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NACA TM 1399 ACOUSTICS OF A NONHOMOGENEOUS MOVING MEDIUM By D. I. Blokhintsey "Akustike Neodnorodnoi Dvizhushcheisya Sredy". Ogiz, Gosudarstrennoe Izdatel'stro, TekhnikoTeoreticheskoi Literatury, Moskva, 1946, Leningrad. Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and tne Sloan Foundation http://www.archive.org/details/acousticsofoh0ui Pr:EFAc'E The practical problems btrought about by the Great War have given rise to theoretical problems. In acoustics interest centers about the problem of the p~r';:?sti tio of sound in a nonhomogeneous moving medium, vbich is the nature of the atmosphere and the water of seas a~nd rivers, as well as about prob lems concerning moving sources and sound receivers. These problems are closely connected; they lie at the boundary between acoustics and hydro dynamics in the broad sense of the word. It is precisely these aspects of acoustics that have been either little developed theoretically and experimentally or are not very popular among acoustics technicians. This is the circumstance that has provided the occasion for the appearance of this book, which is devoted to the theoretical basis of the acoustics of a moving nonhomogeneous medium. Experiments are considered only to illustrate or confirm some theoretical explanation or derivation. As regards the choice of theoretical questions a~nd their treatment, the book does not in any way pretend to be complete. The choice of material was to a. considerable extent dictated by the author's own in vestigations, somIe of which were, previously published and others first presented herein. Certain problems were not worked through to the end but have merely been indicated. The author, nevertheless, included them in the book, on account of the creative interest which they may arouse among investigators in the field of theoretical acoustics. The author expresses his appreciation to N. N. Andreev and S. I. Rzhevkin, who were acquainted with the manuscript of this book, for their useful advice and comments, and also to L.D. Landau, whose consultation made possible the clarification of a number of problems. Institute of Physics, USSR Academy of Sciences N~ACA TM 1399 [:ACA TM 1399 CONTENTS Page CRAPTER I. ACOUSTICS EQUATIONS OF A NONHOMOGENEOUS MOVING MEDIUM. .1 1. Outline of Dynamics of a Compressible Fluid. .. ... 1 2. Equations of Acoustics in Absence of Wind. .. .. .. 7 3. Energy and Energy Flow in Acoustics. .. .. ... .. 13 4. Propagation of Sound in a, Nonhomogeneous Moving Medium 18 5. Equation for Propagation of Sound in Constant Flow .. 25 6. Generalized Theorem of Kirchhoff .. ... .. .. .. 28 CHAPTER II. PROPAGATION OF SOUND IN ATMOSPHERE AND IN WATER .. 35 7. Geometrical Acoustics. .... .. .. .. ..... 35 8. Simplest Cases of Propagation of Sound ... ... 42 9. Propagation of Sound in a Real Atmosphere. Zones of Silence. ...... .. .. .. .. 45 10. Turbulence of the Atmosphere .. .. ... .. .. .. 46 11. Fluctuation in Pha~se of Sound Wave Due to Turbulence of Atmosphere .. ... .. .. .. .. ... .. 53 12. Dissipation of Sound in Turbulent Flow ... .. .. 59 13. Sound Propagation in Medium of Complex Composition, Particular in Sa~lty Sea Water. .. ... .. .. .. 66 CRIAPTE:R III. MOVING SOUND SOURCE. ... .. .. ... .. .. 74 14. Warve Equation in an Arbitrarily Moving System of Coordinates. .. .. .. .. ... .. 74 15. Sound Source Moving Uniformly with Subsonic Velocity 76 16. Sound Source Moving Arbitrarily but with Subsonic Velocity 80 17. General Formula, for Doppler Effect .. .. ... .. 85 18. Sounid of a~n Airplane Propeller .. .... .. 88 19. Characteristics of Motion at; Supersonic Velocity. Density Jumps (Shock Waves). .. ... .. .. ... .. 95 20. Sound Source Moving with Supersonic Velocity and Haing Small Head Resistance. .. .... .. .. ... 102 21. Sound Field of a Sound Source for Supersonic Velocity of Motion .. .. .. .. ... .. .. .. ... 107 iiii ITACA TM 1399 Page CHAPTER IV. EXCITATION OF SOUUDi~ BIY A FLOW ... .. .. .. 11_2 22. General Data on Vortical Sound and Vortex Formation. 112 23. Theory of the K Jrm n Vortex Street Computation of the Frequency of Vortex Formation. .. .. .. .. .. .. 119 24. Pseudosound. Conditions of Radiation of Sound by a, Flow 123 25. Vortex Sound in the Flow about a Long Cylinder or Plate. 129 26. Remarks on the Vortex Noise of Propellers. .. .. .. 134 27. Excitation of Resonators by a Flow .... ... 136 CKAPTER V. ACTION OF A SOUND RECEIVER IN A S'TREAM~ ... ... 145 28. Physical Phenomena in the Flow about a Sound Receiver. 146 29. Shielding a Sound Receiver from! Yortical. Sound Production. 152 30. Shielding of Sound Receiver from Velocity Pulsations of Approaching Flow .. .... .. .. .. .. .. 1_59 31. Sound Receiver Moving with Velocity Considerably Less Than Velocity of Sound. .. .. ... .. ... .. .. 162 32. Sound Receiver Moving with Velocity Exceeding Velocity of Sou~nd. .. .. .. .. .. .. .... .. ... 166 REFERENCES. .. .. .. .. .. .. ...... 175 NACA TM 1399 CHAPTER I ACOUSTICS EQUATIONS OF A NONEOMOGEN~EOUS MOVING MEDIUM 1. Outline of Dynamics of a Compressible Fluid The medium in which sound is propagated, whether it is a gas, a liquid, or a solid body, has an atomic structure. If, however, the fre quency of the sound vibrations is not too large, this atomic character of the medium may be ignored. For a gas it may be shama (ref. 1) that if f << 1/z, where f is the frequency of the vibrations and T the time taken to traverse the free path between collisions, the gas may be considered as a dense medium characterized by certain constants. This method of considering the prob lem is assumed in aerodynamics and in the theory of elasticity. Since the atomic character of the medium is ignlored, the phenomenon of the dis persion of sound cannot, in all strictness, be taken into account. For tunately, in the majority of practical problems, thre dispersion of sound does not have great significance. For this reason, phenomena which require consideration of the atomic nature of the medium will not be considered, and the aerodynamic equations of a compressible gas will be used as the bassis of the theoretical analysis of the acoustics of a moving medium. These equations are first considered without the assumption of any specific restrictions for the acoustics (such as large frequency and small amplitude of vibrations). The equations of the dynamics of a compressible gas express the three fundamental laws of conservation: (1) conservation of' matter, (2) conservation of momentum, and (3) conservation of energy. In order to formulate these laws, a certain system of coordinates x, y, and z, fixed relative to the undisturbed medium, is chosen. Further, t is the time, v is the velocity of the gas in this system (Translator's note: An arrow is used in the typescript to indicate that a symbol stands fora victor v1~, z 'x>'2 y, nd v = are theoomponents ofv along the x, y, and z axes, epciey and P is the density of the ga.In these notations, the law of the conservation of matter, mathema tically expressed by the equation of continuity, assumes the form a +x,1 k 'k) = 0 11 2 NACA TM 1399 where the sumtion, is carried out for k = 1, 2, and 3. The vector Qv3 is the flow density ve~ctor of thet substance. This equation states that the change in amount of substance in any small volume is equal, to thes flow of the substance through the surface enclosing this volume. The vetctor pv may be considered also as the vector of the momen tum density. The change of momentum in any small volume should be equal to the momentum transported by the motion of the fluid through the sur face enclosing this volume plus the force applied to the volume. The momentum flow due to the transport of momentum is a tetnsor with the components: Pv1 k (iik = 112,3). The~ assumption is made that there are no volume forces. Hence the force applied to the volume is equal to the resultant of the stresses applied to the surface of the volume. The tensor of these stresses will be denoted by Tik anld is composed of the scalar pressure p and the viscous components aik Tik = P ik sik (1.2) where 81 1 if 1 = kr, and 81 = if i f k. When applied to a small volume, the~ law of the conservation of mo mentumn can be written ia the form (py1) 8xk + ~~Tik O'l'k) = 0 (1.3) 1 and k = 1, 2, an 3 and again is summed for k = 1, 2, and 3. The equation of the conservation of energy abould express the fact that the change ia the total energy in a small volume, made up of the kinetic energy and the internal energy of a unit volume of the gas, is equal to the flow of the kinetic and internal energy through the surface enclosing this volume, the heat flow through this surface plus the work performed by the stresses acting on this volume. The part of the energy flow vec tor due to the transport of t~he kinetic energy P 1 and the internal energy pE (E is the energy of unit mass of the g~as) 2s (P 1 pE)v. If the heat flow vector is denoted by S(S1,S2,S3) ~and the conservation law is applied to a small volume, a v2 + pE +p pE vk Sk Vi ik = O(1.4) NACA UK. 1399 where the summation is for i and k = 1, 2, and 3. The last term gives the work of the stresses on a unit volume. For an isotropic, homogeneous liquid (or gas), the stresses Sik are connected with the deformations vik according to the Newtonian relations Sii = 2pyvii + 7 div v; Sik =. 2p vik (1.5) where 14 is the viscosity of the gas and vik is the tensor of the deformations Vik x2:ax (1.6) The magnitude 7 can be written in the form 7 = Ct' 2pF~3, where p' is the socalled second coefficient of viscosity (see ref. (1)). With this coefficient, account is taken of the conversion of the energy of the macroscopic motion of a gas into the energy of the ithernal degrees of freedom of the molecules (the rotation of the molecules), a fact which is of appreciable significance only for ultrasonic frequencies. For this reason, ia the majority of cases the assumption may be made that 4' = 0 and 7 = 2CL/3 (the value assumed in the theory of Stokes). The flow of heat expressed La terms of the gradient of the absolute temperature T is Sk = h X = P cvX(17 where x is the coefficient of the heat conductivity of the gas and c, is the specific heat of the gas at constant volume. To the three fundamental hydrodynamic equations, (1.1), (1.3), and (1.4), the equation of state of the gas (or liquid) connecting the pres sure p, the density P, and the temperature T is added p = Z(P,T) (1.8) Equations (1.1), (1. and (1.4) permit a rational determination of the flow of substance Lthe flow of momentum represented by the Lrhis form for vik follows from the assumption of the isotropic character and homogeneit~y of the gas or liquid if a linear relation is assumed between the stress tensor sik and the deformation tensor vi . C 4 NACA TM 1399 tensor Mik, and the flow of energy N, vbich, like the flow of substance, can. be written in vector form. This determination. will be such that the divergence of the flow, taken with Inverse sign, is equal to the deriva tive with respect to the time of thet density of the corresponding mag nitude. In this manner from equatiqfn (1.1) for the flow of substance (equal to the flow of momentum) the following is obtained: S= Pv (1.9) From equation (1.Z), substitution of the value of Sik from equa tion (1.5), gives the~ tetnsor of the momentum flow Mii Pv + p + 7 iv di v p vil Mik = Pvivk 2pvik = Mki; 1 f k (1.10) where, as before, i and k = 1, 2, and 3. The terms of the form pv2> Pvi'k give the momentum flow due to the transport of momentum by the motion of the fluid, and the terms containing P, CL) and 7 give the flow of momentum due to the~ action of the pressure forces and the viscous stresses. Finally, from equation (1.4), substitution of Sik from1 equation (1.5) yields the energy. flow N =~ p~,a + E7 pv +t pI Vv2 + ro vX v3 + 7 div v v (1.11) The first termn gives the energy flow due to the transport of energy by the fluid, the second (S) gives the heat flow, and the term: pv and the sterns with CI and 7 give the part of thet energy flow due to the work of the pressure forces and the viscous stresses. The fundamental equations can also be written ia vector from, by substitution of the value of the tensor Tik from equations (1.2) and (1.5) iu equations (1.3) and (1.4). Equation (1.1) may, however, be as ji + div(py) = (1.12) 2The vector Nr = ( vzr + p representing the flow of energy for an ideal inlcompressible liquid, is called the N. Umnov vector (ref. 3). NACA TM 1399 If use is made of (1.12) equation (1.3) can be written in the form p Vp + Cpd? + 1 CLV div ? (1.13) at 3 where V is the symbol for the gradient and a = 4/ax2 2 2"as 62/322 = 82. Th aniued/dt is the total derivative of the velo city with respect to time and is equal to iv av by v2 (, 4 it~~V Vt V t + v + Crot vX j(.4 The energy equation (eq. (1.4)), with the aid of equation (1.12), assumes the form pdE T+Qpdiv v (1.15) at =E a + ( V)E (.5' where 'Q is the dissipative function Q = Sik Vik (1.16) i,k=1 If this equation is divided by P, it may be interpreted so that a change of energy of unit mass dE/dt is equal to the heat flow XAT/P, the amount of heat divided by the work of the viscous forces Q/P, and the work: of the pressure forces (P div v/) This equation may also be interpreted in terms of thermodynamics. Thle first law of thermodynamics for unit mass of substance yields dE = TdS p dY (1.17) where E: is the energy of unit mass; S, its entropy; p, the pressure, and V, the specific volume (V = 1/P). Thus = T dS dV S T ap (1.18) at at dt dt 2 St 6 NACA TMu 1399 Dn the other hand, +o (p ,V div 3 (1.19) so that 9 = 2 (1.20) ~For adiabatic processes dE p ap (1.21) dt 2 dt from which E = dp (1.22) The magnitude w = E E= (1.23) is termed the heat function. If the process is nonadiabatic, equation (1,18) holds. From equations (1.15) and (1.18) the following is ob tained: TdS & T+B (1.24) St p P The magnitude T(dS/dt) is the increase of heat of unit mass of the gas, which is determined exclusively by the ~heat conductivity and the work of the friction forces. If X a~nd are neglected since the effects produced by them in the overall energy balance are usually small corrections, the following results: dS iS + S (1.25) dt ~ dt that is, the adiabatic motion of the fluid. The Bernoulli theorem holds for this motion if it is also irrotatio,.al (rot v = 0). If v = V (1.28) NACA TM 1399 where cP is t~he velocity potential, from equations (1.13) and (1.14) "C~ a2 =z 2i (1.27) and since, on the basis of equation (1.23), plP = Vw, integration of equation (1.27) gives w iV = =t~P2 (V4)~2 (1.27') If the compressibility of the fluid is neglected, v = P + constant (1.28) PO so that P c = 0gy~[ (VQ)2 + constant (1.29) and in the case of steady flows (80/at = 0) p0 (~2 O 'sat iV p = constant ()=cota (1.30) 2 2 Because the entropy remains constant during the motion for an ideal fluid (h = p = O) introduction of the variables P and S in the equation of state, equation (1.8), La place of the variables p and T, is expedient since with such a choice of variables one of the variables (S) remains constant, whereas the temperature T varies even for an ideal fluid (for adiabatic comlpressionls and expansions of the fluid). The following may be written in place of equation (1.8) p = Z'(P,S) (1.8') 2. Equations of Acoustics in Absence of Wind The equations which determine the propagation of sound La a motion less median can now be considered. The vibrations of the medium are called sonic vibrations or simply sound if the amplitude of the vibra tions is so small that it is possible to neglect all the changes in state of the gas in any small volume are produced in it by the transport (convection) of mass, momentum, and energy. This situation is the cona dition of linearity of the vibrations. Further, these vibrations are 8 NACA. UKP 1399 assumed to occur with frequencies in the hearing range (the region of classical acoustics) or near this range (infra and ultra sound). Nbthe~ maticall~y the above assumption reduces to the neglect of the terms in the aerodynamic equations of a compressible gas which contain second powers or the products of mall magnitudes which determine the deviations of the state of the gas from equilibrium~. Where Jt is the deviation of the pressure from the equilibrium value pO> p is set equal to p P =pO + 8 where PO is the value, of the density for p =pO n T = O, ad fnall 7 =((gis a small velocity). Similarly. for the temperature, entropy, and energy, S = SO + O E = EO + In place of equations (1.12) and (1.13), the following is obtained: P0 ( V + CI A + 3 MVdv((1.31) + 'oO div r = 0 (1.32) The equation of state of the gas, for an ideal gas in the variables P and T is p = P .rT (1.33) where r is the gas constant for unit mass; and in the variables P and S SSO pO c p = Py e (1.34) where cv is the specific heat at canatant volume (cy=r(1) n 7 = c,/c, is the ratio of the specific heats at constant pressure and constant volume. For small changes of state the following is obtained from equation (1.34): P0 PO PO It = 7  +  + ... = 028 + ha + ...h PO cv c NACA UKM 1399 9 For a = 0, only the first term representing small changes in pres sure for small adiabatic compression. or expansion of the gas remains. The magnitude (1.35) is the adiabatic velocity of sound. The second term gives the change in pressure produced byr the addition or decrease of heat. The changes of entropy a obey equation (1.24) which is written by neglecting magni tudes of the second order of anallness as follows: TO =j ,1= cx 0 (1.36) The changes in temperature changes in density and entropy. 8 may be expressed in terms of the From equation (1.17) T = ES (1.37) The energy of an ideal gas is equal to 'cTC E = cT = p! 0 (7 1)P 907 pl S 1 (1.38) from which aE/aS = aE/ao is obtained in the form pg 71 T'CV T = 0 9 e p7 (7 1)c O = P (7 1)pcy (1.37') that is, for small values of P and S P20c 0(Y 1)c, (1.39) where the first term represents the change in temperature during adiabatic compression or expansion of the gas and the second term represents the change in temperature due to the change in entropy of the gas. Substitution in equation (1.38) yields xl= x itxla t x a6; (1.40) I ~ o C = I/ r  P r O NACA TM 1399 Equations (1.31), (1.32), and (1.40) together with the equation of state (1.34) determine the propagation of sound in a motionless medium when account is taken of the viscosity and heat conductivity of the medium. The effects aristag: from the presence of viscosity and heat con ductivity reduce, in a first approximation, to the absorption of the sound by the medium. This absorption is generally not large and its magnitude for a plane wave can be determined without difficulty. If its directions of propagation is along the ox axis, the frequency of the sound equals w, and the wave number vector is equal to kc, (1.41) where (0, O,~ CO are the amplitudes of vibration of the corresponding magnlitudes. Substitution of equations (1.41) in equations (1.31), (1.32), and (1.40) yields imP0O O= ik(c2gO+600) h g ~lPk2O(13 iw60 ikPO 0 = 0 (1.32') i00 = xk2,O xlk250g (1.40') Elimination of the amplitudes gives the relation between .k and o cLg k 2 kpO h x1 k24 S(io + xk2) 34i~k (1.42) If k is set equal to w/c ia, where a is the coefficient of damping of the wave, the velocity of propagation. C' in the first approximoation is equal to C, and thet damping coefficient a is equal to 3 c3 2 a2 2 (1.43) 5= (ei(wtk~x) g ~ i(otk~x) o = Oei(wtk~x) NACA TM 1399 where a2 = PO 90 is the square of th~e isothermal velocity of sound. For air a = 1,i 1013f2cm1, where f = w/2x is the frequency of sound ia Hz (1 BHertz = 1 cycle/sec). B~ence in many cases the effect of the viscosity and heat conlductivitY may be neglected or their effect takren into account by introduction of the absorption. coefficient in the final results. The anallness of the effect of viscosity and heat con ductivity of the air an the propagation of sound is determined not only by the smallness of the coefficients C1 and x but also by the anall ness of the gradients of all magnitudes which vary in the sound propagation. Equations (1.31) and (1.40) show that these gradients enter the equation in the form of second derivatives of 5, o, and so forth (for example, Clb( and x~cn). In the propagation of a wave in free space these derivatives are in order of magnitude equal to 5/h1, o/h2, , and so forth, and become appreciable only for very short wave lengths (as the final equation for the absorption coefficient a shows since a increases proportionally to the square of the frequency. Near the boundaries of solid or fluid bodies which may be considered as stationary, the losses by viscosity and heat conductivity increase. In these cases sharper changes of state of the gas in space occur and the second derivatives of 5, a, and 8 are determined not by the length of the wave but either by the dimensions of the body 2 so that aT ( /22 and AO 0/22 or by the "natural" length d' = l/v~7m (this length is in addition to the lengths X and 3, and is determined from dimensional considerations), where v is the kinematic viscosity (v = IP/), or by the length d" = 7/x/m. In these cases the order of the magnitudes is given by dir 5 (d2 and do c /d2 En general, the losses by viscosity and heat conductivity near the boundary of a solid or fluid body are determined by the least of the three lengthy h, 2, and d(d', d"). Despite the increase in the losses near walls and stationary boun daries, the losses remain small and can be considered a correction to the motion which occurs without losses (except for the case of the propa gation of sound in very narrow channels). An example of the approximate computation of the effects of viscosity and heat conductivity mI~ay be found in the work of the author (ref. 4). In addition to the absorption of sound associated with the heat conductivity and the viscosity of the medium still another molecular absorption of sound exists which was discovered by V. Knudsen (ref. 5) and explained by G. K~neser (ref. 6). The physical character of this absorption lies in the conversion of the energy of the sound vibrations into the energy of inner molecular motion (energy of rotation of the 12 NACA TM 1399 molecules). This absorption likewise increases with the frequency and is of special significance for the ultrasonic range. As the consideration of these problems deviates from the present subject, discussion is limited to the references given. La all those cases where the losses of the sound energy are not of interest, the viscosity and heat conductivity of the air may be ignored. If h and )1 are set equal to 0 in equations (1.3') and (1.40), o = 0, that is, adiabatic propagation of sound is obtained and the equations describing this propagation assume the form DO = V (1.44) ~ pg div ( = O (1.45) x = 028 (1.46) These equations may be solved with the aid of the single function cp which is termed the velocity potential (or simply the potential). The first three equations (1.44) are satisfied by setting *=,,.~(1.47) The wave equation for the potential fromn equations (1.46) and (1.45) is obtained: 1 a2( Aq O (1.48) c i~t2 which, in the presence of bodies, must be solved with the boundary con dition (Onr)e, (on the surface of the body) (.9 where a/an. is the derivative along the normal to the surface of the body nd (9 is the normal velocity of the surface of the body assumed 1 = 0 (on the sulr~fae of the body) otn (1.49') HIACA. TM 1399 For a unique solution of the problem of the sonic field described by equation (1.48) the initial conditions for cp and ac/bt must be formulated in addition to the boundary conditions of equations (1.49) or (1.49'). 3. Energy and Energy Flow in Acoustics For linear acoustics all magnitudes referring to the sound are computed with an accuracy up to the first degree of the amplitude A, which may,. for example, be the amplitude of a piston which excites sound vibrations. Achievement of more accurate solutions of the equations of hydrodynamnics will yield the' succeeding approximation containing terms proportional to A2, and so forth (when account is taken of nonlinzear ph~etomena~). For the pressure p, the density p, and the velocity of motion d, the following series is written: p = pO + "l + "2 *** P = PO + 1 + 62 + "'. v = YO f 1 + ~2 + ... (1.50) The magnitudes pO' 0,' and v0 refer to the motion undisturbed by the sound; the magnitudes sl n 1 are proportional to A, the magnitudes %2> s2> and (2Z are proportional to A2, and so forth. The energy and energy flow contain the squares of the magnitudes 8>1 and l.For this reason caution must be used when the energy and energy flow are computed in linear acoustics, as was pointed out by I. Bronshtein and B. Konstantinov (ref. 7) and also by N. N. Andreev (ref. 8), since these magnitudes, being of the order of A2, may also contain the first degrees of the succeeding approximations <2, 62> and (2 while their contribution will be of the same order as the contribution from the squares of all 61, and (1. The general expression for the energy density of a compressible medium is U pv + pE (1.51) where E is the internal energy of unit mass of the medium. The energy flow N, computed on the basis of equation (1.11) with the viscosity and heat conductivity neglected, is equal to N = Uv + pv (1.52) 14 NAQA UKN 1599 Fr~om the law of the conservation of energy, ; + div N = 0 (1.53) This equation is one of the fundamental equations of hydrodynamics, that is, equation (1.4) for the case of an ideal fluid (p = 1 = 7 = 0). For an ideal gas PE = p/(7 1) (equation (1.38)); hence N= c2 ? + y130[ (1.52') 2 1 For acoustics the initial medium~ is considered motionless (v0 = 0) The energy of the sound "2 = U2 P0 EO and the flow of sonic energy N2 is obtained with an accuracy up to the order of magnitude A2. em of the order of AZ rejected, at+ div NZ = 0 (1.53) where (5 2 + (.4 2 2 1 Inaemuch as 2~. 90 0 12 2 =pO + c2O6 (8 62) + (7 1)c208 O + "l + "'2 + ... (1.55) (cZ = (dp/dp) = 7 pO 0O is the square of the adiabiatic velocity) and "1 = 0061' equation (1.54) may be rewritten in the form (1.54') P02 2 2 O(6 1 6l c E + + ( 2 2 2 1 1 2 20000O o'p0 rPl 1 N2 1 1i ) Tt 7 r 1 (1.54') NIACA TM 1399 1o For a homlogeneous medium. at rest (v0 = 0, cO = constant, and P0 = constant), a new form of the conservation law follows from equation (1.53) in which the energy of the sound and its flow are expressed only in terms of the magnitudes characteristic of linear acoustics (xl 11 anrd 5 1) rnot containing the second approximatio ~n~s (w2> 2~, and 1~2'* The equation of continuity expressing the law of theF ~cIonsevatUion of matter (equation (1.12)), when written with an accuracy up to terms of the order of A2; is ac(8 + 87) t~ 0 Po a (1 2 z i (61 1) = O (1.56) This equation is multiplied by c20/(7 1) and the result is subtracted from equation (1.53). Tuasmuch as 81 = rc1/c2, equation (1.54) yields ael a + div N1 = O (1.57) where 1r 2 2 2000 N1 = WE1 (1.58) The new expressions obtained for the energy of sound and the energy flow "Iare precisely those which are applied ia acoustics. In particular, if the potential cP ((1 = ~79, Rl 0 ~(84/at), see equation (1.47)) of the sound wave is introduced, then l 4 P 992 t 2 N1 0P SE V (1.59) If, as is often the case, the potential (p depends harmonically on the time and is given ia complex form (cp is proportionaal to etiwt), the mean. energy in time and the mean flow in time are equal to P0 cj~ 4 = 2V 2 N1 g4 9* 9 9 p (1.60) R\ACA TM~ 1399 where the signl indicates that the conjugate complex magnlitude should be taken. The expressions for the energy and energy flow equations, (1.54) and (1.58), are physically equivalent because the medium is supposedly homogeneous (in a nonhomogeneous medium equations (1.59) are not valid). In order to show the equivalence of the two forms of the conservation laws, one of which is a consequence of the other (under the given con ditions) the radiation of sound is considered. In figure 1 is shama a sound source Q (solid body), a certain part of whose surface a exe cutes vibrations which. excite sound waves. If the vibration started at the time instant t = O, at the moment t the surface of the wave front will be the surface F (see fig. 1). TIhe entire space between. this surface and thet source Q vill be filled with energy radiated by the sound. With an arbitrasry control. surface S enclostag the sound source, the conservation theorem (1.53) is applied in integral form to the volume V included between S and Q. In order to do this, equation (1.53) must be integrated over the volume and then, the theorem of Gause is used ia transforming the ~integral of div Ng2 to a surface integral. This integral will consist of the integral over the surface S and the surface of the source Q. Although same inconvenience is caused because part of this surface is movable (a), it can easily be circumvented by the consideration that the flow of energy through the surface of the source mnust simply be equal to the source W2. From equation (1.53) the following equation is obtained: at '7 r~r~ 1) 1' 2)+ (7i 7 1)(xl1 8do = W2 (1.61) where n denotes the projection of a n the normal to the surface 8, E2 =4 E~dv is the total energy of the sonic field enclosed within S; and the~ strength of the source Q is evidently equal to W2~Pg i 01 1v+(xEj v d (1.62) where v denotes the projection on the~ normal to the surfac~er If the control surface is passed outside the sonlic field (for example, out side the wave front F, but infinitely near it), from equation (1.61) is obtained dE2 NACA TMv 139.9 that is, the total radiated energy E2 is equal to the work of the source Q. On the other ~hand, if the second form of the conservation law (eg. (1.18)) is treated in the same manner, the folloviag equation results: dE dt = W2; El 2dw~t (1.63') from which it follows that El must be equal to E2* Fr~om equations (1.54') and (1.58), E22 y E261 t 2)dv (1.64) where the integration is over the volume V. The integralJ (81 + 2) is the total change of mass of gas in the volume occupied by the sonic field. This change is equal to zero because the substance could not flow out beyond the limits of the wave front; hence El = E2. If the integral over the time period in equations (1.63) or (1.63') is taken over the entire number of periods of vibration of the source and if the fact is tak~en into account that in this case do pO (1 + 2 ~vdt is equal to zero (since this integral is equal to the algebraically assumed path of a surface element d6 of the source Q in the direction along the normal to 8 for a complete number of periods), and if the energy obtained over part of a period is neglected, E2 =$s El d dn 1v (rrl()v at (1.65) where (nlg~v is the mean value of the energy flow vector. Both forms of the conservation law are identical when expressed in integral form. Despite the complete legitimacy and generality of the expressions for E2 and N~2 containing the elements of nonlinear acoustics, in linear acoustic it is entirely possible and more rational ;rnder the conditions of a homogeneous and stationary medium to use equa tions (1.58) for the energy and its flow. The equivalence of equations (1.54) and (1.58) no longer holds if the medium is noonhomogeneous and in motion. The equations for E2 and NACA TM 1399 $2can easily be generalized to the case of a moving medium. Rather comlicated expressions are obtained which will not be considered herein . As will be shown to sectica 7, it is essential that relatively simple expresgions are obtained for the energy density of sound E and energy flow N resembling expressions (1.58) and containing magnitudes of only linear acoustics in the approxlnation. of geom~etrical acoustics in a non homogeneous and moving medium. 4. Propagation of Sound in a :;cnbomojgeneojus Moving MNedium In the presence of air motion the acoustical phenomena become more complicated. Generally, separation of the acoustical phenomena, La the narrow sense of the word, from the doubly nonlinear processes taking place in a moving medium is not possible. T~hus, for exaple, the flow; pulsating iu velocity if the frequency of these pulsations is sufficiently large, acts on the microphone or ear located in it (not considering phenomena connected with vortex formation on the microphone body itself, see section 28) as a sound of corresponding frequency although the velo city of propagation of these pulsations has nothing in common with the velocity of sound. The relation between the pressure of these pulsationls and their velocity is nonlinear and also differs fundamentally from the~ relation. between the pressure in a sound wave and the velocity of sound vibrations. Finally, the variable nonstationary flow itself can be a source of sound. Phenomena of this kind will be considered later but this section will be concerned exclusively with the problem of the propagation of sound. In order for it to be possible to separate the sound propagated in the medium from the acoustic phenomena arising in the same medium only as a result of its motion, this motion will be assumed to be "soundless", that is, that the motions in the flow are sufficiently slow so that x > 9, (1.66) where z is the time during which appreciable changes occur in the state of the flow (for example, the period of pulsations of the flow velocity) and f is the frequency of the sound propagated through the medium. This condition requires additional explanations. I~t depends on the choice of the system of coordinates to which the motion of the florw is referred. In fact, a general translational motion of the medium has no signi ficance since it simply leads to a transfer of the soun wave. For this reason, it is sufficient that equation (1.66) be satisfied in some one systemn of th~e uniformly moving systems of coordinates. NACA T~M 1399 If, for example, a flow la considered to which the propagation of the velocities is stationary (that is, does not depend on the time, but the velocity of the flow periodically changes in space with the period 2), that. for this flow I = *D. If this flow is considered From the point of view of an observer moving with velocity n, the flow will appear to him nonstationary, the period of the velocity pulsations being equal to The phenomenon of the propagation of sound in the two systems of coordinates will differ only in the transport of the sound wave as a whole with velocity u. Since for the present interest is confined to the propagation of sound, this difference, which can easily be taken into account, is not essential. When the3 statement of the problem is broadened and a sound receiver is considered, entirely different results are obtained in these two reference systems. In the first system, in which the flow is stationary, the sound receiver would assume only one frequency f, the frequency of sound propagation. In the second system, to addition to this frequency f the receiver would also receive the frequency of pulsations in the flow, that is, f' = 1/7;' = afl and the combined frequencies f, = f + af'l n = 1,2,5,... In the following, condition (1.66) is assumed satisfied in any of the possible reference system~s. The effect of the flow on the sound propagation will then express itself in two ways: In the first place, the sound will be "carried away" by the flow and, in the second place, it will, be dissipated in the nonhomogeneities of this flow. In the derivation of the fundamental equations of the acoustics of a moving medium, the effect of the viscosity and heat conductivity of the medium on the sound propagation is ignored. This effect, which can more conveniently be taken into account as a correction, leads to the previous ly considered absorption of sound. The part played by these factors, which determine irreversible processes in hydrodynamics, may be very appreciable in the formation of the initial state of the medium in which sound is propagated. No less essential in this connection is the effect of the force of gravity. Bence the theory of the propagation of sound in a nonhamogeneous and moving medium must have as its basis the general equations of motion of a, compressible fluid. According to equations (1.12), (1.13), and (1.24), these equations are ; + div(py) = (1.67) 3Actually it changes somewhat because of the Doppler effect; see section 5. NACA TM .1399 + rot v +Vv T = + + v67 + V div v (1.68) +s (T + VS (1.69) where v = CL/P is the kinematic viscosity of the medium. Further, equa tion (1.13) was supplemented by the term +$Z, which represents the effect of the force of gravity. The vector g is the vector of the acceleration of gravity directed always toward the center of the earth. Thus P [ is the force of gravity acting on unit volume of thet fluid. Now let sound be propagated in a medium the state of which is des cribed by thae magnitudes vT, p, P, and S. The initial state of the medium (v, p, p, and 8) is considered stable and the sound is considered as a small vibration. All the previously mentioned magnitudes will then receive small increments: C, a, 8, and rJ, respectively, wherea 'i will be the velocity of the sound vibrations; re, the pressure of the sound; 8, the change tu density of the medium; and a, its change of entropy occuring on passing through a sound wave. In order to obtain the equations for the elements of the sognd wave ia equations (1.67), (1.68), and (1.69), v srpacdb +( ,b p + Ir, P, by P + 8, and S, by S + a; by restriction to a lineagC approxi mation, terms of higher order relative to the small magnitudes (, R, 8, and a are rejected. Moreover, as has just been mentioned, the irre versible processes taking place during the sound propagation are ignored, which means that in the linear equations for (, x, 8, and a the terms proportional to the viscosity (IJ or v) and the heat conductivity are rejected. Qa the basis of equations (1.16) and (1.5), the heat Q dissipated in the fluid likewise belongs to the number of magnitudes pro portional to p.By the method indicated, +. rot v, ;3 + rot v+ v(v, () V 9 1.0 + (v, V6) + (E, VP) + P div (; + 8 div v = O (1.71) + (, Va) + (5,VS) = O (1.72) The equation of state, which is given in thet variables P an~d S, is still to be added to these equations. For small changes of pressure st, and ia exactly the same maner as in the preceding section the, follorw ing is obtained: n = 028 + ha; c2 = h = (1.73) NACA TM 1399 Equations (1.70), (1.71), (1.72), and (1.73) are the fundamental equations of acoustics for a homogeneous moving medium (eQ. (1.74)). Their differences from those known ia the literature lie tu the fact that they are true in a medium the entropy of which varies from point to point (VS f 0) and in a flow in which vortices may exist (rot v f 0). The approximations made in these equations, in addition to linearity, consist ia the fact that no account is taken of the irreversible processes in the sound wave so that the sound wave is considered an adiabatic pro cess. This fact is also expressed by equation (1.72). In fact, it fol lows from this equation that d(S + o)/dt = 0, that is, the entropy of a given amount of substance remains unchanged with the passage of a sound wave. The entropy of the substance at a given point of space may vary; oalit f 0. La this sense the sound wave is not isentropic. The linear charac ter of the equations requires that a small disturbance remain small in the course of time (stability of the initial state). Bence it is not possible with the aid of these equtions to describe, for example, such interesting phenomena as the "sensitive flame" of a gas burner, the height of which changes sharply under the action of a sound wave. La other respects the equations are entirely general and it is quite immaterial in what manner the initial state of the median was formed. In bringing about this state, the force of gravity, the heat conductivity, and the energy flow fran. the outside (for example, the sun's heat) may be of considerable significance. The effect of all these factors on the sound propagation is taken into account in equations (1.70), (1.71), (1.72), and (1.73) through the magaitudes v, p, p, and S character izing the initial medium. The equation p = z(P, S) and equation (1.73) are valid only for a singlecomponent medium. In general, the pressure may depend not only on p and S but also on the concentration of the various components. In a coImplex medium it is necessary to take into account the diffusion of the various components. The corresponding uncomplicated generaliza tion of equations (1.70) to (1.73) will be made in section 13, where the case of sea salt water is considered. The choice of the thermodynamic variables P and S that has been made herein is very convenient for general theoretical considerations. For final numerical computations, however, the variables p and T are more convenient. For this reason, formulas are given expressing the mag nitudes (ap/aS)p and VS entering the equations through the variables p and T. VS = (aS/BT) VT + (aS/ap!fgp NACA TM 1399 on the basis of the known thermodynamic relations (BS/aT)p = p/T (c is specific heat at constant pressuree, (aS/ap)T = (av/bT), = Sp/p tP is the coefficient of volune expanaian and PP = .)~Ip . Klence VS = .VT~ .h Vp (1.74) Further, (aP/aT), = (ap/as),(as/br)p and (ap/aT) = (bp/IP)T TPaTp The magnitude ~(~,p and ()T = a2 = 02C c where c2 is the square of the adiabatic velocity of sound and (aS/aT) = c /T. Thus pc2 2 0T (1.75) Du the basis of equations (1.74) and (1.75) and the medium (c2 op,Pp) and its state (p and T as functions of the coordinates) VS and (ap/abS), can easily be found. The system of fundamental equations (1.70)) to (1.73), even if, with the aid of equation (1.73), one variable is eliminated (e.g., 8), contains five unknowns and is therefore very complicated. Nevertheless, if a canplete wave picture of the propagation, of sound is to be obtained, these equations cannot be avoided. The main complica tion lies in the fact that, because the pressure in the medium is a function of two variables (P and T or, preferably P and S), than even in a median at rest where not only vortices of the flow are: absent but where, in general, there is no flow, the right side of equation (1.70) will not be a complete differential of same function, and therefore the sound will be vortical (rot r 0 ). Considerable simplificationrs are obtained when the changes in p, p, and S are small ovetr the length of the sound wave. Ge~ometrical acoustics are considered in Igreater detail in the next chapter. NACA TMI 1399 For the present, certain special cases of the general system which are not reduced to the approximations of geometrical axoustics are considered. The most important special case will be the one for which the initial flow is not vortical (rot v~ = 0) and the entropy of the medium is constant (VS = 0). Under these conditions the pressure In the medium only of the density of the medium so that Vp = c2Vp. (1.72) it follows that for VS = 0, a = 0 so that the propagated isentropically. Then nI = c26 If the potential of the sound pressure is introduced is a function From equation sound will be n _x p (1.76) the right side of equation (1.70) will be equal to VH~. Therefore the velocity potential of the sound vibrations can also be introduced = V@ (1.77) The' sound will be nonvortical in this case. From equation (1.70) ~=n=~+I;f, B~p) d~pdt (1.78) Substitution in equation (1.71) of the magnitude n (for which al/at = (c2/p) (aB/at), VII = 8 Y(c2/p) + (c2/p) 78) in place of 8 yields the following equation for m =E c2i &9 (VDO, Vqp) + (v, V log c2) (1.79) where DO is the potential of pressure (heat function) of the initial flow 80 = (1.80) Equation (1.79) was derived by N. N~. Andreev and I. C'. Rusakov (ref. 10) without the last term, which was erroneously omitted. This equation exhaustively describes the propagation of sound in a medium in which. the entropy is constant. NACA TM 1399 A. M. Obukhov (ref. 11) gives an equation which permits an approxi mate consideration of the presence of vorticity of the flow but never theless makes use of one function, the "quas potential" 3r. Thia quasir potential, is Latroduced by the equation 5 = VlS +,, rot? Vrat (1.81) TIhe quasipotential may be introduced only for sufficiently small vorti city of the initial flow, that is, the assumption must be made that g = Irot 1 << w (1.82) where o is the cyclical frequency of the sound. Moreover the assumption is made that v/c << 1, so that the initial flow may be taken as incompressible (div v = 0). Finally the pressure of the medium is assumed as a function of the density of the medium onlyr. Since ap/ip is considered by A. M. Obukhov as the adiabatic velocity of sound, this implies the assumption thaet the entropy of the medium is constant. In connection, with this assumption, the question arises as to what extent the~ assumptions of the presence of vorticity (rot ? f 0) and the constancy of the entropy (VS = 0) generally apply together. The possibility is not excluded, however, that the influence of thet vortices on the sound propagation is more effective than the influence of an efn tropy gradient. These hypotheses are assumed satisfied and 1( i sub atituted from equation (1.81) into equation (1.70) and, since USi = 0, the right side of equation (1.70) will again be = mT. After simple reductions, the equation, which was found previously, is obtained. 8 d (1.83) P ,dt La~2 thi caehowevetr, it is true only approximately with an accuracy to Expressing 8 in equation (1,71) in terms of H and JI gives the equation of A. M. Obukhov: 2 eJ t2 (Pn0> hp 's V lo yIg c2) at c2 (VA d 90, ro V )t (1.84) NACA TM 1399 This equation holds with an accuracy up to g/o, 51/0 L (k = m/c). The magnitude Bb = rot rot v. I~n this equation, the terms of order v2/c2 can not be taken into account because in the approximations the assumption was made that v/c << 1.4 5. Equation for Propagation of Sound in Constant Flow En many cases the vgilocity of the flow v may be suitably separated Lnto the mean velocity V and the fluctuating velocity u. The effect of these two components of velocity on the sound propagation may be dif ferent. The mean velocity of flow produces the "drift" of the sound wave while the second variable part of the flow velocity leads to the dissipation of the sound wave. This phenomenon will be considered in more detail later. For the present, attention is concentrated on the effect of the mean flow velocity and the equations are considered for the sound propagation, with the variable part of the flow velocity u ignored. The solution obtained under these conditions is of interest not only as a first step toward the approximate solution of the complete problem with the velocity fluctuations being considered but is of value in itself, especially for the theory of a moving sound source. En order to obtain an equation for the propagation of sound in a homogeneous forward moving medium, it is sufficient to put VnO = 0 and VJlog c2 = O in equationa (1.79). Expansion of the total derivative with respect to time Cd29/dt = (a/8t + (v,V)(a(P/at + (v,9) yields 69 22~ ,V (q;r) = (1.85) c2 at2 c2 2 If the Xaxis is taken in the direction of the mean velocity and B is set equal to P/c, (1 0) 82rp a2p a"29 1 aip 20 a2 = (1.85') ax2 dz2 2i2 c For the system of coordinates 5, rl, and C moving together with the stream r = x Vt, rl = y, and ( = z, equation (1.85') is transformed into the usual wave equation a29 a2(p a2( 1 az24 + +t O (1.86) a, g2 at2 c2 h t2 4The result of A. M. Obukhov is. probably more rigorous and could have successively been obtained as the second approximation of geometri cal acoustics (see section 7). Ir 26 NACA TM 1399 as expected, since in this system of coordinates the medium is at rest. Certain important solutions of equation (1.85') are now available. A plane sound wave is first considered. I~n the system of coordin ates r, 1, and at rest relative to the air (hence for an observer moving with the stream), this wave has the potential 9P(E;,q, t) = Ae ci a2 +Z + + 2 (1.87) where al, a2, and a3 are the direction cosines of the normal to the surface of the wave; o the frequency of the oscillations; and c, the velocity of sound. ~Equation (1.87) is a solution of equation (1.86). According to the previously mentioned transformation, the solution of equation (1.85') is immediately obtained if 5 is replaced in equation (1.87) by x vt, rl by y, and t: by z cp(x,y, z,t) = Ae lx + a;y + a (1.88) where W' = e +v a (1.89) Thus the sound frequency tu a stationary system of coordinates will not be a but o'. This change of the frequency of the sound is the acoustical Doppler effect. The effect has an exclusively kinematic origin; it depends only on the choice of the system of coordinates. The entire difference in the propagation of a plane wave in a moving medium as compared with a sta tionary one reduces to this kinematic effect. Later the Doppler effect will be considered more fully; not only the motion of the observer of the sound will be taken into account but also the motion of the sound source itself, which at present does not enter explicitly in the computation. A second important form of the solutions of equation (1.85) is pre sented by sound waves diverging from a certain small point source of sound (or, on the contrary, converging to it; io the latter case a sound "sink" is being dealt with, which is a very artificial but mathematically useful concept). The ma3themnatical expression for the potential of such waesPE1 is a generalization of the potential of spherical waves for a medium at rest. NACA TM 1399 27 This potential of spherical waves is a solution of equation (1.86), having the form XO F (tg (2 q2 4 (2 (1.90) where F is an arbitrary function. The solution with the minus sign is given by waves diverging from a sound source located at the origin of coordinates (5 = rl = ( = 0) and the solution with the plus signl represents the same waves converging to a sound stak at the origin of coordinates. If F is a harmonic function, the following is obtained from equation (1.90) xO e (1.90') that is, a spherical harmonic wave with frequency o. In a moving medium in which the propagation of sound is described by equation (1.85') in stead of solutions of the form of equation (1.90), the more general ex pression is obtained. F t + / X = RMc(1.91) where R = Brx RlE RM = 1 xy 2 + z2 x*= X (1.92) With the substitution of X from equation (1.91) into equation (1.85), it is not difficult to show that equation (1.91) is in fact the solution of equation (1.85), which moreover transforms into a solution of the form of equation (1.90) for Y = 0 (p = 0). The solution (eg. (1.91)) for a moving medium thus has the same value which equation (1.90) has for a stationary medium; it represents waves diverging from a point source or waves converging to a sink. 5The origin of this solution is clarified in detail ia section 15. an arbitrary point of the space Q, with the coordinates xQ,y4,z&, so that x =x x y = yQ yp, and z = zQ zp. 28 NACA TM 1399 6. Generalized Tfheorem of Kirchhoff In the theory of the propagation of waves, an important part is played by the theorem of Kirchhoff, which permits expression. of the oscillations at any point of space La terms of the oscillations at the surfaces bounding the space considered (including also th1e surface at infinity). This theorem is derived for a moving medium, starting from equation (1.85') (ref. 12). This equation, if the coordinate system x",y,z contracted in the xdirection is introduced x* = x ; = Y; a = s (1.93) assumes the form 1 a29 AW  c2 t2 2P 1 a2p ~~1~~ O atx (1.94) where a = a2/axec2 + 2/as2 2 a2/ z The singular solution X (eQ. (1.91)) likewise satisfies equation (1.94) 2P 1 a2X. c' 292 = 0 1 a2x 2 2 cz at (1.95) The solution X contains the arbitrary fiction F which, because of later utilization of the solution for the proof of the thetorem of in terest, is specialized. R 8 t +  c X R , gx1c + R+ R = (1.96) dw~2 2 2z~ from which the potential 9 where R is thet distance coordinates xq,Yp,zp, at thet potat P, with the is to be determined to NACA TM 1399 The function 8(r) is determined such that aS f(r) 8(5)d5 = ~f(0) if b > 0, a < 0 (1.97) vb a if > O f(5)6(5)d( = O Equation (1.97) is assumed valid for any faction f(r) so that 8(r) is everywhere equal to zero except at the point ( = 0, where 6(() = Hence B (t + R/c)/RY represents a converging spherical impulse (shock) concentrated about R = ct. A certain surface is considered (see fig. faces S1 and S2; the S enclosing the volume D in the space xu,y, z 2 where the surface S is formed by two sur volume 2is crosshatched). After equation (1.95) is multiplied by tP and equation (1.94) by X, one equation is subtracted from the other and the result is integrated over the volume and over the time t1 to t2. Integration over the fourdimensional volume g2(t2 1l) yields 9   ( dG X  = 0 7 2 2 (1.98) Applicationa of Green's transformation results in d9(@AX XA) = dS 9  X (1.99) where a/in denotes the derivative along the external normal to the surface S enclosing the volume SZ. At the point P the transformation (eq. (1.99)) will fail because at this point X becomes infinite. The point P is surrounded by a small surface I, and the volume AS 2 at S dQ(PAX Xllp) + 12d1 dG 2 t NACA TM 1399 enclosed by it is excluded from the volume of integration P in equation (1.98). The surface C (see fig. 2) is considered as part of the sur face S. The normal to the small sphere Z is denoted by N and di rected toward the interior of the volume. If Green's transformation, equation (1.99) to equation (1.98), is applied, the following results: 2 at 2~ dt ,  X + t d2 , XI ~  ZB I fr;t2 ""S, c I nl1 2 3tl B d2 9 az  X ' (1.100) The second integral on the right permits carrying out the with respect to time integration dd( O 9 (1.101) +* so that t1 + R/c < O and at t1 and t2 are equal to X; hence I2 = 0. The first in But if tl tends to oD and t2 to t2 + R/c > O, then both X and ax/St zero on account of the form chosen for tegral on the right is considered (1.102) Integration by parts of the second term with respect to time and use of the property. of 8 (eq. (1..97)) yield SdS~  X ~ L2 IE 2 at dR = I dG X 9 *)) 11~~~ ~~ 2R 1 as dS9 89  NACA TM 1399 i~t * 9t= 1 1 c R+ c (1.103) where (p, aq/in, and abp/at are taken at the instant t = R/c. ICa a similar manner the third integral on the right in equation (1.100) gives a2,x at ax+ I3 = 12 2  Sda ~ ~) dSx~ R ) = 22 281 B 1~)2 (1.104) t=_ c =p 12 1 ~ B .~ dSx where dSx is the projection of the area n dS on the flow velocity VO (on the xaxis). The integral in equations (1.100) on the3 left is trans fonmed exactly as the first and, since in this case a/a& is identical with a/aRIC, II; , S1 2t s ~~ NIAQA TM 1399 IO J 2 cR~~a R R 1.05 c and, since dC = 4nR"2 dR*C, as the radius of the sphere R* approaches zero, the following is obtained: IO = t=0 110' Thus on the left the value of the potential at the point P at the in atant of time t = O is obtained. Since this instant is arbitrary, if the time origin is everywhere shifted forward by t and all the inte grals Il, 2>, and 13 are collected, the potential at the point P at the instant of time t will be 12 2 _dSx (1.106) where the brackets indicate that the magnitude enclosed by them is taken at the instant of time t R/c. For YO = 0 (p = 0), RxM = r and R = r and this equation trans forms into the usual equation. of K~irchboff for a medium at reat. If the potential depends harmonically on the time so ~th (1.107) NACA TM 1399 then substitution of equation (1.105) in equation (1.104) yields for the amplitude an = ~ cljcf dS  2ipk 1e ikR 4~i2 W Sx (1.108) where k = w/c is the wavenumber vector. If, from the nature of the physical problem, it may be assumed that the disturbances giving rise to the vibrations start within the surface S1 and not at an infinite time barck, they do not~ have time to be propagated to the surface S2 at a great distance from S.For this reason, if S2 is shifted to infinity, the values 'Pt a'P/in, (p/at can be assumed equal to zero in it. The volume S1 then takes up the entire space with the exception of S1 in the interior. If the presence of an infinitely removed surface is "for gotten," it is natural to call the normal n the interior normal since it is directed inwards from the surface S ihnwihtesucso vibration are concentrated according to twti hc the prsn asmton.c Unde this condition equations (1.104) and (1.106) may be assumed to give the expression of the potential at any point of space in terms of the values cP, AP/i3n, and afit on the surface S1 within which (or on it) the sound sources are concentrated. In~ conclusion, a certain generalization of this theorem is considered for "volume" sources of sound. It is assumed that equation (1.94) has a right side which is considered as a volumee sound source." The strength of this source is denoted by Q. Equation (1.94) can then be written La the form 1 a2(p 28 1 a2( 69c2 at2 2 c 92 69 = 45Q (1.94) Such equations are encountered, for example, in the problem of the dissi pation of sound by a turbulent flow (see section 12). If t~he same opera tions which were applied to equation (1.94) are applied to this equation, an expression is obtained for cp differing from equations (1.106) and (1.108) by a volume integral. The additional term, on multiplication of equation (1.94) by X, will be 14 = 4n 2 dts dG "X (1.109) NACA TM 1399 Integration over t yields (on account of the 8 function form of X) 14 =4 6Q R (1.109') Hence, in place of equations (1.106) and (1.108), there are obtained I ~ia~~ d S 4 I 2P 2 1y P c tdSx (.0' and ikR QS +eC ikR =S ~ d o S2i k eiR Rn dSx (1.108') 4 1 B c if the strength of the source depends harmonically on the time Q = Qeiut (1.110) The theorems derived herein are used in the theory of wave propagation from a moving source, in particular from an airplane propeller, and to the~ problem of the occurrence of vortical sound in the motion of bodies in the air. NACA TM 1399 35 CHAPTER II PROPAGATION OF SOUND IN ATMDSPRERE AND IN WATER 7. Geometrical Acoustics In the study of the propagation of sound in the atmosphere or in water, the state of the medium generallyr changes little over a distance equal to the length of the sound wave 1.In the background of this slow change of state of the medium there can also exist smaller changes, but these give rise to secondary effects which may be considered sepa rately (see section 12). The main features of the sound propagation picture are determined by the slow changes in the state of the medium (for example, changes in the force of the wind and in the temperature and density of the air with increasing distance from the ground surface). Under these circumstances the application of the methods of geometrical acoustics is suitable. The fundamental equations of geometrical acoustics are derived in this section (ref. 13). A start will be made from the fundamental. equations of the acoustics of a moving, nonhomogeneous medium (section 4). These equations are ar+ rot (xy] + (rotx] vx]+ C, +) = J + 2 (2.1) + (t;, Op) + (v, VB) + p *div ( + 8 div v= 0 (2.2) gg+(~ (7; ) + (5, VS) = 0 (2.3) It = c28 + ha (2.4) The change in v, p, p, and S is assumed small over the distance of a wavelength of sound. Use is made of this fact for the construction of an, approximate theory of the propagation of sound: ( =~,l O e@Oe@ g 0ei a ~ i' (2.5) cf = cut kgO8 (2.6) NACA TM/ 1399 where (o is the frequency of the sound; kO = mcOc = 2r(/XO is the wave number in the medium, the state of which is assumned normal (00 is the normal velocity of sound); and kgO is the phase of the wave. The magnitudes Ob, x07 Og, and oGO are assumed to be slowly varying functions of the coordinates and, possibly, of the time. The number kO will be assumed large so that the phase kOS, on the contrary, varies rap~idly. The solutions for 30, ngO O, and GO will be sought in the form of series in the reciprocal powers of the large number ik : 8o = 8' + + * O O ikO II = + 1 +* * O 0 ikO (2.7) Substituting equations (2.5) and (2.3) and making use of equation (2.6) in equations (2.1), (2.4) result in (2.2), and (2.8) (2.8') (2.8") where q = cO T'7, 98) S= 5 + [(EOxrot v] + [VXrot (O] 0(V FO VxO V p x0~ hoO P c 1~ a"O h v800 )(' 1 1o 4CvV b4 = 2 at 2 5 ~ V (OhO v, O c c e V(h, o0)] (VP, 0) "0* O O~2G div O b5 = 2 (~ (7 00 (0, VS) (2.9) (2.10) (2.10') (2.10"l) ikO 0s 98 ~= d ikOC~ cPx'O GO 0, VS)= b4 ikOgOa = b5 NJACA TM 1399 Substituting (O, nO, and GsO from equations (2.7) into equations (2.8) and. (2.8') and collecting the coefficients of the same powers of ikO give for the zeroth approximation (the coefficient of the zeroth power of ikO) Qg6 VB p O (2.11) q/c2(n0 had) p((O> VS) = 0 (2.11') qGO = O(21" and for the first approximation (the coefficient of the first power of ikO> q/c2(n"; ha i) p(g Pg) = b4 21' q aj = bS (2.12") where b', b and bS are the values of b ~b4, and b5 on substi tuting in them the zeroth approximation of ~t, ni and GOt from equations (2.11), (2.11'), and (2.11"). From equation (2.11"), it follows that at = O, that is, in the zeroth approximation of geometrical acoustics the sound is propagated without change of entropy (isentropically). Solving equations (2.11), (2.11'), and (2.11") gives, in the first place, the equations connecting the velocity of the oscillations with the pressure 5 O pl;; (2.13) and as the condition of the simultaneity of equations (2.11) and (2.11'), the equation of the surface of constant phase (0= constant) is c~l= 2 (2.14) For v = 0, as is seen from equation (2.9), q2/c2 = e0/c2 = .2 where I4 is .the refraction index for sound waves. The equation NACA TM 1399 rV0 2 C2 is called the "eikonal" equation. For v / 0, the ratio q/c may likewise be considered the index of refraction of thne me~dium~, but it now depends also on the direction of propagation of the waves. The siT~uation is similar to that in crystal optics, but more com~ plicated because for acoustics the medium is not only anisotropic but also nonhomogeneous, since the position of the axis coincides with the position of the wind or flow which changes from point to point. Sub stituting in equation (2.14) the value of q from equation (2.9) and solving equation (2.14) for IVOI = 8/an, where aefin denotes dif ferentiation along the direction of the normal to the surface of con stant phase (O = constant), give ve (2.15) where v, is the projection of the velocity of the wind on the normal to the wave. With aefin known, the phase velocity. of the waves Vi can be determined. The equation of the moving phase surface is 45 = cut = kOS = constant. Differentiating this equation with respect to time results in be an be W kO aJE dt kO a;n Vf = O (2.16) On the basis of equation (2.15) there is then obtained Vf = c + vn (2.17) that is, the phase velocity of the waves is equal to the local velocity of the sound plus the projection of the velocity of the wind on the normal to the wave. This kinematic relation is clarified in figure 3; equation (2.1_7), which was obtained as a consequence of the strict theory, was put at the basis of a geometrical theory of sound propagation as one of the initial assumptions by R. Emden (ref. 14). It is important, however, not only to find the geometry of the wave field but also to compute the magnitudes characterizing the intensity of the sound. The equation for the determination of the sound pressure XO is obtained from the equations of geomnetrical acoustics (2.11) and (2.12). This magnitude is generally. measured in a test. The equations of the second approximation (2.12) are used to obtain this equation. Th~e left sides of these equations agree with equations 211. If thie notations Ob = (xlr x2, x ), "O = x41 and "O = x5 are introduced and equations (2.11) are written in the form aik* t 0 i = 1, 2, 3, 4, 5 (2.18) NACA TMI 1399 equlatiojns (2.1_2) can be written in the form aik x~ = bi i = 1, 2, 3, 4, 5 (2.18) k=1 By a known theoremof algebra, equations (2.18) will have solutions x" only when the right sides are orthogonal to the solutions yk of the adjoint system of equations: Rik Yk = O where ~ik = aki (2.19) k=1 The condition of orthogonality is bkk=O (2.20) k=1 With aik determined from equations (2.11), (2.11'), and (2.11") and aik; transformed, yk is obtained from equations (2.19) in the form h y' = pO .q =S y g 75 TZ q (2.21) Substituting by b4, and b5 from equation (2.10) in equation (2.20) and making use of equation (2.15) give the condition of orthogonality (eq. (2.20)) in expanded form: 2 g + 2n0 div VS + 2VWO s,, log p qc2] O = 0 (2.22) where the velocity VS is given by (see fig. 3) VS = en + v (2.23) n being the unit vector along the normal to the surface of constant phase. Dropping the strokes of nO and (O), because the zero a~pproxi mation is concerned in what follows, equation (2.22) is multiplied by It and an equation for the square of the pressure amplitude is obtained: NACA TM 1399 b + div (Sn2 (iS' 9 log pqe2)x2 (2.24) which together with equation (2.13) 'i = 99  (2.25) pq completely solves the problem of obtaining tlhe sound pressure n and the velocity of the sound vibrations r. Equation (2.24) may be considered also as.a certain conservation law. In fact, the mean kinetic energy of the sound vibrations T is defined by the equation T = (p + 6)( + r )2 p2 g,()(.6 where the remaining terms are rejected either as magnitudes of third order smallness or as magnitudes which within the framework of the lin eair theory should, on the average, give zero (for example, p( Z, P)). Since 8 = x/c2 (compare eq. (2.4)), 1 2 2 2 T~~ =v 7 v .2 (2.27) PQ pqc Adding the mean potential energy of the second order U 1 x2 U (2.28) 2 2 pc results, on the basis of equations (2.9) and (2.14), in s = T +U = (2.29) pqc If equation (2.24) is divided by pqe2 CO, then after simple reductions, gg+ div(sVS) = (2.30) that is, the law of conservation of the average energy in geometrical acoustics. This law, like the law for E1 and 1f (see section 3), is remarkable in that it contains only magnitudes characteristic for linear acoustics. It is valid for any nonhomogeneous and moving medium pro vided only that the length of the sound wave is sufficiently small that the approximations of geomnetrical acoustics are applicable. The magnitude EVg is evidently the mean energy flow ~= E3S (2.51) It follows immediately that the sound energy~ is propagated with the velocity, VS = n + v, different from the! phase velocity Vf. The vel~oc it~y V is called the ray velocity. This velocity is equal to the geometric sum of the local sound velocity on and the wind velocity v. It coincides with the velocity of weak explosions according to Hadamard (ref. 15). On the basis of equations (2.23) and (2.25), the energy flow may also be represented in the form N (+ 2 T (2.31') Fo = O, q = c0; and the previously derived (section 5) equation for the flowu N = 1x( is obtained (the expression Nl = "l 1 differs, how, ever, from N = rc since the latter vector represents the average value in time of the energy flow while Njl is its instantaneous value). If the proccess is stationary, so that the mean energy of the sound field doeE, not change (at least where the sound field has already filled the space), from equation (2.30), div (iVS) = O (2.30') From this equation it follows that, if tubes are constructed the lateral surf'acnes of which are formed by lines along which the ray velocity is directed ("ray tubes," fig. 4), the product e VSys(s is the cross section of the tube) is constant eVSs = constant (2.52) Substituting the value of e from equation (2.29) gives "2VSs = Il2slel pqcZ (2.53) where alr 81,sl' l1 P' 91, and C1 are values of these magnitudes at anyi chosen section of the tube. This equation permits computation of the pressure of the sound at any part of the ray tube as soon as it is kntown at any section of it. To obtain the geometry of the ray tubes, however, a solution of the problem of geome3trical acoustics (equation of the eikonal (2.14)) is required.' NACA TM 1399 42 NACA TM 1399 8. Simplest Cases of Propagation of Sound A. Propagation in an isothermal atmosphere. In an isothermal atmosphere at rest, the velocity of sound is constant (since it depends only on the temperature). Thus c = cO = constant. The magnitude q = CO (since =).Hence, from equation (2.33) for the conditions considered, rr2s = X2s1 P/ (2.34) In the special case of a plane wave, the cross section of a tube is constant (s = sl) and rr = nl 1 1pp~ /2 (2.35) that is, the pressure of the sound is directly proportional to the square root of the density of the medium. The ratio p/pl in an isothermal atmosphere is determined by the barometric formula pIpl = eX*H (2.36) where x = N~g/RT, H is the altitude, M is the molecular weight of the air, g is the acceleration of the force of gravity, R is the constant gram. molecular weight of the gas, and T is the temperature. From equations (2.35) and (2.36) it is seen that the pressure will decrease with altitude by the exponential law. If the wave .:'s not plane but spherical, the cross section of the tubes increases as the square of the distance from the source r2. Hence for a spherical wave in place of equation (2.35), x =s1911/ (2.35') TIhe velocity of the sound vibrations t, in contrast to the pressure, will increase. In fact, for a plane wave 90 = n (n is the unit vector in the direction of the normal to the wave) and therefore from equations (2.25) and (2.35) there follows (+ = sl 91 1/2 = n* sl (p 91 1/2 (2.37) The mean energy flow N = ag = n (2.38) lel1 remains constant. NACA TM~ 1399 In a similar manner, for the spherical wave, x l F (2.37') 2 2 rl al N = n(5 = n (2.58') r2 plel. where n is again the unit vector along the normal to the wave, that is, in the direction of a ray issuing from the source. B. Case of the presence of a temperature gradient. Let the tem perature T be a function of the altitude y. The velocity of the sound c will then vary according to the law c = T E= 'CrTF (2.39) and the index of refraction of the sound wave CL Will be C1 = =i (2.40) The equation of the surface of constant phase (equation of the eikonal) in the absence of wind will, according to equation (2.14), read +ay = 2 (2.41) (The xaxis is directed horizontally (fig. 5) in the plane of the sound ray and therefore it is assumed that O does not depend on z.) The cosine of the angle 4 between the xaxis and the normal to the wave will be cos c = Ex ~ ao y~ (2.42) Let a8/ax = cos cPO, where 90g is the value of cp for y = O, that is, on the ground surface, where T = TO. From equations (2.41) and (2.42), NTACA TM .1399 cos cP = cos cpO J (2.43) From this equation it is seen that, if, as is generally the case, the temperature drops with the altitude, cos cP will decrease in absolute magnitude and therefore thie ray will be deflected from its initial direc tion upward (fig. 5). By use of equation (2.43), if the temperature distribution over the layers is known, the entire curve of the ray can be constructed. C. Propagation of sound for a stratified wind. The case of a medium of constant temperature and density wherein there is a horizontal wind (let it be directed along the xaxis) the force of which varies with the altitude is now considered. Let the velocity of the wind be v = v(y) (2.44) Then according to equation (2.49), the magnitude q is equal to q = cO v(y)~ (2.45) and on the basis of equation (2.14), the equation of the eikonal will be where y(y) = v(y)/cO' The velocity of thae wind at the ground surface itself (y = 0) will. be assumed equal to zero (y(0) = 0). Assuming, also, as in (B), that the initial angle of the normal to the wave is equal to c90, a8/ax is set equal to cos tp0 and from equation (2.46) is obtained cos 90 cos =llcsPY (2.47) Fran this equation it follows that if the ray6j is directed along the wind (y cos 90g > O), then as the velocity of the wind increases with 6In the presence of a wind, as was already pointed out, the line of the ray differs from. the line of the normal. Since, however, v/c<1, this difference is not large. NACA TM 1399 the altitude, cos cP increases in such a manner that the ray is deflected toward the earth. (fig. 6), while a ray traveling against the wind is deflected upward. This upward deflection is one of the reasons for the imupairment of heating in a wind. Consider a ray which in the absence of wind almost glides over the surface of the earth (fig. 7). In the presence of a wind the force of which increases with the altitude, this ray is deflected upward and passes by the receiver P. This does not mean, of course, that at P nothing will be heard since other rays will arrive there, but the intensity of the sound will be considerably weakened (small number of rays). If the force of the wind drops with the altitude, the same conclusion will hold for the propaga tion of the sound along the wind direction. In those cases where not only the force of the wind but also its direction varies from layer to layer, the picture of the sound propaga tion bEComeIis considerably more complicated because the rays will be curves of' double curvature. 9. Propagation of Sound in a Real Atmosphere. Zones of Silence Under the conditions of the real atmosphere all the factors con sidered (wind, temperature gradient) act simultaneously and in a very complicated manner since the variation of the temperature, force, and direction of the wind may be very different. In the general case the direction cosines of the normal to the wave a, P, and y are again determined from equation (2.14). Since cO/c = Jif~Tf and ii~a = 1c ~ 0 = 1 1 (2.48) a8 rTO 1 dz Tt (T V~vO For their determination, it is thus necessary to know the function O from equation (2.14). NACA TM 1399 As could have been seen from the equations of the preceding section, an essential part in the propagationn of sound is played not so much by the temperature and the force of the wind as by their change. It is found that negligible gradients of the temperature or of the wind force lead to considerable curvature of the sound rays. Several illustrations borrowtJed from the paper by R. Emden (ref. 14) are presented. In figure 8 is represented the case of the propagation of sound in an atmosphere in which the temperature drops by 6.2o in 1 kilometer; on the ground surface up to an altitude of 370 meters there is assumed a calm, but further on the velocity of the wind increases by 4 meters per second per kilometer. In this case there is formed a wide "zone of silence" lying to the right of the sound source. The sound reaches the surface of the ground only at a considerable distance from the sound source (beyond 159 kilometers). Similar regions of sound shadows are seen in figure 9 where sound rays are shown propagated in ~an atmosphere in which up to a height of 910 meters the temperature drops by 30 while the wind increases by 2.13 m~eters per second, and higher up the temperature drops by 3.650 in 1 kilometer and the wind velocity likewise drops by 3.28 meters per second in 1 kilometer. Zones of silence were first observed in the last war when it was found that the audibility of an artillery cannonade was greater at places further removed from the sound source than in its neighborhood. Very brilliant and detailed computations of the propagation of a soundwave front in a nonhomogeneous atmosphere in the presence of wind may be found by the reader in the work of S. V. Chribisov (ref. 16) in which examples of zones of silence are likewise given. The velocity of propagation of weak explosions (according to Ht~adamard) which figures in the work of Chibisov agrees (ref. 12) with the ray velocity VS introduced in section 7. Since it is not possible to enter into more detail in regard to the computational problems of air seismics, the discussion of these problems is limited to the illustrations given and to the references cited. 10. Turbulence of the Atmtosphe~re The propagation of sound in a medium the state of which changes little over the distance of a soundwave length was considered in the preceding section. In the real atmosphere such a method of treatment gives only the main features of the sound propagation. As a matter of fact, in addition to the slow change of state of the atmosphere from one layer to the next, there are also more :rapid changes brought about by accidental fluctuations in the velocity of the wind, namely, th~e turbulence of the atmosphere. These changes may be very rapid and their effect on the sound projpagation catn by no means always be considered by the methods of geometrical. NACA TM 1399 acoustics since the dimensions of the region in which an. appreciable change of state of the medium occurs may be entirely comparable with the length of the sound wave. Before considering the effect of these phenomena on the sound propagation, the fundamental laws of turbulence are considered. The theory of turbulence forms a very extensive and as yet far from fully developed field of hydrodynamics and aerodynamics. At the end of this chapter the reader will find references to the fundamental literature on this subject. The work of A. N. Kolmogorov, M. D. Millionshchikov, and A. M. Obukhov in recent times has greatly contributed to the development of the theory of turbulence. The scope and purpose of this book do not permit any detailed consideration of these works. The discussion is restricted to what is most required for present purposes without pretense of mathematical rigor. The velocity in a turbulent flow v(x) is a random function. The entire velocity field of such a flow may be represented as a system of distulrbances ("vortices") of different scales.~ The largest vortices are defined by the dimensions of the entire flow as a whole L. The meaning of the magnitude L may be very different. For example, it may be the height of a layer of air above the surface of the ground, the dimensions of the body, or, it the turbulence is brought about from the initially laminar flow about the body, the dimensions of the pipe from~which the stream issues, and so forth. 'These largescale disturbances break up into smaller vortices and the dimensions of the smallest are determined by the viscosity of the medium, since very sharp changes in the motion of the medium rapidly die down precisely on account of the viscosity (compare with the dissipative function Q introduced in section 1 from which it is seen that the energy, of' the flow converted into heat because of the action of the viscosity is greater the greater the gradient of the flow velocity). Such a picture of the distribution of the velocities of a turbulent flow over different scales of disturbances with successive conversion of the energy of the large disturbances into the energy of small distur bances and finally into heat was first clearly described by Richardson. In order to characterize mathematically the spectral distribution of the velocity of the turbulent flow v~x) over the different scale disturbances, the velocity v~x) is expressed as a Fourier type integral vi ( )=eXx) id( (2.49) NACA TM 1399 where vi(x) denotes a component of the velocity of the turbulent flow (i =1,2,3 are the numbers of the axes ox, oy, oz), qi(ql' P2' 12) is the wave vector belonging to the scale 2 = 2n/q, and dS1(() i an element of volume in space of wave number q. Finally, U (dBl( )) is the (infinitely small) Fourier amplitude defining the magnitude of the ve locity pulsations of scale 3. It is an additive function of the vJolume Uig ( 51 2) = Ui P1) + Ui 21) (2.49') If vi(x) were a continuous function of the point x: there could be written: Ui (d1 i(+q) *dgr(Vi(q)); the "density" of th~e velocity in space Q and the additive property would t~hen be trivial since vid~ + idG U( ) + (2 (2.49") The density vi, however, may not exist while the additive? property, as a more general one, may be maintained (for example, discontinuous functions). In particular in this case, Ui(dS1) is a random function (in the space Q) and cannot, in general, be assumed as continuous. Hence it is necessary to make use not of the Fourier integral but of the more general expression (2.49)7, The following assumptions are made relative to the statistical properties of Ui: (1) The velocity fluctuations associated with the different scales are statistically independent so that the mean of Ui(a1) U (;92) is equal to zero 7With regard to the mathematical basis of the expression of a random function as an integral (2.49), see A. N. K~olmogorov (ref. 18). In the following discussion, the presenitation of A. M. Obukhov (ref. 19) is followed (essentially). The same results, but by a somrewh~at different method, were obtained also by Kolmogorov (ref. 20). IJACA TM 1399 Ui 21) Uk 9(s2) = 0 (2.50) if t~he volumes 511 and 512 do not overlap (which means that the Ui and Uk belong to different q). The asterisk denotes the conju gate complex magnitude. (2) For coinciding volumes it is assumed that Ui 1)U (91) =ik S1) (2.51) is an additive function of the region 3. Physically this means that the intensities associated with the different scales of turbulence are combined. Since 9ik is a certain mean magnitude, it may be a smooth function and may be expressed in terms of the "density"' Wik: lik(B i likq)ag (2.52) The value ikshall be called the spectral tensor since it dietermines, as will be seen, the distribution of the energy in a tur bulent flow over the different scales of the fluctuations 3 = 2sn/q. If interest lies not in the complete velocity of the turbulent flow but in only that part of it T;rP) which refers to the velocity flue tuations having a scale less than 2 = 2xr/p, the expression for rfP01) is obtained from equations (2.48) if the integration with respect to q is extended over the range q > p: S x) = ei ', )U(dG(4) (2.53) The "moments of correlation" MP (x', x") are determined by the ik equation M kCI)xr, x) = v (x~') *v~kx)(.4 that is, as the mean of the product of two velocity components v~i and vk taken at two different points x' and x. The set of magnitudes Mkxx")(i, k = 1, 2,t 3) forms the tensor of the correlation moments. For homogeneous turbulence, that is, such that the states of the flow at different points of space do not differ from one another, the tensor of the correlation moments will depend only on the distance between the NACA TM 1399 points x' and x", that is, on p = x' x". Subtstituting vp(x) from equation (2.53) into equation (2.54) gives N kb)= ip( xpU( d2( q') ~ ei~q, x) U (dz( ")) (2.55) Use is made of ing to different q (2.49), (2.51), and the statistical independence of Ui and (condition (2.50)) and of the additivity (2.52)) to obtain Uk belong (conditions M k($ ) = e ) fik@)da (2.56) The motion of the fluid is considered incomrpressible so that div V = 0. From equation (2.53) there then follows: >:,i o (2.57) Cxi Applying this relation twice to equation (2.54) (differentiating once with respect to x' and again with respect to x"+I) results in a2M k x~', xi = 0 i, k=i (2.58) From the preceding and from equation (2.56) it then follows that the spectral tensor tik( i) must have the form 9ik ik ik 2 fq) ~]= (bitq (2.59) This tensor is now connected with the energy distribution in a turbu lent flow over the fluctuations of different scales 3. T~he energy shall be considered as referred to unit mass so that the measure of energy will be v2/2. The mean energy E(p), referring to the velocity fluctuations the dimensions of which are less than I = 2xr/p, will be equal. to NACA TM 1399 3 53 i=1 1 3 E(p~) = v()2 = v() i=1 L b ~ii (_d (2.60) or on the basis of equation (2.59), E(p) = da f;(q)q~dq (2.61) E(p), use is made of dimensional con not only homogeneous but also isotropic the mean). The turbulent motion of a certain constant supply of energy energy of solar radiation giving rise For determining the form of sidera~tions. The flow is assumed (of course again statistically in such a flow must be maintained by from outside, for example, by the to the motion of air currents. This same energy, since a stationary state is considered, is dis sipated in turbulent motion, being converted because of the action of viscous stresses into hea~t. The energy dissipated shall be denoted in unit time (per unit mass of gas) by DO. (It is equal to the supply of energy from outside.) The dimensions of Dg are L2T3(cm2/sec3)~ In a developed homogeneous and isotropic turbulence its spectral state must be determined by the supply of energy which maintains the turbu lene, ha is Ep) FDO~).Representing F in the form Dn pm, a dimensional equation for determining n and m is obtained in the form. L2r2 r= (L2T3)nLm (2.62) from which n = 2/3 and m = 2/3. nondimensional combinations from DO The impossibility of forming any and p leaves E(p) = constant D2/3 p2/3 (2.63) A more detailed ~analysis by A. M. Obukhov (loc. cit.) shows that contan = E x2/3 where x is a certain nondimensional number of the order of 1; thus, in the notation of Obukhov, E(p) = /2 * Since p = 23x/2, E(P) 2 2/3 p2/3 (2.65') (~?2/3 NACA TM 1399 This law, established by A. M. Obukrhov (ref. 19) and A. N, Kolmogorov (ref. 20) is usually briefly referred to as the "'213" law. From the law it follows that the energy of homogeneous and isotropic turbulence is concentrated mainly in the region of la~rgesea~le flue tuations of the velocity. The value of the energy E(3) is restricted by the maximum scale of the turbulence L determining the dimetnsion of the flow as a whole. For atmospheric turbulence L is the height of observation above the earth's surface. Differentiating equation (2.61) with respect to p and using equation (2.63) give f(p) = pll/33 2 D 2/ (2.64) and therefore the spectral tensor is equal to ikil kT1/ (2.65) In concluding, the meansquare difference of the velocity component taken at two different points of space is comrputed: (vf[(') v22"))2 = 2((v (xt) 2 p ,) p( ")") (2.66) On the basis of equation (2.54), (U@(x')~ v (")2= 2 i(0) ip (2.67) from which, with the aid of equation (2.56), there is obtained (v x') ih v'(")) = 21 e42 k( (2.67') Introducing the new nondimensional variables a = glP 92 ' and y = q3P (g = da dBayp a9 and (q, p) = ag/p + prllp + yg/pI where 4, 9, and (~ are the projections of p) and using equation (2.65) result in (VPx') vIG)2(x)) K2p2/3 (2.68) whee te cnstntK2 is of the order of magnitude of 7(e eq. (2.64)). A. M. Obukrhov (ref. 19) gives an estimate of the value of r from the fact that the energy of the atmospheric turbulence is derived from the energy of the solar radiation. According to Brent (ref. 21) 2 percent of the sun's energy is converted into the energy of atmospheric turbulence and in this way is dissipated, being converted into heat. This gives DO = 5j(erg/sec3), which leads to the value Y = 2.4. All the results given refer to isotropic and homogeneous turbulence. A wind blowing under actual conditions may perhaps be considered as an isotropic turbulence provided all the gigantic air flows in the atmos phere as themselves are not considered turbulence phenomena of the air envelope about the earth. Such a point of view is possibly justified in meteorology and geophysics, but it is unsuitable for an observer who has little time at his disposal for following the changes in weather (at least in relation to the wind). Hence for short intervals of time in the course of which there is observed a prolonged constancy of the mean wind, it is convenient to consider the turbulence as superimposed on the mean wind (and the change of "mean" wind will lie outside the small scales of time in the course of which the observation is conducted, for example, in the course of minutes or hours). For such an approach the preceding derived equations may be assumed valid in a system of coordinates mov irng together with the mean wind. The value~ of the constant y or K2 in equation (2.68) may then depend, however, on the absolute magnitude of' the mean wind velocity vO. This evidently has also been observed in tests (see the following). 11. Fluctuation in Phase of Sound Ware Due to Turbulence of Atmosphere Very interesting tests on the propagation of sound under the actual conditions of a turbulent atmosphere were conducted by V. A. Krasilnikov (ref. 22). His tests, the main features of which shall be described in this section, are of interest from two points of view. In the first place, they provide a method for the study of atmospheric turbulence; and in the second place, a circumstance which bears a direct relation to our subject, they throw light on the laws of sound propagation in a turbulent atmosphere. They also have a bearing on the accuracyr of operation of directionfinding acoustical apparatus. The test of Krasilnikov consists essentially of the following: At a point Q is placed a sound source reproducerr, fig. 10) at some distance from two microphones MI and M TIhe distance M1M2 3 is the base of the directionalfinding pair. The distance QB from the source of the sound to the center of the base is denoted by L. If the base were turned at a certain angle to QB different from 900, NACA 'TM 1399 RACA TM 1399 then on account of the different distances QM1 and QM2 the soud wave would arsrive at the microphones Ml and M2 with different phase. By determining that position of the base MIM2M (by an objective method or by the binaural effect) for which. this difference in phase is equal to zero, the direction to the source Q ma be determined. On this principle are based acoustical direction finders. Such difference in phase may, however, also be obtained for the "correct" position of th~e base NMI (at angle 900 to QB) if the physical conditions of the sound propagation along the two rays QMI and M2are different. Such difference in conditions is obtained as a result of the turbulence of the wind. The velocity of the wind, on which the wave phase depends, is a random function of the point of space. On account of these random differences in the velocity of the wind along the two rays QMI and QM2, the difference in phase of the waves arriving at Ml and M2 is likewise a random magnitude. This phase difference was determined in the tests of Krasilnikov; in particular, its meansquare value $2 was found. As has been shown (section 7), the phase velocity of sound in the presence of a wind is equal. to Vf = c + v,, where c is the vel~ocity of sound and vn is the projection of the wind velocity on the normal_ to the wave. In this case the directions of the normnals for the ralys QM1 and QM2 differ little from the direction QB, which is taken for the xaxis. The projection of the wind velocity on this axis is denoted by v, and Vf = e + v is obtained. The phase of th~e wave passing from Q to Ml will be c1 = aS c dx'1 = 90O Id (269 (terms of the order of v 1/c2 and the differences between dx and ds1 = dx/cos 9 are neglected; see fig. 10) where v1 denotes the value of the velocity on the ray QM1. A similar expression will be obtained for the phase in the microphone M2. For the difference in phase, =I <2 t1 L 2 UK = IV ax (2.70) NACA TM 1399 where v2 is the value of the projection of the velocity in the second rayr (M2) on the axis. The mean value of Jr is, of course, equal to zero. The measure of w ill be Z. From equation (2.70), L 2rZ = C 2O L ax' dx"t nvlx ')av(x") (2.71) The averaged magnitude under the integral sign is equal to SY(x')av~x") = [v(x') v2(x')] (v1x") v2(x")] =v1(x')v1(x"t) + V2(x')v2(x"I) V1(x')v2(x"l) vl(x")v2(x') On thze basis of equations (2.57), (2.66), and (2.68), v2 1 '2 = Xr23 where rl2 is the distance between the points 1 and 2. Use is made of equation (2.73) to obtain Av(x') Av(x") = l22/5 + r2/3 r2/3 r2 2 1'l"2'2" 2'l" 2": from equation (2.72). In figure 10 it is seen that (2.72) (2.73) 3 I' (2.74) r'2 l" = r2'2" = (xl x2)~2 (1 + 62) r2'l"c = r2"I (xl x2)2 + (xl + x2)2 6 In this manner there is obtained from equations (2.71), (2.74), and (2.75) (9<<1) (2.75) NACA TM 1399 #2 ~I~ K2 dx1 dx2 x x2)2 + (xlt +Z x2 921/3 (xl x2 2/3(1 + 62)1/3 (2.76) Setting x = x/L and y = x2/L gives equation (2.76) in the form 1 1 ~2 2 2 ~ K2839/ B dy x 2 + (x,~ + ) I3/ (1 + 9)1/ (2.76') If in the preceding double integral are introduced the variables ( =X x ad n = x + y, then for 9 O, it does not depend on 8 and converges to a value of the order of 1. Hence 2 = contan K2 L8/3e5/3 (2.77) Denoting the length of the base MIM2 by 2 and remembering that 8 = 1/2L result in I= constant K ~ L1/275/6 (2.78) Thus, the meansquare fluctuation of phase of the direction finder is proportional to the sound frequency w, to the square root of the distance from the source, and approximately (exponent 5/6) to the length. of the base. The test data of Krasilnikov (loc. cit.) very well con firm both the dependence on cu (the tests were conducted in the range from 1000 to 5000 hertz) and the dependence on 2 (,25/6). It is of interest to remark that the constant K according to the data of Krasilnikov is proportional to the mean velocity of the wind v. The same result was reached by Gedicke (ref. 25) and Findesen (ref. 24), who measured the turbulence of the atmosphere near the ground. This is in agreement with the remark herein on the fact that the turulence of the atmosphere, if the observation times under consideration are not too large, must not be considered isotropic (section 10). NAC"A TM 1399 The question of the error of the direction finder will now be considered. Let the direction at the source make the angle a with the direction of the base. Then the difference in phase at M1 and M2 'in the absence of turbulence will be S2~x2 o (2.79) 'The error be in a due to the random fluctuations of I will be Xe=*S (2.80) 2x3r sin a .At large values of a (cLr/2) for the meanroot deviiation of 8m2 there is obtained 6a ZrZ "~ =constant KCL1/221/6 (2.81) Making use of the data of his tests, Krasilnikov determined the numerical value of the constants entering equation (2.81) as follows: =0.3 1/6 ( ~()1/2i~ (2.82) where a is in degrees, I and L in meters, and the mean velocity of the wind is in meters per second. For example, for 2 = 1 meter, vG = 2.7 meters per second, and L = 2000 meters, there is obtained =\~ Zo The value, if cam~pared with the errors observed in practice of acoustical direction finders, is somewhat exaggerated. The fact of the matter evidently is that acoustical direction finders generally operate in a range of frequencies of 200 to 500 hertz. For these low frequencies the approximation of the geometrical acoustics on which the preceding computations are based may not be suitable. Krasilnikov ibidd.) also conducted interesting observations on the random variability of the phase in time. The measurements were in this case conducted with the aid of a single microphone M; the values of the phase 't a~t two instants of time separated by a small interval at were compared. The results were worked out for the case where the mean wind was perpendicular to the ray joining the source Q and the mderophone M (fig. 11). The computation was conducted on the basis of the hypothesis (section 10) on the isotropic and homogeneous character NACA TM 1399 of the turbulence in a system of coordinates moving together with the wind. In the time interval at the phase at the point M changes by ut L by dx (2.83) where av is the change in velocity during the same time. Hence St2=dr axd' dx"*. av(x') nv(x") (2.84) The principal change in the velocity is due to the transport of turbu lence by the mean wind so that the change of the velocity v in the time 6.t may be represented as the result of the displacement of the turbulence by a small distance 8 = v *at. Thnen nf(x')av(x")> = [v(x', O) v(x', 8)] [ v(x"t, 0) v(x"t, 8)] = v(x') O)v(x", OJ + v(x', 6) v(x"l 8) v~x', 8)v(x", 0) v(lx', 0)v(x", 6) (2.85) Making use of the "2/5" law gives Av~x')dv~x" = 2(x' x)2/5 2[(x' x")22 1/5 ,/ = K2 r2, 2)1/3 2/3i ) 2 =(x' x")2 (2.86) Substitutingl equation (2.86) in equation (2.84) and applying to the obtained double integral the same considerations that were applied to the integral (2.76) result in At constant K2L8/3 2~ 2 (2.87) where the constant is found to be a .Thus A = Kx/E L4/3 3 v At516 (2.88) NACA TM 1399 Test data give the relation (v *at)4/6 rather than ( At)5/6, It is as yet difficult to explain the source of this divergence. Equation (2.88), since att L, v, nt, and o, are known from tests, permits determining the constant K in the "2/3" law. For v = 6.5 meters per second there is obtained from tests K= 11(cm2/3/sec) The iurbulelnce measurements at the height of 2 meters above the earth ccoriucted by A. M. Obukhov and N. D. Ershova give (for v = 3 m/sec) the value K = 3.1 centimeters2/3 per second. Gedicke (ref. 25) obtains for K (at v = 0.65 mn/see and height 1.15 m) the value 2.05 centimeters2/3 per second. It follows that the order of magnitude of K is in all cases obtained as that of unity. The increase of the constant K with the velocity of the wind is a fact, however, which shall have to be taken into consideration in another connection. 12. Dissipation of Sound in Turbulent Flow It is a wellknown experimental fact that in the presence of wind the audibility of sounds is markedly decreased. This decrease in audibility is not a consequence of the curvature of the rays in a wind with velocity gradient considered in sections 8 and 9j it has a more complicated character and is connected with the turbulence of the wind. The first to point out these phenomena in connection with the occurrence of acoustical fading vere Dahl and Devick (ref. 25). The same phenomenon of acoustical fading was investigated by Y. M. Sukh~arevskii in measure ments on mountains (Elbruz expedition of the USSR Academyr of Sciences, 1940). The general impairment of audibility in a wind has also been pointed out by Stewart (ref. 26). ]From the experimental viewpoint the problem was investigated most thoroughlyr by Sieg (ref. 27) who showed the existence in a wind of an additional damping of sound exceeding the damping associated with the molecular properties of the gas (viscosity, heat conductivity, and Kneser effect). The results of Sieg may be essentially reduced to the following: In the frequency interval 250 to 4000 hertz ina weak wind (1 to 2 m/see or at an almost complete calm) considerable fluctuations in the sound intensity (fading) are not observed, but the intensity of NACA TM 1399 the sound drops with increasing distance, the darrping coefficient a being equal to 1.5 to 2.2 decibels at 100 meters Sieg does not find any dependence of the coefficient a on the frequency. It should be borne in mind, however, that the accuracy of Sieg's observations is notI large; the directional characteristics of the source were not taken into account, andIj the conditions under which. the points for the various frequencies were taken were not identical. For this reason this result doec not appear entirely reliable it gives rather the order of magnit~ud of a which in the interval 250 to 4000 hertz does not change. In the case of a strong gusty wind the coefficient of damping decreases, reaching a magnitude of 5 to 9 decibels at 100 meters (flor a wind with gusts of 7 to 17 m/sec). Under these conditions the dependence of a on the frequency becomes more marked, a being equal to 5 decibels for 250 hertz, 8 decibels for 2000 hertz, and 9 decibels for 4000 hertz (at 1_00 m). Under the same conditions, fading is observed the fluctuations of the intensity attain 25 decibels. Both these effects are explained without forcing by the theory of the propagation of sound in a turbulent flow refss. 28 and 29). In considering the propagation of sound in a turbulent flow, it is first of all necessary to bear in mind that those fluctuations of the velocity of the stream having the scale I which is considerably greater than the length of the sound wave X do not lead to the dissipation of the sound. They bring about only changes in the shape of the rays and therefore a general fluctuation of the sound intensity at the location of the receiver (fading). The effect of these largeseale pulsations may be considered by the method of geometrical acoustics. Hence the velocity of a turbulent flow must be decomposed into two components v (mcoopnn)adu iro component): v = l '~x U(d2l(q)) `. 90i (2.89) u: =iqx) U(dS1(q)) whee includes the mean velocity of the flow YO. The magnitude qO = k/ where k = 2ngh, is the wave number of the sound wae and CI is a nondimensional number >>1. The dissipation of sound from a parallelepiped L3 where L;>1 and L 8There is here subtracted the molecular absorption (Kneser effect with account taken of the humidity of the air). It has a considerablle value starting with frequencies of 1000 hertz. The classical absorp tion due to the viscosity and the heat conductivity is of significance only for frequencies greater than 10,000 hertz. IIACA TM 1399 Uhder this condition the velocity v may be considered approximately constant in the volume. In a local system of coordinates which move with the velocity v, the frequency of the sound f varies in it (Doppler effect) only by the small amount f *v/c, but the frequencies of the turbulent fluctuations in this system are equal to v =u(3)/2, where 3 is the scale of the pulsations and u(2) is the velocity of the pulsations associated with this scale. codn oth 25 au = constant 2/5 < sound through a turbulent flow, only the instantaneous picture of the turbulence and not its process with time is of significance. For the same reason it is not to be supposed that the damping of sound in a turbulent flow is conditioned by the existence of turbulent viscosity. The tensor of the turbulent stresses with which the concept of turbulent viscosity is associated is obtained as a result of the averaging of the turbulent pulsations for the given mean flow. This averaging presupposes that all the changes in the mean flow occur more slowly than the random pulsations of velocity produced by the turbulence. For a sound wave the situation is the reverse (v4=f). The effect of the turbulent flow on the sound wave should reduce to the dissipation of sound in a manner similar to the dissipation of light passing through a turbid medium; in both cases random changes of the velocity of the wave propagation occur. An estimate of the magnitude of this dissipation is now made. A start will be made from the equation of A. M. Obukhov, approximately taking into account the presence of vortices. The quasipotential of the ~ound waves is denoted by and the total velocity of the flow by V = v + u to obtain from equation (2.84) (for 900 = O, glog c2 = O, v/c << 1) Atr + V, (~) ~ Vt, nAVsdt = (2.90) Passing over to a local system of coordinates in which v = O results in 9It should be remarked that there exists a minimum scale of turbu lence 3 = min = 1/y3 K p/DO# DO is the supply of energy, is the viscosity of the medium, p is its density, and x a number = 1. See A. M. Obukhov (ref. 19). On account of this, the inequality v4:f may be violated only for f of the order of several hertz. NACA TM 1399 Thie right side of this equation will be considered as the disturbance. By rejecting it completely, the zeroth approximation WOI representing the fundamental vave, is obtained as 0 = Aeilartk$1': (2.92) where ul is the unit vector in the direction of propagation of thez fundamental wave k Z= mo/c. TIhe complete solution will be r = JrO + cp (2.93) where cp is the dissipated wave. For large distances R from the parallelepiped considered, q, is of the form B i(catkR) 4 = e (2.94) The amplitude of the dissipated wave B is determined by use of the method of the theory of disturbances and the substitution of 90g in the right side of equation (2.91) in place of \I. There is then obtained L69 = Vat O ,Ad .5 The~ solution of the wave equation (2.95) having the form of equation (2.94), as is known, is equal to q(1, t) =4r e av (2.96) where dv' = ax' *dy' *dz' and r is the distance between thie points x (point of observation) and x' (source of dissipated wave). Let n be the unit vector in the direction of the dissipated ra~y (fig. 12), R the distance from the center of the parallelepiped, and 8 the angle of dissipation (angle between n1 and n). Then, as follows from the sketch, r =R (x', Cn) (neglecting terms of the order of x'/R). Substituting in equation (2.96) Q from equation. (2.95) and using equation (2.92) give for R +* * NACA TM 1399 S= *ectkR* 2k2+a' n1)e ,Kx') dv' (2.97) where the vector K is equal to K kn n); K = 2k sin 2 (2.98) and u' is the value of the velocity u at the point x'. Thus the amplitude of the dissipated wave B is equal to B = ~c r(2uk2 p* 1e$x'&'(.9 The coefficient of damping a is expressed in terms of the amplitude of the dissipated wave. The flow of sound energy N into the base of the parallelepiped L2 ~is proportional to A2L2, while the flow of ene~ry dissipated from the. parallelepiped is obtained by integration over a distant sphere of radius R and is proportional to R2 Bi ~ 2a5, where dS1 denotes integration over all the directions of dissipation. Since interest lies not in the instantaneous value of the dissipation but in the mean value, R2 B 2d2 must be taken in place of the previous expression, where the bar over 1B 2 denotes the averaging over th~e velocity fluctuations of the turbulent flow. The mean decrease of the energy flow in passing through the parallelepiped L" will be 65J = aNL (2.100) from which a = NINL, and since LN = a R2 IB 2dG (B is the factor of proportionality) and N = pA2L2, A2L3 NABCA TM 1399 From equation (2.99) it follows that dv"ei ()x (Zu'k2* + u')(2u "k2 + nu") 3 (2.102) where p x x is the radius vector between the points x" and x and ul is the projection of u on nl. Introducing in place of x' and x" the relative coordinates p and the coordinates of the center of gravity x =x "rslsi BI 2 1AL3L i(K p), Cq4k41(p~) + 4k2aM11) C2ML11(p) (2.103) where is the moment of correlation. MP k~) introduced in section p = q0. Now equations (2.56) MllP 0 e^ This moment is identical with the 10 (see eq. (2.54)) for i = k = 1 and (2.65) are used to find that (T4 pYq 11/3d4 1 2 5 moment and (2.1L04) The multiplication of the expression under the integral in equation (2.104) by q2 and by q4, respectively, is obtained by simply applying to M11(p) the operators D and C\2. Substitution of the moment (2.104) in equation (2.103) leads to integrals of the form 16sr2c2 flACA T,l 1399 dql dq2 3 *q ei(Kq,p iqdql22 43 o(K1 1)8(K2 92)x 8(Kg3 q3) F~q) = (2n)3 F(K;) =O for K> qO for K< q0 (2.105) Hence He~re 8(x) is the symbol of the 6function (see section 6). is ob; citainedl as K1 K 11/3 (2.106) of equation (2.107) 1 + 1 yK11/5 dS a =k4 S where the integration over the angles is extended to the values K> qO' Settling sin 6/2 = F and dP = sin 9 de dP = 45 dS 69 shows that t~he integration over E = K/2k is extended f'rom E 1/2Cl to r = 1. Carrying out this elementary integration yields a p/3p 4121/52 (2.1_08) (2.109) where B= (2x) 1/3 1+252)152 2)3+0( 4 The .n .:.itude 27iyl/2pl/3 is the velocity of the turbulent pulsations, the scale of which is less than X. Thus the coefficient of damping of' thie sound waves in a turbulent flow is proportional to the "square of .he tlach number (M, = u(X)/c) for the velocity of the turbulent pulsa .ctln ofl scale less than X and3 inversely proportional to the length of = (2n)s_ ~ llB2 2A'Lk4 K2 K4 r~l2 %7A L k2c 4k4 ) (fojr = qO, Otherwise lB 2 = O). From this, on the basis 66 NJACA TM 1399 the sound wJave X. TPhe negnitude 241/~l2, on the basis of the estimate of A. M. Obukhov given in section 10, is equal to 3. The data of V. A. Krasilnikov (section 11) and also of A. M. Obukhov and N. D. Ershov (section 11) give, for a moderate wind, 2411/2 = 6. As already pointed out, the turbulence of the wind must not be considered isotropic so that, in general, 241/2Z is an increasing function of the wind velocity.i If use is made of the as yet not very reliable test data presented in section 10, it is necessary to assume y proportional to the wind velocity. This explains the increase in the coefficient of damping a with the wind velocity. The dependence of the coefficient a on the length of the sound wave is obtained in the form h1/3, that is, a very weak dependence but, on the basis of what has been said, this; dependence does not contradict the test data of H. Sieg. In order to estimate the value of the numerical factor CLI use is again made~ of Sieg's data for a weak wind. In this cash 24y1/2 = 6. The coefficient a is equal to 1.5 decibels in 100 meters, which in absolute units gives a = 105centime~ters1. For f = 500 hertz ( 1= 68 cm) there is obtained 4 2 10. This value of CL should be considered as entirely reasonable. 13. Sound Propagation in Medium of Complex Composition, Particular in Salty Sea Water In the theory of sound propagation presented, the medium was assumed homogeneous in its composition. In practice, however, it is necessary to deal with cases where the composition of the medium varies from point to point (air, for example, the humidity of which is differ ent at different places or sea water with variable saltiness). All the theorems of geometrical acoustics that were derived in sections 7, 8, and 9 retain their validity for media of variable cam position.10 The initial general equations of the acoustics of a non homogeneous and moving medium must, however, be modified. The need for modifying these equations is dictated by the fact that in a medium of complex composition the pressure p de~pends not onlyr on the density of the medium p and the entropy S but also on the con centrations Ck of the individual components forming the! mediumn (for examIple, on the concentration of the water vapor in the air, the con centration of salt dissolved in the water, and so forth). Hence the equation of state must be written not in the form p = Z(p, S'), as previously, but in the form 10Provide~d, of course, that the fundamental hypothesis of geomnet rical acoustics on the smoothness of all changes in state of the medium is not violated. NACA TM 1399 p = Z(p, S, C) (2.110) Here p is the density of the medium and C is the concentration of the second component in it; C = p"/ps, where p" is the density of the disso~lved component, and p' is the density of the solvent (p = p' + o" = p'(1 + C)). Further, to the hydrodynamic equations it is necessary to add equations governing the changes in concentration of the dissolved com ponent. These changes are produced by convection, diffusion, and the action of the gravitational force. In order to write down 4he cor responding equations, the flow of the dissolved component J" is noted as J'" =vp'C + i (2.111) i = p'DIVC p'D2VT + p'ugC (.1' where Dl is the coefficient of diffusion, D2 is the coefficient of thermodiffusion,, u is the mobility of the solvent in th~e field of gravity, and g is the acceleration of gravity. The first term in equation (2.111) ~vp'C represents the part of the flow due to the con vection of the substance, and the second term i, the part of the flow due to the irreversible processes (diffusion, thermodiffusion, and motion in the gravity field with friction). On the basis of the law of conservation of matter, a(p'C) + St + div J" = 0 (2.112) The density of the pure medium p' is subject, of course, to the equation of continuity gt div(p'v )= o (2.113) The required equation for C is obtained from equations (2.112) and (2.113) ~+ (vVC) = div i (2.114) For the total density p = p'(1 + C) there is obtained from equations (2.112) and (2.113) NACA TM 1399 ;5+ div(py) = div i (2.115) TIhe fundamental dynamic equation of hydrodynamics av V2 ~t+ [rot v, v] + VZ. = +g + v ny + Y Vdiv v (2.116) P 3 remains unchanged. The equation of entropy will be written in the abbreviated form as  ;Si + (VW) = J1 (2.117) where J denotes the changes in entropy due to t~he irreversible processes occurring in the~ motion of the fluid (Jr contains term proportional to v, h, D1' D2, and u) and also the possible supply of heat from without. Equations (2.1_10), (2.114), (2.115), (2.116), and (2.117) form a complete system of equations for a mediumt in which some component is dissolved (vater vapor in air, salt in water, and so fort~h). In the propagation of sound all the magnitudes characterizing the medium receive small. increments so that v is replaced by v + r, p by p + x, p by p + 8, S by S + a, and C by C + C, where E denotes a small change in concentration of the dissolved component that occurs in the medium on the passage of a sound ware. Substituting these changed values in equations (2.110), (2.114), (2.115), (2.116), and (2.117), restricting to a linear approximation, and rejecting the added terms proportional to v, X, Dy_ D21 and u, that is, leaving aside the irreversible processes accompanying the sound wave, givell ~+ (ot, j (] + rt,]+ V6, ) = + (2.118)  + VB) + (7(y Vp) + p div( + 8 div v O (2.119) gt+, (v g) + ((7 VS) = (2.120) 11Tlhe diffusion of the salt may give an absorption of sound in addition to that due to the viscosity and heat conductivity. NACA TM 1399 ae+ (v, VL) + ((,VC) = O J = ,2 6 + ha + gT (2.121) (2.122) where c2 )SCh= pp,S (2.123) The square of the adiabatic velocity of sound for constant concentration of the solution is e2, These equations must be col for the propagation of sound in variable composition. If by C of the water vapors in the air, propagation of sound in a humid nsidered as the fundamental equations a nonhomogeneous and moving medium of there is understood the concentration these will be the equations for the atmosphere . The same equations may also be considered as the equations for sound waves propagated in salty sea water. For this, C must be con sidered as the concentration of the salt dissolved in the water. In the presence of entropy gradients (VS pl 0), as in the presence of gradients of the concentration of the dissolved component (VC pl O), the right side of equation (2.118) is not a total differential of some function. Hence even in the absence of vorticity (i.e., for rot v = ) the sound will be vortical (rot ( / O). Because of this the system. of equations (2.118) to (2.122) cannot be reduced to an equation for a single function (for example, to an equation for the sound potential, to an equation for the sound pressure, and so forth). In order to change to the equations of geometrical acoustics it is noted that equation (2.121) does not differ formally from equation (2..120). Hence, following the same method which was used in section 7 for deriving the equations of the geometric acoustics of a medium of constant composition, and assuming, in addition to equations (2.5) and (2.7), 0 0+ . r, = ZO ei (2.124) result in ZO= O (2.125) NACA TM 13199 that is, in the first approximation of geometric acoustics the sound is propagated not only isentrop~ical~ly but leaves unchanged the composition of the medium (20r = 0). All the remaining conclusions with regard to geometric acoustics previously obtained likewise remain in full force. The effect of the nonhomogeneity of composition of the medium is in this approximation reduced to the effect on. the velocity of sound in the medium c and on the density of the medium p. The sound will be propagated within the ray tubes with velocityi 1,= h 4 c=7 SC (2.126) and the pressure x will be subject to the law 2 i~ constant (2.127) pqc (compare section 7, eq. (2.32)). The particular case when the medium. is at rest is now considered. This case is of special interest for water in which the velocity of sound is large while the velocity of flow is small. For a medium at rest (v = O), from equations (2.118), (2.119), (2.120), (2.121), and (2.122), ar Vx y In ~ he gZ\ + 2 2 (211' ata 2 +p div( 2.1' = (y VS)(2.120') = ((,VC) (2.121') Setting rr/p = H and making use of equations (2.120') and (2.1_21_') give the equations for n and %: azii an vqg' an Vp (P ) 218 at2 ~ E pc2 E Pc2(p, 218 MACA TM 1399 2a + div ( + = (2.1_29) where Vp' = hVS + g VC = Vp c2pp (2.130) Substituting aD/at from equation (2.129) in equation (2.1_28) gives the equation for the velocity of the sound vibrations =% a2 .( div r + +~~i D i P~(P~ (2.151) This is the equation for the propagation of sound vibrations in a medium at rest in which the density, temperature (entropy), and concen tration of the dissolved substance vary. It is seen from the equation that for the computation of 5 it is sufficient to know c, p, and p as point functions, where c is the adiabatic velocity of sound and p is the total density of the medium. Equation (2.131) does not reduce to an equation for the potential or the pressure. After has been found from equation (2.131), the sound pressure is found from equation (2.129) as =J 8 = div ( + aPLiL t (2.132) In certain special cases equation (2.131) may approximately be replaced by the simpler wave equation. In fact, a medium for which the term in equation (2.121) containing Ve2 is much greater than the terms containing Vp' is assumed. Then, rejecting the terms with Vp' and setting ( = 94 ( is the velocity potential of the sound vibrations), the usual wave equation is obtained: a2~= 2 n.9 (2.133) NACA TM 1399 in which, however, c varies from point to point. The term with Ve2 is Vc2 div ( and in order of magnitude is eqaltoO2 " equalto y k k is the wave nuniter). The greatest term containing pp' is VPs div (fp, in order of magnitude equal to pp' *kS/p. Hence the terms containing Vp' may be rejected and the~ term containing Ve2 retained if 702 ,,]01 p (2.134) In order to obtain the condition satisfying this inequality, c2 and p' are considered as functions of p, T, and C. Then W e22 ;C OT~p,T (2.135) *VC ~ PC ~p,C * VT + Here (aP/ap)p,C = 1/a2 (a,2 is the square of the isothermal velocity of sound), (ap/aT)plC = pB (B is the coefficient of volume expansion), = )Tyrp is the relative change of and (ap/aC) =p px, where x volume of the fluid (gas) with change in the concentration of salt (or vapor, respectively). Since a 2 C c and ep cy = a2 2T from equation (2.136) lp a.242T Op + e28 VT + c27tVC p p *c, (2.157) Y t ,C T, c * V (2.156) NAC'A TM 1399 These equations, on the basis of experimental data, permit solving the problem of satisfying (or not satisfying) inequality (2.134). In particular, for salt sea water, this inequality is evidently satiiSfied. In fact, for water P = 2 104 at 180 C, and at 40 C, B = Th magitud x (av/aC )p,T for a solution of NaC1 or KCl a~t 150 is about 0.15 to 0.20. According to the measurements of A. Wood refss. 30 and 31), the velocity of sound in sea water at t = 16.950 and saltiness of~ 35.02 percent (that is, at C=.5*02) is equal to 1526.5+0.3 meters per second and is governed by the equation c = 1450 +t 4.206t 0.0366t2 + 1.137 103(C 3.5 102) whence (ac2/aC) = 2c 1.137 105 = 1.42 c2 It is seen that ac2/aC>>xc2. Further, (bc2/aT)plC = 2c 4.2 = 5.8 103 ,2 and pc2, = 2 104 ,2, that is, (ac2/8T)p,C~ ~ 02 Thus the magnitude Ve2 for salt sea water considerably exceeds the magnitude 9%p'lp. Hence the wave equation (2.133) may be assumed to describe the propagation of sound in calm sea water in an entirely satis factory manner. IUACA TM 1399 CHAPTER III MOVING SOUND SOURCE 14. Wavie Equation in an Arbitrarily Moving System of Coordinates In a system of coordinates (x,y,z,t) associated with thE? air at. rest, the wave equation for the acoustic potential is 2 2 2 2 (9  O* a= + (3.1) c2 at2 b2 a 2 a,2 It is assumed that the position of a moving source of sound is determined by the coordinates y = Y(t)> (3.2) In this case it is convenient to introduce a. system of coordinates ((5,q,(,T connected with the sound source =' x X(t) = 7 Y(t) S= z Z(t) v = t (53.) In this system of coordinates the velocity of a wind VO has the components dX YOx dt ~ x V = (3.4) VOy dt Vy dZ. V v Oz dt Z Equa't~ionr (3.1) is then transformed to the system of coordinates (5(1,517). For this purpose 9p(x,y,ztt) = cp(( + X(z), r + Y(z), r + Z~t); z) NACA TM 1399 so that "j (3.6) that is, Vxy ~~~( .22 (3.6') Hence, the wave equation (3.1) in the system of coordinates 5, n1, will bet 1 a cP 2 byc 2 2 c,~\a2 dt~,=,(3.7) or, if in place of the velocity of the source v, the velocity of the wind VO is introduced, then 1 22 1~~o *~ 2 O, V9 = O(3.7') This equation may be considered as the equation for the propagation of sou~nd in a medium moving with velocity VO0(t), depending on the time but not depending on the coordinates. In fact, it almost agrees with the previou~sly, (Chapter I, section 5) derived equation (1.85) governing the propagation of sound in a medium in which the wind blows with constant velocity VO. The difference lies only in the presence of the last term. containing the acceleration diO/dt. If it is assumed, however, that the v.elocity, of the wind. V, is a function of the time, an equation accur ately, agreeing with equation (3.7') would be obtained in section 5. The assumption of the presence of such wind is, of course, an artificial one, but it is compatible with the equations of the hydrodynamics of an in compressible fluid. These equations, in the presence of external volume forces of,? are aV Dp 38 sq (VV)V= o+ g; div V = 0 38 With the assumption that V and p do not depend on the coordinates, there is obtained ~ ~ g (3.9) dt It follows that such motgjon igi realized in a fictitious field of gravity: having an acceleration g =dV0/dt. Thus, in considering the sound field of a moving source, the source is assumed as stationary but it is then necessary, in general, to assume that the acceleration of a variable wind is conditioned by the "force of gravity" producing the acceleration dv 8  dt (3.10) 15. Sound Source Moving Uniformnly With Subsonic Velocity velocity v less The velocity v is coordinates fixed An arbitrary sound source moving with constant than the velocity of sound c will be considered. directed along the xaxis. Changing to a. system of to the sound source q = = x vt (3.11) yields a particular case of equation (3.7): 1 a20 2 89 2 a2q 69 + = 2 2 22 2 and introducing, as was done in section 5, a system tracted along the xaxis of coordinates con x vt 5= z 7= t yields, in place of equation (3.12), = O 1 2 c2 ~2 1 a2p c at ag* 2p + ,/1 P2 3C a2 n a5~K2 (3.13') V NACA TN 1399 (3.13) NACA TM 1399 This equation agrees with equation (1.94),12 and the generalized theorem of Kirchhoff (see section 6) may be applied to it. It is evidently suf ficient to restrict this report to the consideration of the sound of fre quency t in the system attached to the source), so that S=eiwt (3.14) On the basis of equation (1.108), ikR e (3.15) 2ipk 4rt7/1~Z; S MT 1 02 eikR .dS where 9p is the value of the potential at the point of and the surface S encloses the source. Further observation P, RMc = J/ 4 q2 4 g2 R=g +R (3.16) where R* signifies the distance (in the system 5*, 1, r;) from the point of' oscervation P to the point of the surface S(Q): (3.17) The waveF field far from the surface S(R" j *) is now considered. For large distances from the point P from the surface, as is seen from f'iiuret 13, (3.18) R RP~ + Rg cos 8pg + r.. where 8is the distance OP, RQ is the distance OQ, and 9p& is the angle between OP and 0Q. On the basis of equations (3.18) and (3.17), P5{ + Sq cos 8p& + ~+ *** == Rp + a + *** (3.19) Pg~ + R 3 *15 + R R = 12It is necessary to bear in mind that p is now v/c, whereas in section 6, p denotes VO/c; thus p in section 6 and here differ in sign becausee V0 = vS S 11 = ng ip KACA TM 1399 where py$S + RT; (3.20) B@ + ER~C cos 8pa Substituting the value of R (eq. (3.19)) in equation (3.15) and neg lecting terms of the order 1/R 2 yields = e ik eikA *dS + ipk eikA (3.21) The expression in braces depends only on the dimensions and form of the surface and the angles determining the direction of the radius v:ector OP. These angles are different .depending on whether they are taken in the contracted system ( 7), r or in the initial system 5, 9, 5 (they. differ by a. magnitude of the order of. P2). Let them be 9, X in the system (a~nd 6 ,3 X in the contracted system, respectively). With the system r, 1, (, the following may be written: eikRp q((,4,() e (6,x) (3.22) where (~ in R and R8 must be expressed in terms of (p. On the basis of equation (3.14), the following is obtained for (p: v~qnyty ) e. (e,x) (3.23) There Q(9,9p) is the integral. 45rQ(9,X) ia ik e i A *d +eikA dS( .4 The magnitude Q(8,X) determines the force of the sound source (it has the dimensions of the volume velocity (cm3/sec)) and its direction. If Q(9,'P) is developed in a series of spherical, functions ~Pm(cos 9)eimX where 3 = 0,1,2,3,***, and m = O dd, j:2, f3*** 3:3, then n +1 Q(9,9) C QLm *P l(os 8) imy (3.25) 2=0 m=3 NACA TM 1399 When all the coefficients QIm, except Q0 = Q, are equal to zero, then a source of zero order results in q((p,qp,5pr t) = e R 0 (3.26) If, for example, only QlO is different from zero, then, since Pl = cors 9, q((5'p~qp't ) = e Rw '1 cs6(3.27) that is, a dipole source where the dipole is oriented along the Fa~xis. The terms with 2 > 1 give multiple radiation. Consideration will now be given to the dependence of on the dis tanice. It is evident that the surfaces of constant amplitude Jr diverg ing in direction by angles included in Q(9,(p) will be the surfaces R*t = constant (3.28) P~ut Rp = + 92 + 2, that is, the surfaces of constant amplitude will be the ellipses (fig. 14) 2 ~2 2 =constant (3.29) The surfaces of constant phase will be a = mt = constant (3.30) From this it is seen that the phase velocity along Rp is equal, to the veloc ity of sound c. It is now assumed that the wave field (8 is observed from the point of view of a stationary observer. On account of the motion of the sound source, Rp and, therefore, the wave phase a will then depend on the time t in a more complicated wa~y thanr simple proportionality to t. Kence the observer will not~ perceive this sound field a~s a. field of harmonic vibrations (although in the system attached NACA TM 1399 to the source harmonic vibtrations were a~ssumned). changes in the manelntude Rp are not too rapid, be determined for the stationary observer as the a with respect to the time Nevertheless, if the the frequency w' can derivative of the phase dR da U)' s (3.31) The comutation of the derivative dRp/dt, on the basis of equations (3.20) and (3.18), yields p+4*/R t J/1 02 Z d( l dt (1 02! I dRp c dt (3.32) 1 P2 whence (3.33) This formula gives a~n expression for the change of frequency caused by; the motion of the sound source, that is, the Doppler effect produced by the motion of the source. If the observer is located ahead of the source, the following is obtained from equation (3.33): U) CU' = 1P cr~ = R~) (5f~ =EP) and, if behind the source, CD U]' = 1+B (3.33') Equations (3.33) and (3.33') are the simplest formulas for the Doppler effect. Formula (3.33) gives the numerical expression of the Doppler effect for any position of the observer. If magnitudes of the order of $2 are nglected, the following is obtained from formula (3.33): aW' = U)(1 + B cos 9) (3.34) where 9 is the angle between the velocity of the source and the direc tion OP toward the observer. 16. Sound Source Moving Arbitrarily but with Subsonic Veloeity The computation carried out in the preceding section shows that the field at a great distance from a uniformly moving source has the for of a field produced by a point source concentrated at the point 0 (see fig. 13), and the nature of the source is entirely concealed in the function Q(9,9) determining the force and direction of the source. On the basis of this result the theorem of K~irchhoff may be avoided, which, although it can be formulated also for a, nonuniformly moving surface, obt~ains in this case a. form which is very complicated a~nd unsuitable for applications. With the assumption that the source moves along the trajectory x = X(t) y = Yft) (.5 z = Z(t) (.5 The truie nature of the source will be disregarded and the assumption will be made that the vibration is produced by a certain volume force concentrated at the location of the point source. The result will not depend on assumption (ref. 32). This assumption of the method of pro ducing the vibrations is expressed by the fact that in the wave equation an expression determining the strength of the source is introduced on the right side: 1P p = 49Q(x,y,z,t) (3.36) In order to express the fact that the force Q is applied only at the locations of the source, use is made of the 8 functions introduced in section 6 Q1(x:~y,z,t) = F(t) *B(x X(t)) 8(y Y(t)) 8(z Z(t)) (3.37) The magniltude F(t) gives the dependence of the force on the time in the system attached to the source. Due to the introduction of the 8 functions, which are everywhere equal to zero except at the points where their argument becomes zero, the force Q vill be different from zero only at the place where the source is located at the instant of time considered. The solution of equation (3.36) is evidently equivalent to the solution of equation (3.7) with a stationary right side : that is, to the finding of a singular solution of equajtion (3.7'). The solution of the wave equation (5.26) with the right side present, as is known reads (see section 6) cp(x,y,z,t) r(x',y',ze' t r/c) dv' (3.38) NACA TM 1399 NT~ACA TN 1399 where r Z= 1/% X x')2 ) + (Y Y) z'_Z)Z is the distance from the sound source (the point (x',y',z')) to the point of the~ observer (x,y,z). The evident physical sense of this solution consists in the fact that the disturbance formed at the point (x',yT',Z') does not at once reach the point (x,y,z) but is retarded by the time r/c; there fore the disturbance at the point (x,y,z) at the instant of time t is determined by the disturbance a~t the point (x',y',z') which was presen~.t at the instant of time t r/c. Substituting now the value of equation (3.37) in equation (3.38) yields 'P(x,y,z,t) = 8(' (X] (y' [Y)8(' [Z])dx'dyrggs(3.39) where the brackets denote that the magnitude enclosed is taken at the time t r/c. In order to carry out the integration, new variables which a~re argumen~:l.tsi of the 8 functions a~re introduced in place of x',Y',z': B = y' (Y] C = z' (Z] (3.40) and dx',dy',dz' are transformed by the known formulas of integral calculus B B ;SBi a ax' ay' az' dx'dv'dz' = *dA dB dC (3.41) =I dA dB dC formulas (3.40), and a8z[Z1 (z) ar r (3.42) The determinant I is readily computed from there is obtained I = 1 a (X] (x' x) a Y] (y' y)  *r r rr 1c ~~ NACA TMJ 1399 where [v3R] is the projection of the velocity of the source 9 i h direction of r taken at the instant of time t /.The value of I is now substituted in equation (3.39) and the integration with respect to A, B, and C is carried out. On the basis of the properties of the 8 fulnctions, the result of the integration should simply be equal to the value of the function under the integral at the point A =B =C =O (see section 6), that is, 'p(x,y,t) =C \ I) A=B=C=0 (3.43) where the sum is taken over the points where .A = B = C = 0. These points are easily determined. From the conditions A = B = C =O the following results: (x x) = X] x (Y' y) = (Y] y(3.44) (z' z) = [Z] z By, taking the square of these equations and combining term by term, an equaztion for obtaining the value of r at the point A B =C = is obtained. This value is denoted by R. By the method indicated the following equation results from equation (3.44): R2=xX 22+ Zt 2 (3.45) or f(R) =O (3.46) where 2~ 2 2 "f(R = x t +yYt + Zt R(3.47) Since R > 0, only the positive root of equation (3.46) is to be taken. On the basis of the equivalence of equations (3.44) and (3.46), the sum oser the points A = B = C = O in equation (3.43) goes over into the su~m O..er the positive roots of equation (3.46). The distance r = R is the effective distance. Its physical meaning is illustrated by figure 15. where the trajectory of the source Q and the point of observation P are shown. If at the instant of time t the source is at point Q, the disturbance at the point P originates from the position Q', which it occupied at the instant t R/c, where is the distance Q'P; the instantaneous distance, however, r = CxX(t))2 + (py(t))2 +(zZ(t))2 is equal to QP. Substituting in equation (3.33) the value r = R yields NACA TM 1399 ~(ry~~t=C F(tR/C)RJP (3.48) where, as is easily verified by equations (3.42) and (3.47), /1~[R~ 02 R R 1 (3..49) If the velocity of the source is less than that of sound, there will be only a. single positive root of equation (3.46). In fact, in order that the equation f(R) = O have a second positive root, f(R) must pass through a.n extreme value, that is, afldR must become zero. From equation (3.49) it is seen that in this case [vR] must be equal to c, which is impossible. Hence, for v < c, 'P(x,y,z,t) = (3.50) R ~1 02 where R is the only positive root of equation (3.46).13 The case v >c will be considered separately (section 20). From equation (3.50) it is seen that the wave field for all motions of the point source is expressed only through R" and R, but the functions R (x,y,z,t) and R(x,y,z,t), since they are obtained from equation (3.46), are, of course, different. In particular for a uniform motion with velocity v along the xaxis f(R)~~ e t + z2 R1 (3.51) 13In section 5 the solution has the form F(t + R/c)/]R F. The dif ference between them, and equation (3.40) is only an apparent one. In the first place, the factor ./1 02Z did not enter for the reason that in. section 5 there was no interest in the absolute strength of the source. Further, equation (3.51) has also a formal leading solution. Thus, in equation (3.40), Q(x',y',zc',t +t r/c) can b~E taken. The chosen sign + yields, in place of equation (3.40), O = F(t + R')IR 1 ~ 02Z R 41 pZ2 ) CR]'/c where [vR] is the value of v:R at the instant t + R/c. In equation (3.46) the sign before R would likewise change. The value of R would be R" (see fig. 15). From this it is seen that if equation (3.46) has the solution RI = R, italso has the solution R2 = R". Hence, in order to obtain a lagging solution of equation (3.46), it is necessary to take R > 0 if starting from Q(x',y',z',t r/c) while it is necessary to take R < 0 if start ing from Q(x',y',z',t + r/c). But this root is precisely equalto R1 From equation (3.46) the already familiar result is obtained BE*PS; + R R* = / 3r2 2+ 2 (3.52) x vt The solution obtained (eq. (3.50)) represents the field of a zer~o source. By combining such sources, however, with. the proper phases and disposing them according to a known method, a. wave field having a~ny directional characteristic can be represented. For example, two zero sources of the same strength but of oppostie phase placed at a small distance from each other (2< R) will give a dipole. If the source began to function at a. certain instant of time, for example, t 0: (that is, if F(t) = 0 for t( < ), there would be present a wavle front, that is, of a surface which would be reached by a distur bsnce starting out from the source. Fro~m each position of the source a wave starts out at time t at the distance R = et. Substituting this value of R in equation (3.46), the equation of the wave front is obtained: x X0 2+ YO z () =ct2 (3.53) that is, a, sphere of radius et with center at the point where the source began to function (that is, at x = X(0), y = Y(0), z = Z(0)). Thus, for v < c, the moving source is at all times located within the sphere formed by the wave front (fig. 16). The results obtained for the sound field of a moving source are, in manly respects, in agreement with the known results of LenardWichert for the electromagnetic field of a moving point charge (electron). 17. General Formula for Doppler Effect If the source of sound is assumed harmonic and having in its own sysrtem the frequency cu, the form of cp (eq. (3.47)) is restricted: 'P(x,y,z,t) = Q e Q e (3.54) R 71 62 R 1 62 NACA TM 1399 NIACA TM 1399 Fromrr the instantaneous frequency co' perceived by a certain observer not moving together with the source, the derivative of the phase a with respect to the time is understood da 1 dR m)' a 1 (3.55) dt c dtj This formula must be considered as the most general formula for express ing the Doppler effect. It was Tnresented earlier for uniform motion; it remains true also for the general case of motion. In section 15, however, the question of the limits of validity of this formla wads not considered. For an observer not attached to the source, the spectrum of the wave field q(x,y,z,t), notwithstanding the harmonies of the source, will appear as continuous and the intensities of the indiviidual frequencies will be determined by the amrplitudes Y(x,y,s,ua) in the expression

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