CT73 for the high frequency TMM. ............. .................237 F9 Ordinary coherence function between the two microphones for the TMM measurement of the rigid termination. ................ ...............237........... ... F10 The measured FRF for the rigid termination for the TMM ................. ................238 F11 Ordinary coherence function for the high frequency TMM measurement of the rigid term nation. ............. ...............238.... F12 The measured FRF for the rigid termination for the high frequency TMM...........239 F13 Ordinary coherence function for the TMM measurement of the SDOF specimen.239 F14 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 soundabsorbing liners. The standard twomicrophone 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 soundhard and soundsoft 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 degreeoffreedom 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 11. During these flight segments, the configuration of the aircraft is altered from the clean cruise configuration via the deployment of highlift devices and landing gear. Also, during takeoff the engines operate at full power, further increasing the noise levels. In order to reduce the flyover noise, the power to the engines is reduced after takeoff, 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 12. 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 13 for a Boeing 767300 with GEAE CF680C2 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 11: 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 highbypassratio 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 FfowesWilliams 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 14. The first generation of jet aircraft relied on propulsion from a single, high velocity j et from the aft of the engine (RollsRoyce 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 lowbypassratio turbofan engine (RollsRoyce 1996). OII, JVertical tail eness Slat Tip vortices Nose landing/ o a gear Horizontal tail Engines lni Flap/sideedge \/ Man ladingvortices gear Figure 12: 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 13: Component noise levels during approach, cutback, and takeoff for a Boeing 767300 with GEAE CF680C2 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 lowervelocity j et from the fan and the mixing of the two j ets. Modern turbofan engines use highbypassratio inlets, with a bypass ratio of approximately three or greater (RollsRoyce 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 highbypass turbofans is now restricted by problems related to the large weight and large frontal area, such as drag (RollsRoyce 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 14: Comparative overall noise levels of various engine types (adapted from Rolls Royce 1996). Since the use of highbypassratio 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 farfield (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 15. 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 16, typical single degreeoffreedom (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 degreeoffreedom (2DOF) liner as shown in Figure 17. 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, semiempirical 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 fullsize engine testing can be done to verify the noise level predictions and to certify the engine. Stators Exhaust Inlet Figure 15: Engine cutaway showing the acoustic liner locations (adapted from Groeneweg et al. 1991; RollsRoyce 1996). Honeycomb layer Perforate face .' '"' sheet Solid back sheet I: Figure 16: 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 normalincident 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 17: An example of a 2DOF liner (adapted from RollsRoyce 1996). The TwoMicrophone Method (TMM) (Seybert and Ross 1977; Chung and Blaser 1980; ASTME105098 1998; ISO105342:1998 1998) and the MultiPoint Method (MPM) (Jones and Parrott 1989; Jones and Stiede 1997) are two techniques to determine the normalincidence 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 (Figure 211). Two microphones are flushmounted 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 leastsquares 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 E105098 1998; ISO105342:1998 1998). The size is limited in order to prevent the propagation of higherorder modes and thus maintain the plane wave assumption. For square crosssection 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 higherorder 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 mmsquare duct, the bandwidth is increased to 13.5 k or to 20 k decomposition method (MDM) is that the higherorder 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 obliqueincidence information can used to verify the local reactivity assumption, which states that the acoustic impedance is independent of the angle of incidence (Dowling and FfowesWilliams 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 crosssection of the waveguide and test specimen to increase the cuton frequency for the first higherorder 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 higherorder 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 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 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 TwoMicrophone Method. * Application of a Modal Decomposition Method for measuring normalincident acoustic impedance in the presence of higherorder 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 higherorder 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 zaxis aligned with the axis of the tube, and the origin located at a corner of the tube as shown in Figure 21. The daxis 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 21: 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 FfowesWilliams 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 timeharmonic 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= Jl, 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 soundhard or rigid and therefore do not vibrate or transmit sound. Practically, the soundhard 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 soundhard 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 21. 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 realvalued 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 halfwavelengths in the x direction and ydirection, respectively. The frequency at which a mode makes the transition from evanescent to propagating is known as the cuton frequency. Below the cuton frequency, the mode is evanescent. Above the cuton frequency, the mode is propagating and present along the entire length of the waveguide. The cuton 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 crosssection. 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 cuton frequencies for the different modes are given in Table 21. Notice that the cuton frequency for the first higherorder mode is approximately 20 kHz. This implies that only plane waves are present below this frequency and that the TMM can be used. Table 21: Cuton frequencies in k 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 cuton frequencies for the different modes are given in Table 22. 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 xdirection and ydirection, respectively. The (1,1) mode has a half wavelength in both the xdirection and the ydirection. The mode shapes are given in Figure 22 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 22: Cuton frequencies in k 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 22: 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 higherorder modes to be considered as plane waves traveling at an angle inside the waveguide, as shown in Figure 23. From Equation (2. 11), as the frequency is increased the phase speed approaches the isentropic speed of sound but the phase speed at the cuton frequencies for each mode tends to infinity. The incidence angle with respect to the waveguide wall normal as seen in Figure 23, 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 23 and from specular reflection, ,,, is complementary to 0,,,, by ~,= 90o = cos cc= cos 1 2 (2.13) V ve front Figure 23: 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 crosssection 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 crosssection 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 24, Figure 25 and Figure 26, respectively. A bandwidth of 13.5 k (0,1) and (1,1). Figure 24 shows the phase speed, Figure 25 shows the angle of incidence the mode makes to the sidewall of the waveguide, and Figure 26 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 cuton 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 incidentwaves. 1000 700 600 500 400 300 0 Figure (O, 1) } (1, O) (1, 1) 2000 4000 6000 8000 10000 12000 24: 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 25: 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 26: 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 rightrunning 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 kde"' 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 crosssection. Figure 27 shows the attenuation of the higherorder modes in the large waveguide for modes (0,1), (1,0), up to (3,3) up to their cuton frequency. The mode can be determined by comparing the cuton frequency in the figure to those listed in Table 22. 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 28 shows the attenuation of the higherorder modes in the small waveguide for modes (0,1), (1,0), up to (3,3) up to their cuton frequency. The mode can be determined by comparing the cuton frequency in the figure to those listed in Table 21. The amplitude of the first evanescent wave ((0,1) and (1,0)) is reduced by 3.8 dB with the frequency lowered from the cuton frequency by only 16 Hz for the large waveguide and only 25 Hz for the small waveguide. The attenuation of other higherorder modes is larger. Figure 29 shows the attenuation of the first higherorder 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 27: Attenuation of higherorder 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 28: Attenuation of higherorder 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 cuton 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 nonnegligible amplitude of the higherorder modes propagating from the specimen to the microphone. The absolute amplitude of the first higherorder 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 higherorder modes upon reflection. The data shown in the Eigures in this section demonstrates that the attenuation of the higherorder modes is not an instantaneous effect. At the cuton frequency, the higherorder modes have an infinite speed and are felt throughout the entire duct. As the frequency decreases away from cuton, 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 higherorder 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 29: Attenuation of the first higherorder 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 210. 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 210: Reflection and transmission of a wave off an impedance boundary. 2.2 TwoMicrophone Method The TMM (Seybert and Ross 1977; Chung and Blaser 1980; ASTME105098 1998; ISO105342:1998 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 211. The notation used here follows the ASTM E105098 standard (ASTM E105098 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 211: 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 ey~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 E105098 standard (ASTME105098 1998). The only difference between this form and the form presented in the ISO 105342:1998 standard (ISO105342: 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 (ISO105342:1998 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 E105098 standard, which is the distance from the surface of the specimen to the closest microphone (ASTME105098 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 higherorder 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 noslip 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 noslip 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 noslip 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 crosssectional 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 x106 m2 S, Pr = 0.708 (Incropera and DeWitt 2002), Table 23 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 23: Minimum frequencies to keep effects of dispersion and dissipation <5%. Waveguide crosssection 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 TWOMICROPHONE 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 soundhard boundary that is representative of a highimpedance 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 (1RR)" +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!,, (yys (yy) F,, 1, =1a, (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 31. 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 singleinput/singleoutput 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 quadspectral 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 pseudorandom source turned off to estimate the noise power spectrum. Table 31: 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, soundhard 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 halfwavelength 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 rootsumsquare 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,) ((1R )+R~) (1R ) +Ri) J = 4,( 2( (3.27) Notice that each term has the same denominator and a singularity exists (i.e. when (1R )2 + R = 0) for a soundhard 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 soundhard boundary and an "ideal" impedance sample are carried out using the analytical method outlined in Section 3.2 for the overall uncertainty estimate. Timeseries 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 32. The test frequency of 5 k bounded for this set of microphone locations. A parametric study of the effects of sensor signaltonoise 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 noisefree. 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 31 and the real and imaginary parts of the FRF are assumed to be correlated, as shown in Appendix B. The simulations used either a zeromean periodic random signal for a broadband periodic source or a sinusoid for singlefrequency 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 32: 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 SoundHard Sample The first specimen studied is a soundhard 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 32 shows the absolute uncertainty in the reflection coefficient as a function of the uncertainty in 1, s' and T at 5 kHz. Figure 33 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 signaltonoise 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 31: 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 34(a) as a function of frequency. The uncertainty results for the multivariate method and the Monte Carlo simulation are shown in Figure 34(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 32: Absolute uncertainty of Ro,oo, due to the uncertainties in 1, s, and Tfor Roo,oo = 0.999 at f5 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 34(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 SigaltoNoise Ratio [dB] Figure 33: Absolute uncertainty E,, ,, due to the SNR for E,, ,, = 0.999 at f5 k 3.3.2 Ideal Impedance Model The second simulation corresponds to the ideal impedance model given in Figure 37 of the NASA CR1999209002 (Bielak, Premo and Hersh 1999), designed using Boeing's MultiElement 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 34: Estimated value for the (a) reflection coefficient and (b) total uncertainty for the soundhard 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 35. 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 soundhard 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 36(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 37(a) shows the absolute uncertainty in the reflection coefficient as a function of the SNR, and Figure 37(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 soundhard 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 35(a). The estimates for the normalized specific acoustic impedance are included in Figure 35(b). The uncertainty results for the multivariate method and the Monte Carlo simulation are shown in Figure 38(a) for the reflection coefficient and in Figure 38(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 35: 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 36: Absolute uncertainty of (a) Ro,oo, and (b) ~,a due to the uncertainties in 1, s, and T for the ideal impedance model at f5 k 58 10( 104 1066 30 35 4 45 5 56 60 65 70 10" 101 1033 30 35 4 45 5 56 60 65 70 SigaltoNoise Ratio [dB] (a) 102 10 10 30 35 40 45 50 56 60 65 70 102 103 104 105 30 35 40 45 50 56 60 65 70 SigaltoNoise Ratio [dB (b) Figure 37: 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 38). Figure 39(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 39(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 "boomerangshaped" 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 39: The confidence region contours for the resistance and reactance for the ideal impedance model at 5 k uncertainty and a SNR of 40 dB for a frequency of 5 k given in Figure 310, 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 310: Confidence region of the ideal impedance model at 5 k 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 211i. 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 crosssection measuring 25.4 mm on a side. The walls of the waveguide are constructed of 22.9 mm thick aluminum (type 6061T6). The cuton frequencies for the higherorder modes, given in Table 22, 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 rootsumsquare 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 crosssection measuring 8.5 mm on a side. The walls of the waveguide are constructed out of at least 12.7mm thick aluminum (type 6061T6). The cuton frequencies for the higherorder modes, given in Table 21, 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 rootsumsquare 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 16bit 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 surfacemounted, 10002 platinum resistive thermal device (Omega SRTD1) 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 pseudorandom 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 100120 dB (ref. 20 CIPa) for all frequency bins. Then the fullscale 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 timeseries 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 rootsumsquare 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 31). 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 leastsquares 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 higherorder modes by adapting the standingwave 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 leastsquares 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 FourierLommel 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 higherorder modes since all of the data were near the cuton 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 leastsquares 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 cuton frequency of higherorder 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 timeharmonic, 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 rigidwalled square duct. The dispersion relation and an expression for the cuton frequencies are given in Section 2. 1.1 in Equation (2.8). 4.1.1 Complex Modal Amplitudes The experimental procedure flushmounts 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 cuton 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 Bnkd (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 submatrices 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 realvalued matrix equations, each with a coefficient matrix that has a nonzero 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 leastsquares 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 RL 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 higherorder 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 offdiagonal 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 samemode reflections as (Blackstock 2000) specimen cos ) 1R,nm {,,,, = "" '" ,(4. 11) Z, 1Rmnm 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 FfowesWilliams 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 normalincident 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 higherorder 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 (ISO105342:1998 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 crosssection 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 cuton 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 flushmounted in the duct wall at two axial locations near the specimen to resolve the incident and reflected waves. Fourier transforms of the phaselocked, 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 41, with eight microphones flushmounted 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 crosssection measuring 25.4 mm on a side. The walls of the waveguide are constructed of 22.9 mmthick aluminum (type 6061T6). The cuton frequencies for the higherorder modes, given in Table 41, 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 signaltonoise ratio. The locations of the eight microphones are provided in Table 42. The independent sources for the MDM are generated via the four different restrictor plates shown in Figure 42, each one designed to emphasize one or more of the first four modes. Figure 41: Schematic of the experimental setup for the MDM (some microphone connections are left out for clarity). Table 41: Cuton fre uencies in k 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 42: 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 42: Schematic of the four restrictor plates. (The dotted line represents the waveguide duct crosssection.) a Briiel and Kj ur Pulse Analyzer System, which also acquired and digitized the eight microphone signals with a 16bit 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 10002 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 pseudorandom 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 signaltonoise 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 soundhard 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 timeseries data. The timeseries data are then processed using the MDM described above. The rootmeansquare (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 42. 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. 