mn^^'^'^x 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1399 
 
 ACOUSTICS OF A NONHOMOGENEOUS MOVING MEDIUM 
 
 By D. I. Blokhintsev 
 
 Translation 
 
 'Akustika Neodnorodnoi Dvizhushcheisya Sredy". Ogiz, Gosudarstvennoe 
 Izdatel'stvo, Tekhniko-Teoreticheskoi Literatury, Moskva, 
 
 1946, Leningrad. 
 
 UNIVCRSIW OF FLORIDA 
 DOCUMENTS DEPARTMEI^ 
 1 20 MAf^STON SCIENCE UBRARY 
 P.O. BOX 11 7011 
 GAINESVILLE. FL 3261 1-7C 
 
 February 1956 
 
to( n r^ ^^ ^^^ '^^-^^ 
 
 MCA TM 1399 
 
 ACOUSTICS OF A NONHOMOGENEOUS MOVING MEDIUM 
 By D. I. Blokhintsev 
 
 "Akustika Weodnorodnoi Dvizhushcheisya Sredy" . Ogiz, Gosudarstvennoe 
 Izdatel 'stvo^ Tektmiko-Teoreticheskoi Literatury, Moskva, 
 
 1946, Leningrad. 
 
Digitized by tine Internet Arclnive 
 
 in 2011 witln funding from 
 
 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
 http://www.archive.org/details/acousticsofnonhoOOunit 
 
WACA TM 1399 
 
 PREFACE 
 
 The practical problems brought about by the Great War have gj.ven 
 rise to theoretical problems. 
 
 In acoustics interest centers about the problem of the propagation 
 of sound in a nonhomogeneous moving m.edium, which is the nature of the 
 atmosphere and the water of seas and 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 and 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 ^ some 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 W. 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 
 
MCA TM 1399 iii 
 
 CONTENTS 
 
 Page 
 
 CHAPTER 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 Phase 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 Salty Sea Water 66 
 
 CHAPTER III. MOVING SOUND SOURCE 74 
 
 14. Wave 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. Sound of an Airplane Propeller 88 
 
 19. Characteristics of Motion at Supersonic Velocity. Density 
 
 Jumps (Shock Waves) 95 
 
 20. Sound Source Moving with Supersonic Velocity and Having 
 
 Small Head Resistance 102 
 
 21. Sound Field of a Sound Source for Supersonic Velocity of 
 
 Motion 107 
 
iiii NACA TM 1399 
 
 Page 
 
 CHAPTER IV. EXCITATION OF SOUND BY A FLOW 112 
 
 22. General Data on Vortical Sound and Vortex Formation. . . . 112 
 
 23. Theory of the Karm^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 
 
 CHAPTER V. ACTION OF A SOUND RECEIVER IN A STREAT^ 145 
 
 28. Physical Phenomena in the Flow about a Sound Receiver. . . 146 
 
 29. Shielding a Sound Receiver from Vortical Sound Production. 152 
 
 30. Shielding of Sound Receiver from Velocity Pulsations of 
 
 Approaching Flow 159 
 
 31. Sound Receiver Moving with Velocity Considerably Less Than 
 
 Velocity of Sound 162 
 
 32. Sound Receiver Moving with Velocity Exceeding Velocity of 
 
 Sound 166 
 
 REFERENCES 175 
 
NACA TM 1399 
 
 CHAPTEE I 
 
 ACOUSTICS EQUATIONS OF A NOWHOMOGENEOUS 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 shown (ref . l) that if f « 1/t, 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 ignored, the phenomenon of the dis- 
 persion of sound cannot, in all strictness, be taken into account. For- 
 tunately, in the majority of practical problems, the 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 
 basis 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: (l) 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 
 for a vector), v-, = v , Vp = v , and v^ = v^ are the components of v 
 along the x, y, and z axes, respectively, and p is the density of the 
 gas. In these notations, the law of the conservation of matter, mathema- 
 tically expressed by the equation of continuity, assumes the form 
 
 |.5|;(Pv,).0 (1.1) 
 
NACA IM 1399 
 
 1 
 
 where the summation is carried out for k = 1, Z, and 3. The vector Pv 
 is the flow density vector of the suhstance. This equation states that 
 the change in amount of substance in any small volume is equal to the 
 flow of the substance through the surface enclosing this volume. 
 
 The vector 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 tensor 
 with the components: py-.y, (i,k = 1,^,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 T^j^ and is 
 composed of the scalar pressure p and the viscous components b^-^ 
 
 Tik = P • ^ik - Sik 
 
 (1.2) 
 
 where ^ik ~ -'- ^^ i = k, and 
 
 ik 
 
 if i / k. 
 
 When applied to a small volume, the law of the conservation of mo- 
 mentum can be written in the form 
 
 ^ 
 
 (pv.) 
 
 1' 
 
 ^ ('J'ik + P^l^k) 
 
 (1.3) 
 
 i and k = 1, 2, and 3 and again is summed for k = 1, 2, and 3. The 
 equation of the conservation of energy should express the fact that the 
 change in 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 the kinetic energy p • Z_ and ':he internal 
 
 2 
 
 energy 
 
 T?. 
 
 pE (E is the energy of unit mass of the gas) is (p _ + pE)v. 
 
 If the heat flow vector is denoted by 3(3-1^,82,33) and the conservation 
 law is applied to a small volume. 
 
 P - + PE 
 
 
 P - + PE I vj^ + S^ 
 
 ^ (v,T.^) = 
 ox, ^ 1 ik' 
 k 
 
 (1.4) 
 
NACA TM 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 
 liq.uid (or gas), the stresses S^jj. are connected with the deformations 
 Vj_j^ according to the Newtonian relation^ 
 
 ^ii = ^i^vj^i + 7 • div v; S^-^ = 2n • vj_]j: (1.5) 
 
 where [j. is the viscosity of the gas and y^s, is the tensor of the 
 deformations 
 
 T / Sv . Sv, 
 
 The magnitude y can he written in the form 7 = p. ' -2|-l/3, where 
 \x' is the so-called second coefficient of viscosity (see ref. (l)). 
 With this coefficient, account is taken of the conversion of the energy 
 of the macroscopic motion of a gas into the energy of the internal 
 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, in the majority of cases the assumption may he made 
 that |j ' = and 7 = -2|a/3 (the value assumed in the theory of Stokes). 
 
 The flow of heat S expressed in terms of the gradient of the 
 absolute temperature T is 
 
 S^^ = \ . ^; \ = p • c^x (1.7) 
 
 °^k 
 
 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, (l.l), (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 (l.l), (1.3), and (1.4) permit a rational determination 
 of the flow of substance L, the flow of momentum represented by the 
 
 •4?hls form for v^j^ follows from the assumption of the isotropic 
 character and homogeneity of the gas or liquid if a linear relation is 
 assumed between the stress tensor Sj^j^ and the deformation tensor v-y^. 
 
MCA TM 1399 
 
 tensor Mj^j^, and the flow of energy W, which, like the flow of substance, 
 can he 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 the density of the corresponding mag- 
 nitude. In this manner from equation (l.l) for the flow of substance 
 (equal to the flow of momentum) the following is obtained: 
 
 t = pt (1.9) 
 
 From equation (1.3), substitution of the value of Sj^j^ from equa- 
 tion (1.5), gives the tensor of the momentum flow 
 
 2 "* 
 
 '^ii ~ "^"^i + P + ^ ■ ^'^^ V - 2n • Vj^j^ 
 
 %k = P^i^k - 2MVijj = Mki; i ?^ k (1.10) 
 
 where, as before, i and k = 1, 2, and 3. 
 
 The terms of the forTn Pv£, PVj_vjj give the momentum flow due to 
 the transport of momentum by the motion of the fluid, and the terms 
 containing P, [i, and y give the flow of momentum due to the action 
 of the pressure forces and the viscous stresses. 
 
 Finally, from equation (1.4), substitution of S^j^ from equation 
 (1.5) yields the energy flow 
 
 W 
 
 I p L_ + pEJ v+^+pv + |J.^ Vv2 + Trot v X v] \ 
 
 7 • div V • V (l.ll) 
 
 The first term 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 terms with \i and y give the part of the energy flow due to 
 the work of the pressure forces and the viscous stresses. 
 
 The fundamental equations can also be written in vector from, by 
 substitution of the value of the tensor T^j^. from equations (1.2) and 
 (1.5) in equations (1.3) and (1.4). Equation (l.l) may, however, be as 
 
 ^ + div(pv) = (1.12) 
 
 2The vector N = ( P — + pjv, representing the flow of energy for 
 an ideal incompressible liquid, is called the N. Umov vector (ref. 3). 
 
NACA TM 1399 5 
 
 If use is made of (1.12) equation (1.3) can be written in the form 
 
 p — = - Vp + fiAv + i nv div V (1-13) 
 
 dt 3 
 
 where V is the symbol for the gradient and A = c^^/Sx^ + S /hj^ + 
 h^/hz^ = v^. The magnitude dv/dt is the total derivative of the velo- 
 city with respect to time and is equal to 
 
 i^lfMv^v'.lf.V^.Crotvxvj (1.14) 
 
 The energy equation (eq. (1.4)), with the aid of equation (1.12), assumes 
 the form 
 
 p ^ = \ . AT + Q - p . div V (1.15) 
 
 dt 
 
 || = |.(7,,)E (1.15.) 
 
 where Q is the dissipative function 
 
 3 
 
 Q = ^ S.^ . Vij, (1.16) 
 
 i,k=l 
 
 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/p). 
 
 This equation may also be interpreted in terms of thermodynamics. 
 The first law of thermodynamics for unit mass of substance yields 
 
 dE = TdS - p . dV (1.17) 
 
 where E is the energy of unit mass; S, its entropy; p, the pressure, 
 and V, the specific volume (V = l/p). Thus 
 
 (it dt ^ dt dt p2 dt Vi.^o; 
 
NACA TM 1399 
 
 On the other hand. 
 
 3o that 
 
 |£. ^+ (?,») p = - p . dlv ? (1.19) 
 
 ,.il^__.^,l^ (1.20) 
 
 For adiabatic processes 
 
 P 2 dt 
 P 
 
 dE p_ dp 
 
 dt ~ p2 ■ dt 
 
 (1.21) 
 
 from which 
 
 E = / i£ - P (1.22) 
 
 a P p 
 
 The magnitude 
 
 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 : 
 
 T ^ = ^ AT + ^ (1.24) 
 
 dt 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 and |j are neglected since the 
 effects produced by them in the over-all energy balance are usually small 
 corrections, the following results: 
 
 ^ = ^ + C^, VS) = (1.25) 
 
 dt dt 
 
 that is, the adiabatic motion of the fluid. The Bernoulli theorem holds 
 for this motion if it is also irrotatioi.al (rot v = 0) . 
 
 If 
 
 v = - V* (1.26) 
 
NACA TM 1399 
 
 where <p is the velocity potential, from equations (1.13) and (1.14) 
 
 M * k (")' 
 
 ^ (1.27) 
 
 P 
 
 and since, on the "basis of equation (1.23), p/p = vw, integration of 
 equation (1.27) gives 
 
 "'/ i?-s? -!<'*>' 'i-^^'' 
 
 If the compressibility of the fluid is neglected, 
 
 w = i + constant (1.28) 
 
 PO 
 
 PQ ....2 
 
 so that 
 
 P = Pq 1^ - 2^ (^'i')'' + constant (1.29) 
 
 and in the case of steady flows (S'i'/St = 0) 
 
 p = constant - -ii (v?) = constant - — — (1.30) 
 
 Because the entropy remains constant during the motion for an ideal 
 fluid (a = i-i = 0) introduction of the variables P and S in the 
 equation of state, equation (1.8), in 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 compressions 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 in a motion- 
 less medium 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 con- 
 dition of linearity of the vibrations. Further, these vibrations are 
 
NACA TM 1399 
 
 assumed to occur with frequencies in the hearing range (the region of 
 classical acoustics) or near this range (infra and ultra sound). Mathe- 
 matically the above assumption reduces to the neglect of the terms in 
 the aerodynamic equations of a compressihle gas which contain second 
 powers or the products of small magnitudes which determine the deviations 
 of the state of the gas from equilibrium. Where n is the deviation of 
 the pressure from the equilibrium value pq, p is set equal to p^ +. jr, 
 
 and 
 
 = Pq +& 
 
 where 
 
 is the value of the density for 
 
 P = Pr 
 
 T = Tq, and finally v = ^(5 is a small velocity). Similarly for the 
 temperature, entropy, and energy, 
 
 T = Tq + e 
 
 S = Sq + o 
 
 E = Eq + e 
 
 In place of equations (1.12) and (1.13), the following is obtained: 
 
 (1.31) 
 
 as 
 
 Pq • ^ = - y-st + [X ■ A£, + - [iV div ? 
 
 + p„ div £, = 
 
 St +^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 
 
 ^0 "v 
 
 p = p^ . e 
 
 (1.34) 
 
 where c-y is the specific heat at constant volume (cy = r/(7-l)), and 
 7 = c /cy 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): 
 
 - _ ^ PO c, PO Zf. ^ V, ^0 
 
 "-/ — 0+ — (J + •..= c^o + ha + . • . • h = — 
 
 Pq S ' % 
 
MCA TM 1399 
 
 For 0=0, only the first term representing small changes in pres- 
 sure for small adiahatic compression or expansion of the gas remains. 
 The magnitude 
 
 f¥ 
 
 7 £0 (1.35) 
 
 is the adia"batic velocity of sound. The second term gives the change in 
 pressure produced by the addition or decrease of heat. The changes of 
 entropy a ohey equation (1.24) which is written hy neglecting magni- 
 tudes of the second order of smallness as follows: 
 
 Tn ^ = — Ae; \ = pc^x (1.36) 
 
 Bt Pq ' ^ 
 
 The changes in temperature 6 may he expressed in terms of the 
 changes in density and entropy. From equation (1.17) 
 
 -M 
 
 (1.37) 
 
 The energy of an ideal gas is equal to 
 
 E = c^T = ^ L- = Zo . P^ . e"" ""^ (1.38) 
 
 ^ (7 - 1)P Po7 7 - 1 
 
 from which 3e/SS = Se/So is obtained in the form 
 
 p7 (7 - l)Cy (7 - l)PCy 
 
 that is, for small values of p and S 
 
 . _ PO ^ , Po 
 
 pgc^ Pn(7 - 1)< 
 
 a + '■• (1.39) 
 
 where the first term represents the change in temperature during adiahatic 
 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 (1.36) yields 
 
 , ^=XAa+XA6; x.=x 1^-1^:1^ (1.40) 
 
 dt 1 ' i Pq 
 
10 
 
 WACA 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 arising 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 
 direction of propagation Is along the ox axis, the frequency of the 
 sound equals oo , and the wave number vector is equal to k. 
 
 C= 4oe 
 
 l(ocrt;-k-x) 
 
 >v 
 
 6 = SQei^'^*-^-^) } 
 
 o = a e 
 
 i((jot-k-x) 
 
 (1.41) 
 
 -/ 
 
 where 4q, Sq, cTq are the amplitudes of vibration of the corresponding 
 magnitudes. Substitution of equations (1.41) in equations (1.31), (1.32), 
 and (1.40) yields 
 
 icoPo^Q = ik(c26Q + hOQ) - - ^k^^^ 
 ico&o - ikpQ^Q = 
 
 icoa^ 
 
 xk^c 
 
 V^^O 
 
 (1.31-) 
 (1.32') 
 (1.40') 
 
 '0 -- 
 
 Elimination of the amplitudes gives the relation between k and co 
 
 coPq = k 
 
 kp 
 
 
 
 x.k^ 
 
 ^ (io) + xk2) 
 
 + I i^kS 
 
 (1.42) 
 
 If k is set equal to co/c - ia, where a is the coefficient of 
 damping of the wave, the velocity of propagation C' in the first 
 approximation is equal to C, and the damping coefficient a is equal 
 to 
 
 a = £li. 
 3 
 
 CO 
 
 PC^ 
 
 (1.43) 
 
WACA TM 1399 11 
 
 where a^ = Pq/Pq ■'-^ ^^^ sq.uare of the Isothermal velocity of sound. 
 For air a = 1,1 • 10"-'-^f cm"-'-, where f = ut/Zsi is the frequency of 
 sound in Hz (1 Hertz = 1 cycle/sec). Hence in many cases the effect 
 of the viscosity and heat conductivity may he neglected or their effect 
 taken into account by introduction of the absorption coefficient in the 
 final results. The smallness of the effect of viscosity and heat con- 
 ductivity of the air on the propagation of sound is determined not only 
 by the smallness of the coefficients |j. and x but also by the small- 
 ness of the gradients of all magnitudes which vary in the sound 
 propagation. 
 
 Eq.uations (1.31) and (1.40) show that these gradients enter the 
 equation in the form of second derivatives of ^, ^, and so forth 
 (for example, p.A5 and xAa) . In the propagation of a wave in free 
 space these derivatives are in order of magnitude equal to t/\ , c/^ , 
 •••, 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 K, cr, and & are determined not by the length 
 of the wave but either by the dimensions of the body Z so that 
 
 AC = C/Z^ and Ao = c/l^ or by the "natural" length d' = VWco (this 
 length is in addition to the lengths X and Z, and is determined from 
 dimensional considerations), where v is the kinematic viscosity 
 (v = M-/p), or by the length d" = -yx/co. in these cases the order of the 
 magnitudes is given by AS; = ^/^'^ and Ao = o/d'^ . 
 
 In 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 lengths \, 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 may 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. Khudsen (ref. 5) 
 and explained by G. Kheser (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. 
 
 In 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 'X. and |a are set equal to in equations (1.3') and (1.40), c = 0, 
 that is, adiabatic propagation of sound is obtained and the equations 
 describing this propagation assume the form 
 
 Pq • 1 = - Vn (1.44) 
 
 ^ + p„ . div I = (1.45) 
 
 ot U 
 
 rt = 0^6 (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) 
 I = - Vcp 
 
 The wave equation for the potential from equations (1.46) and (1.45) is 
 obtained : 
 
 Acp - ^ • -f = (1.48) 
 
 c^ St^ 
 
 which, in the presence of bodies, must be solved with the boundary con- 
 dition 
 
 -m 
 
 ^„ (on the surface of the body) 
 
 •On 
 
 (1.49) 
 
 where S/^ is the derivative along the normal to the surface of the 
 body and ^Qn 1^ "the normal velocity of the surface of the body assumed 
 as small. In place of equation (1.49), for stationary bodies 
 
 $i- = (on the surface of the body) 
 
 °" (1.49-) 
 
WACA TM 1399 13 
 
 For a unique solution of the prolilem of the sonic field described 
 by equation (1.48) the initial conditions for (p and ^/St 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 
 hydrodynamics will yield the' succeeding approximation containing terms 
 proportional to P? , and so forth (when account is taken of nonlinear 
 phenomena). For the pressure p, the density p, and the velocity of 
 motion v, the following series is written: 
 
 p = Pq + ir-L + rtg + • . . 
 p = pQ + 8-|_ + bg + • . . 
 
 V = Vq + ^3^ + Cg + • • • (1.50) 
 
 The magnitudes p^, Pq, and v„ refer to the motion undisturbed by the 
 
 sound; the magnitudes jt-, , &-,, and .^i are proportional to A, the 
 
 magnitudes rto, &2^ ^^"^ ^2 ^^^ proportional to k^ , and so forth. The 
 
 energy and energy flow contain the squares of the magnitudes &-[_, E,^) 
 
 and jr-, . 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 A?, may also contain 
 the first degrees of the succeeding approximations Jtg, ^g, and gg 
 while their contribution will be of the same order as the contribution 
 
 —7 
 
 from the squares of jt-, , &-, , and t-, . 
 
 The general expression for the energy density of a compressible 
 medium is 
 
 U = £2- + pE (1.51) 
 
 where _E is the internal energy of unit mass of the medium. The energy 
 flow W, computed on the basis of equation (l.ll) with the viscosity and 
 heat conductivity neglected, is equal to 
 
 W = Uv + pv (1.52) 
 
14 NACA TM 1399 
 
 From the law of the conservation of energy, 
 
 ^ + div N = (1.53) 
 
 This equation is one of the fundamental equations of hydrodynamics, 
 that is, equation (1.4) for the case of an ideal fluid (n = X =7=0). 
 
 For an ideal gas PE = p/(7 - 1) (equation (1.38)); hence 
 
 ^ =2^t + ypv (1.52') 
 
 27-1 
 
 For acoustics the initial medium is considered motionless (vq = O) 
 The energy of the sound eg = Ug - Pq • Eq and the flow of sonic energy 
 Ng is obtained with an accuracy up to the order of magnitude A^ . Terms 
 of the order of A^ rejected, 
 
 — 1 + div N^ = (1.53) 
 
 where 
 
 PQ^I «1 + H 
 
 eg = -T— + 
 
 ^2 - ^ - 1 ^^1 - ^2-) + y - 1 
 
 (1.54) 
 
 Inasmuch as 
 
 p -0 ^ (i)^ . (e. -.) 4(&\ -!^ 
 
 '0 
 
 + c2(6^ + Sg) + 1 (7 - I)c2s2 = Pq + „^ + TTg + ... (1.55) 
 
 (cq = (dp/dp) = 7 • Pq/p,-, is the square of the adiabatic velocity) and 
 «! = c^5, , equation (1.54) may be rewritten in the form (1.54') 
 
 
 ^0^0 ,^ t s ^^A 
 
 ^2 -fri^h- ?2) -7-7T (1-5^') 
 
NACA TM 1399 ^5 
 
 For a homogeneous medium at rest (vq = 0, Cq =" constant, and Pq = 
 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 (7t-[_^ 5-j_, 
 and £-, ) , not containing the second approximations (ng; ^2' ^^^ ^2^ • 
 The eq.uation of continuity expressing the law of the conservation of 
 matter (equation (1.12)), when written with an accuracy up to terms of 
 the order of A^ , is 
 
 S(6i + &p) 
 
 ^^ ^ + Pq div (K^ + Sg) + div (b^C^) = (1.56) 
 
 This equation is multiplied by Cq/(7 - 1) and the result is subtracted 
 from equation (1.53). Inasmuch as 5-|_ = Tt]^/c|, equation (1.54) yields 
 
 ^ + div N;l = (1-57) 
 
 where 
 
 Po?i 4 
 
 \ = -^^ 
 
 ->• 
 
 SPqCq 
 
 B^ = Ti\ (1.58) 
 
 The new expressions obtained for the energy of sound and the energy flow 
 ^-^ are precisely those which are applied in acoustics. In particular, 
 if the potential (p {'%^ = - v<P, rt-j^ = Po(Scp/St) , see equation (1.47)) of 
 the sound wave is Introduced, then 
 
 3 <"' ,„2 
 
 1 /acpy 
 
 2„2 \^) 
 
 2pQWQ 
 
 % = - pQ ^ V(p (1.59) 
 
 dt 
 
 If, as is often the case, the potential cp depends harmonically on the 
 time and is given in complex form (cp is proportional to e-"- ), the mean 
 energy in time and the mean flow in time are equal to 
 
 £l = -^ V^ • VCP^ + —^^— • cp(p* 
 
 4p2c2_ 
 
 V(p - (p • v^P''*' } (1.60) 
 
 •} 
 
16 NACA TM 1399 
 
 where the sign * indicates that the conjugate complex magnitude 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 shown 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 = 0, at the moment t the surface of the wave front 
 will be the surface F (see fig. l). The entire space between this 
 surface and the source Q, will be filled with energy radiated by the 
 sound. With an arbitrary control surface S enclosing 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 Gauss 
 is used in transforming the integral of div Wg to a surface integral. 
 This integral will consist of the integral over the surface S and the 
 surface of the source Q. Although some inconvenience is caused because 
 part of this surface is movable (ct), it can easily be circumvented by 
 the consideration that the flow of energy through the surface of the 
 source must simply be equal to the source Wg . 
 
 From equation (1.53) the following equation is obtained: 
 
 Pn7 ->■ ^ .1 
 
 TT^TT)- (^1 + ^) ^ XT^ K^l)n h^ = ^2 (1-61) 
 
 -> 
 where n denotes the projection of ^ on the normal to the surface S, 
 
 Eg = / Egdv is the total energy of the sonic field enclosed within 
 S; and the strength of the source Q is evidently equal to 
 
 ^2 = J^[^0^\ ^ \\ + («^l)v]'^^ (1-62) 
 
 where v denotes the projection on the normal to the surface a. If 
 the control surface is passed outside the sonic field (for example, out- 
 side the wave front F, but infinitely near it), from equation (1.61) 
 
 is obtained 
 
 dEg 
 
 d^ = vvg, ^2 = I "2 
 
 Jo 
 
 = Wp; Ep = / Wpdt (1.63) 
 
NACA TM 1393 17 
 
 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 (eq.. (1.18)) is treated in the same manner, the following equation 
 results : 
 
 dT = ^2' ^1 =J V^ (1-63') 
 
 from which it follows that E^^ must he equal to Eg. 
 Prom equations (1.54') and (1.58), 
 
 E2 - El = T"!^ r (^1 + ^2)'^^ (1-64) 
 
 where the integration is over the volume V. The integral / (5 + 52)dv 
 
 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 E-|_ = Eg . 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 
 
 taken into account that in this case da • p . / (^ + ^g) at is 
 
 equal to zero (since this integral is equal to the algebraically assumed 
 path of a surface element d& 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. 
 
 ^2 = % = J" ^^J" d<^(«i?i)v = («il)v ^^ (1-65) 
 
 where (n]_5)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 Eg and Ng containing the elements of nonlinear 
 acoustics, in linear acoustics it is entirely possible and more rational 
 binder 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 nonhomogeneous and in motion. The equations for Eg and 
 
18 NACA TM 1399 
 
 iJp can easily "be generalized to the case of a moving medium. Rather 
 complicated expressions are obtained which will not be considered 
 herein. 
 
 As will be shown in section 1 , it is essential that relatively simple 
 expressions are obtained for the energy density of sound E and energy 
 flow W resembling expressions (1.58) and containing magnitudes of only 
 linear acoustics in the approximation of geometrical acoustics in a non- 
 homogeneous and moving medium. 
 
 4. Propagation of Sound in a Nonhomogeneous Moving Medium 
 
 In the presence of air motion the acoustical phenomena become more 
 complicated. Generally, separation of the acoustical phenomena, in the 
 narrow sense of the word, from the doubly nonlinear processes taking 
 place in a moving medium is not possible. Thus, for example, the flow, 
 pulsating in 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 freq^uency 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 pulsations 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 
 
 T » ^ (1.66) 
 
 where x 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. It depends on 
 the choice of the system of coordinates to which the motion of the flow 
 is referred. 
 
 In fact, a general translatlonal motion of the medium has no signi- 
 ficance since it simply leads to a transfer of the sound wave. For this 
 reason, it is sufficient that equation (1.66) be satisfied in some one 
 system of the uniformly moving systems of coordinates. 
 
NACA TM 1399 19 
 
 If, for example, a flow is considered in which the propagation of 
 the velocities is stationary (that is, does not depend on the timej hut 
 the velocity of the flow periodically changes in space with the period 
 I), then for this flow tt = ". If this flow is considered from the point 
 of view of an observer moving with velocity u, the flow will appear to 
 him nonstationary, the period of the velocity pulsations being equal to 
 X' = Z/u. 
 
 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 the 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 freq.uency f, the freq^uency of 
 sound propagation. In the second system, in addition to this frequency-^ 
 f the receiver would also receive the frequency of pulsations in the 
 flow, that is, f ' = 1/t' = u/l and the combined frequencies f ^^ = f + 
 nf, n = 1,2,3,... 
 
 In the following, condition (1.66) is assumed satisfied in any of 
 the possible reference systems. 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. Hence the theory of the propagation of sound 
 in a nonhomogeneous and moving medium must have as its basis the general 
 equations of motion of a compressible fluid. 
 
 are 
 
 According to equations (1.12), (1.13), and (1.24), these equations 
 
 ^ + div(pv) = (1.67) 
 
 ■z 
 
 Actually it changes somewhat because of the Doppler effect; see 
 
 section 5. 
 
20 WACA TM 1399 
 
 ^ + rot V, V + V— = - ^ + g + vAv + ^ V dlv V (1.68) 
 ^ ^ (^ OS) = A . ^ Q (1.69) 
 
 at ^ •' "^ p T ^ pT - 
 
 where v = n/p ia the kinematic viscosity of the medium. Further, equa- 
 tion (1.13) was supplemented by the term +g, 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 • g 
 is the force of gravity acting on unit volume of the fluid. 
 
 Now let sound he propagated in a medium the state of which is des- 
 cribed by the magnitudes v, p, p, and S. The initial state of the 
 medium (v, p, p, and S) is considered stable and the sound is considered 
 as a small vibration. All the previously mentioned magnitudes will then 
 receive small increments: 5, it, 5, and o, respectively, where K will 
 be the velocity of the sound vibrations; n, the pressure of the sound; 
 &, the change in 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 sound wave 
 in equations (1.67), (1.68), and (1.69), v is replaced by v + ^, p, by 
 p + rt, P, by p + 5, and S, by S + o; by restriction to a linea^ approxi- 
 mation, terms of higher order relative to the small magnitudes £, it, 6, 
 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 ^, rt, 6, and o the terms 
 proportional to the viscosity {\i or v) and the heat conductivity 
 are rejected. On 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 (i. By the method indicated, 
 
 § . [rot ^ I] . [rot I v] . v(v, t) =^ - f . ^^ (1.70) 
 
 p2 
 
 
 + (v, V6) + (^, vP) + P • div C + S div v = (1.71) 
 
 ^ + (r, va) + (J,VS) = (1.72) 
 
 The equation of state, which is given in the variables p and S, 
 is still to be added to these equations. For small changes of pressure 
 rt, and in exactly the same manner as in the preceding section the follow- 
 ing is obtained: 
 
 rt = c25 + ha; c^ = ( ¥\ , h ^ fh] (1.73) 
 
 m^ 'M 
 
NACA TM 1399 21 
 
 Eq_uatlons (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 in the literature lie in the fact 
 that they are true in a medium the entropy of which varies from point to 
 point (VS ^ 0) and in a flow in which vortices may exist (rot v ^ O). 
 
 The approximations made in these equations, in addition to linearity, 
 consist in 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 + CT)/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; 
 Sa/at ^ 0. 
 
 In 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). Hence 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. 
 
 In other respects the equations are entirely general and it is quite 
 immaterial in what manner the initial state of the medium was formed. In 
 bringing about this state, the force of gravity, the heat conductivity, 
 and the energy flow from 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 magnitudes 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 
 single -component medium. In general, the pressure may depend not only 
 on p and S but also on the concentration of the various components. 
 In a complex 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 (Sp/SS)p and VS entering the equations through the variables 
 p and T . 
 
 VS = (Ss/St) VT + (^/ap)TVp 
 
 ir 
 
22 NACA TM 1399 
 
 on the basis of the known thermodynamic relations (Ss/ST)p = c /t (c is 
 specific heat at constant pressure)^ (Ss/dp)rp = - (Sv/St) = - p /p /p 
 
 Is the coefficient of volume expansion and Pp = - i/^J j. 
 Hence 
 
 VS =^VT - !£ . vp (1.74) 
 
 Further, 
 
 (Sp/aT)g = ( dp/as )p(Ss/^)p and (Sp/dT)^ = - (ap/ap)T( ^SDp 
 
 The magnitude 
 
 i^i^-m^-'^-'i^x 
 
 c- 
 
 where c^ Is the square of the adiahatic velocity of sound and 
 
 (^/Bt) = c /T. Thus 
 
 (il 
 
 2 
 
 ^^ • P T (1.75) 
 
 Cp P 
 
 On the basis of equations (1.74) and (1.75) and the medium (c^^Cp^Pp) 
 and its state (p and T as functions of the coordinates) VS and 
 (^p/dS) 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.q., 5), contains 
 five unknowns and is therefore very complicated. 
 
 Nevertheless, if a complete 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), then 
 even in a medium at rest where not only vortices of the flow are absent 
 but where, in general, there is no flow, the right aide of equation (1.70) 
 will not be a complete differential of some function and therefore the 
 
 sound will be vortical (rot E, f 0) . Considerable simplifications are 
 obtained when the changes in p, P, and S are small over the length of 
 the sound wave. Geometrical acoustics are considered in greater detail 
 in the next chapter. 
 
NACA TM 1399 23 
 
 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 = O) and the entropy of the medium 
 is constant (VS = 0) . 
 
 Under these conditions the pressure in the medium is a function 
 only of the density of the medium so that VP = c VP . From equation 
 (1.72) it follows that for VS = 0, a = so that the sound will be 
 propagated Isentropically. Then 
 
 jt = c2& 
 
 If the potential of the- sound pressure is Introduced 
 
 n = i (1.V6) 
 
 P 
 
 the right side of equation (1.70) will be equal to -vn. Therefore the 
 velocity potential of the sound vibrations cp can also be introduced 
 
 I = -VP (1.77) 
 
 The sound will be nonvortical in this case. From equation (1.70) 
 
 ^ = n = ^+ i^, Vcp) =g (1.78) 
 
 Substitution in equation (l-7l) of the magnitude n (for which 
 
 on/at = (c^/p) • (S6/at), Vl'i = 6 . V(c2/p) + (c2/p) • V5) In place of 5 
 
 yields the following equation for cf.: 
 
 ^ = c2 . Acp + (vrin, V(p) + I? (v, V log c2) (1.79) 
 dt"^ ^* 
 
 where YIq is the potential of pressure (heat function) of the initial 
 flow 
 
 ^0= f f (1-80) 
 
 Equation (1.79) was derived by N. N. Andreev and I. G. Eusakov (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. 
 
24 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 "quasipotentlal" \|/. This quasi- 
 potential is introduced "by the equation 
 
 P,t 
 5 = - Vt + / frot vx Vt)dt (1.81) 
 
 The quasipotentlal may he introduced only for sufficiently small vorti- 
 city of the initial flow, that is, the assumption must be made that 
 
 Q = |rot ^1 « CO (1.82) 
 
 where co 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 only. 
 Since Bp/Sp is considered by A. M. Obukhov as the adiabatic velocity 
 of sound, this implies the assumption that 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 v j^ 0) and 
 the constancy of the entropy (VS = O) generally apply together. The 
 possibility is not excluded, however, that the influence of the vortices 
 on the sound propagation is more effective than the influence_of an en- 
 tropy gradient. These hypotheses are assumed satisfied and ^ is sub- 
 stituted from equation (1.81) into equation (1,70) and, since ^ = 0, 
 the right side of equation (1.70) will again be = - Vn. After simple 
 reductions, the equation, which was found previously, is obtained. 
 
 n.I=|| (1,83) 
 
 In this case, however, it is true only approximately with an accuracy to 
 2^/(o2, fl/ca • v/c. 
 
 Expressing & in equation (1.71) in terms of n and '"I' gives the 
 equation of A. M. Obukhov: 
 
 p 
 
 — I = c2A\|r + i'^Q, v\|r) + ^ (v, V log c^) + 
 dt 
 
 c2 
 
 r (V\^, Av)dt - ( VHq, r ([rot v, V\|r)dt| (1.84) 
 
NACA TM 1399 25 
 
 This equation holds with an accuracy up to S^/oo, Ss/oo ^1^,1 (k = w/c) . The 
 
 magnitude AaT = - rot rot v. In this equation^ the terms of order v^/c^ 
 can not he taken into account hecause in the approximations the assumption 
 was made that v/c « 1. 
 
 5. Equation for Propagation of Sound in Constant Flow 
 
 In many cases the velocity of the flow v may he suitably separated 
 into the mean velocity V and the fluctuating velocity u. The effect 
 of these two components of velocity on the sound propagation may he 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. 
 
 In order to obtain an equation for the propagation of s£und in a 
 homogeneous forward moving medium, it is sufficient to put VFIq = 
 and Vlog c^ = in equation (1.79). Expansion of the total derivative 
 
 with respect to time fd^cp/dt = fS/Bt + (v,v)(S(p/St + (v,V(p))") yields 
 
 If the X-axis is taken in the direction of the mean velocity and p 
 is set equal to 'V/c, 
 
 (1 - p2) ^ , ^ . ££ 1 ^ . 2P a!^ ^ , (1.85') 
 Sx^ b/ dz2 c2 at2 c dtdx 
 
 For the system of coordinates ^, t\, and ^ moving together with 
 the stream g = x - Vt, t] = y, and 'C = z, equation (1.85') is transformed 
 into the usual wave equation 
 
 ^.^,% 1.^.0 (1.86) 
 
 a^ 51)2 a^ c2 ^2 
 
 4 
 The 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) . 
 
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. In the system of coordin- 
 ates £,, T\, and ^ at rest relative to the air (hence for an observer 
 moving with the stream), this wave has the potential 
 
 a C + CL r] + a^A 
 
 icolt - — —] 
 
 <p(C,Ti,^,t) = Ae V ^ J.o^+o^+<4 = l 
 
 ^' ^ ^ ^ (1.87) 
 
 where a^, og, and Og are the direction cosines of the normal to the 
 surface of the wave; co 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 ^ is replaced in equation 
 (1.87) by X - vt, T) by j, and ^ by z 
 
 i Icjo't - - (a,x + agy + a,^z) 
 
 <p(x,y,z,t) = Ae L J (1.88) 
 
 where 
 
 W 
 
 = CO ( 1 + I o^^ (1.89) 
 
 Thus the sound frequency in a stationary system of coordinates will not 
 be (£) 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; in the latter case a sound 
 "sink" is being dealt with, which is a very artificial but mathematically 
 useful concept). 
 
 The mathematical expression for the potential of such waves 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 
 
 ^0 = 
 
 il^.. 
 
 = \/k^ + ti2 + ^2 (i_9o) 
 
 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 (C = tj = ^ = O) and the solution with the plus sign represents 
 the same waves converging to a sound sink at the origin of coordinates. 
 If F is a harmonic function, the following is obtained from equation 
 (1.90) 
 
 ic.(t±|J 
 
 Xq = 2 _ (1.90') 
 
 that is, a spherical harmonic wave with frequency co. In a moving medium 
 in which the propagation of sound is described by equation (l.85') in- 
 stead of solutions of the form of equation (1.90), the more general ex- 
 pression is obtained^. 
 
 _llLli) 
 
 where 
 
 X = ^ y ^ (1.91) 
 
 E . ^'^* J- S*, B» = V^*^ + / + z^, X* . J (1.92) 
 
 VTT? VTT 
 
 P= 
 
 With the substitution of X from equation (1.91) into equation (1.85), it 
 is not difficult to show that equation (l.9l) is In fact the solution of 
 equation (1.85), which moreover transforms into a solution of the form 
 of equation (1.90) for V = (P = O). 
 
 The solution (eq. (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. 
 
 The origin of this solution is clarified in detail in section 15, 
 
28 NACA TM 1399 
 
 6. Generalized Theorem 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 in terms of the oscillations at the 
 surfaces hounding the space considered (including also the surface at 
 infinity) . This theorem is derivedi for a moving medium, starting from 
 equation (1.85') (ref. 12). This equation, if the coordinate system 
 x*,y,z contracted in the x-direction is Introduced 
 
 X* = ^ ; y = y; z = z (1.93) 
 
 -4 
 
 1 - p2 
 
 'assumes the form 
 
 Acp - 4 . ^ - _JL i ^ = (1.94) 
 
 c2 at2 Y77T2 c ^^^ 
 
 where 
 
 A = a2/ax*2 + a2/3y2 + a2/dz2 
 The singular solution X (eq. (1.91)) likewise satisfies equation (1.94) 
 
 ^^ - — -T - . -.n^t^ - ° (1-95) 
 
 9 
 
 a2x 
 
 at^ 
 
 2(3 1 a2x 
 
 
 ^i-i>'' 
 
 The solution X contains the arbitrary function F which, because of 
 later utilization of the solution for the proof of the theorem of in- 
 terest, is specialized. 
 
 <- - I) 
 
 X =-W^; E = £SlJ-Jl (1.96) 
 
 B* 
 
 VT 
 
 pS 
 
 where R is the distance ^7x■*2 + y^ + z from the point P, with the 
 
 coordinates x*,y , z , at which the potential cp is to be determined to 
 an arbitrary point of the space Q, with the coordinates XQ,yQ,ZQ, so 
 
 Q 
 
 that X* = Xq - Xp, y = jf^ - yp, and z = z^ - Zp 
 
MCA TM 1399 29 
 
 The function 5(5) ia determined such that 
 
 X 
 
 b 
 
 f(C) • 5(5)dC = f (0) If b > 0, a < 
 
 J f(C)5(?)dc =0 if ^ > 
 
 Eq.uation (1.97) is assumed valid for any function f(4) so that 
 B(4) is everywhere equal to zero except at the point ^ = 0, where 
 &(?) = *"• Hence & (t + r/c)/r* represents a converging spherical 
 impulse (shock) concentrated about E = - ct . 
 
 A certain surface S enclosing the volume Q in the space x*,j,z 
 is considered (see fig. 2 where the surface S is formed by two sur- 
 faces S-|_ and Sg; the volume S is crosshatched) . 
 
 After equation (1.95) is multiplied by cp and equation (1.94) by 
 X ^ one equation is subtracted from the other and the result is integrated 
 over the volume 9. and over the time t-^ to tg . Integration over the 
 four-dimensional volume si'^z ~ ^l) yields 
 
 r dt j d2((pAX - XA<p) + ^ ) dt/dS [x 
 
 — — - (P 
 
 ^■^.S^^'^^S'i^-^^)-" <--' 
 
 Application of Green's transformation results in 
 
 J da ((PAX - XA(p) = J dsU ^ - X ^ 1 (1.99) 
 
 where b/ba denotes the derivative along the external normal to the 
 surface S enclosing the volume ffi. 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 Z and the volume AS 
 
30 NACA TM 1399 
 
 enclosed by it is excluded from the volume of integration Si in equation 
 (1.98). The surface Z (see fig. 2) is considered as part of the sur- 
 face S. The normal to the small sphere Z is denoted hy 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: 
 
 C vJt 
 
 ;^-x^^^(^^- 
 
 
 C-L<^^ 
 
 X ^^' ) 
 
 (1.100) 
 
 The second integral on the right pennits carrying out the integration 
 with respect to time 
 
 ^2- 
 
 But if t-^ tends to -« and tg to +• so that t-^ + R/c < and 
 ±2 + I^/c > 0, then both X and ^x/St at t-^ and tg are equal to 
 zero on account of the form chosen for X; hence Ig = 0. The first in- 
 tegral on the right is considered 
 
 h 
 
 X|'"/4^i=>^'^i-^i-^^]<--> 
 
 Integration by parts of the second term with respect to time and use of 
 the property of 6 (eq. (1.97)) yield 
 
NACA TM 1399 31 
 
 II 
 
 X-^^)-..-|-^(^),^.,- 
 
 1 _1_ ^f^cp^ 
 
 c n* mM) 
 
 (1.103) 
 
 where (p, S(p/Sn, and Scp/St are taken at the instant t = - R/c. 
 
 In a similar manner the third integral on the right in eq.uation 
 (1.100) gives 
 
 I3 
 
 
 
 2P 
 
 i P (^ . ± dS (1.104) 
 
 -* ->■ 
 
 where dS is the projection of the area n dS on the flow velocity V^^ 
 
 (on the X-axis). The integral in equations (l.lOO) on the left is trans- 
 formed exactly as the first and, since in this case S/Sk is identical 
 with b/b'B*, 
 
32 NACA TM 1399 
 
 
 t = -ii 
 
 1 J_ SR (b(i\ 
 
 SR ^ ^=-R 
 
 (1.105) 
 
 and, since dH = 4nR^ • dR*, as the radius of the sphere R approaches 
 zero, the following is ottained: 
 
 Iq = -4.(P^^q (1.105') 
 
 Thus on the left the value of the potential at the point P at the in- 
 stant of time t = is obtained. Since this instant is arbitrary, if 
 the time origin is everywhere shifted forward by t and all the inte- 
 grals I]^, Ig, and Ij are collected, the potential at the point P 
 at the instant of time t will be 
 
 1 iTacp 
 
 E^3tJ^<^s - 
 
 , ,^/^^h (1.106) 
 
 J; 2P 
 
 4jt 
 
 where the brackets indicate that the magnitude enclosed by them is taken 
 at the instant of time t - R/c . 
 
 For Vq = (p = 0), R* = r and R = r and this equation trans- 
 forms into the usual equation of Kirchhoff for a medium at rest. 
 
 If the potential depends harmonically on the time so that 
 
 (p = ie^"^ (1.107) 
 
NACA TM 1399 33 
 
 then substitution of equation (1.105) In equation (1.104) yields for the 
 amplitude 
 
 dS - 
 
 (1.108) 
 
 where k = co/c is the wave-numher vector. If, from the nature of the 
 physical problem, it may he assumed that the disturbances giving rise to 
 the vibrations start within the surface S]_ and not at an infinite time 
 back, they do not have time to be propagated to the surface Sg at a 
 great distance from S-, . For this reason, if Sg is shifted to infinity, 
 the values ^, b^/ba, 3v/Bt can be assumed equal to zero in it. The 
 volume SJ then takes up the entire space with the exception of S-j_ 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-, within which the sources of 
 vibration are concentrated according to the present assumption. Under 
 this condition equations (1.104) and (1.106) maybe assumed to give the 
 expression of the potential at any point of space in terms of the values 
 cp , bip/bn, and Scf/St on the surface S-, within which (or on it) the sound 
 sources are concentrated. 
 
 In conclusion, a certain generalization of this the'orem is considered 
 for "volume" sources of sound. It is assumed that equation (1.94) has a 
 right side which is considered as a "volume sound source." The strength 
 of this source is denoted by Q. Equation (1.94) can then be written in 
 the form 
 
 Acp 1 |!i - -^m= . i ^ = - 4.Q (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 the 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 
 
 I4 = - 4rt I "' dt I d2 Q • X (1.109) 
 
 x; - s 
 
I^ = - 4rt [' dS2 Q ^ ' ^ (1.109') 
 
 c 
 
 34 NACA TM 1399 
 
 Integration over t yields (on account of the 5 function form of X) 
 
 Hence ^ in place of equations (1.106) and (1.108)^ there are obtained 
 
 1 P ^<p") 
 c J R^^J 
 
 vrT7 
 
 dS^ (1.106') 
 
 and 
 
 R* 
 
 -ikR 
 
 
 n - f. 
 
 if the strength of the source depends harmonically on the time 
 
 Q = Q^ei^^ (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 in 
 the prohlem of the occurrence of vortical sound in the motion of bodies 
 in the air. 
 
NACA TM 1399 35 
 
 CHAPTER II 
 
 PROPAGATION OF SOUUD IN ATMOSPHERE 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 generally changes little over a distance 
 equal to the length of the sound wave \. In the "background of this 
 slow change of state of the medium there can also exist smaller changes, 
 hut 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 
 -»• 
 ^ + [rot Jxv] + [rot ^x|] + v(^, J) . . ^ + ^ (2.l) 
 
 5I + it, VP) + (v, ys) + p • div C + & • div V = (2.2) 
 
 |^+ (v, ya) + (I, ys) = (2.3) 
 
 jt = c^S + 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: 
 
 I = Iq . ei$ n = rtQei* 5 = bQe^^ a = cTQe^^ (2.5) 
 
 $ = cDt - kQ0 (2.6) 
 
36 MCA TM 1399 
 
 where oo is the frequency of the sound; kg = oo/cg = Sk/Xq is the 
 wave number in the medium, the state of which is assumed normal (cq is 
 the normal velocity of sound); and Rq® is the phase of the wave. The 
 magnitudes ^q, Uq, 6q, and oq are assumed to be slowly varying 
 functions of the coordinates and, possibly, of the time. The number k^ 
 will be assumed large so that the phase kQ9, on the contrary, varies 
 rapidly. The solutions for |"q, n^, Bq, and Qq will be sought in the 
 
 form of series in the reciprocal powers of the large number ik : 
 
 -+ II 
 
 ^0 = ^6 + ik^ + • • • '^o = «6 + ik^ + • • • 
 
 ikQ ik 
 
 (2.7) 
 
 Substituting equations (2.5) and (2.6) in equations (2.1), (2.2), and 
 (2.3) and making use of equation (2.4) result in 
 
 ikQqaQ = bg 
 
 ^4 
 
 where 
 
 (2.8) 
 
 (2.8-) 
 
 (2.8") 
 
 (2.9) 
 
 q_ = Cq - Cr, V0) 
 t = - ^ + [l^xrot V] + [vxrot J^] - V(v, J^) - 
 
 v^o vp "0 - ^^oq , , 
 
 
 V(h, oq)] - (vp, Co) - P • tiiv Cq 2 • div v (2.10') 
 
 c 
 
 ^5= - ^r ~ ^^' ^°0) - feo^ ^S) (2.10") 
 
NACA TM 1399 37 
 
 Substituting ^q, jIq, and Oq from equations (2.7) into equations (2.8) 
 
 and (2.8') and collecting the coefficients of the same powers of ikQ 
 
 give for the zero approximation (the coefficient of the zero'^^ power 
 of iko) 
 
 I 
 
 q|6 - ^0 • ?^ = (2.11) 
 
 q/c2(rtQ - ha^) - p(|q, v®) = (2.11') 
 
 qOQ = (2.11") 
 
 and for the first approximation (the coefficient of the first power of 
 
 y ~ " (2.12) 
 
 q^o " ^0 * — = ^ ' 
 
 q/c2(rt^ - ha^) - pUq, ye) =K (2.12') 
 
 q • a5 = b^ (2.12") • 
 
 -♦• -»• 
 
 where b', b^, and b^ are the values of b, b., and b,- on substi- 
 tuting in them the zero^'^ approximation of ^q, jtA, and a' from 
 equations (2.1l), (2.11'), and (2.11"). 
 
 From equation (2.11"), it follows that Oq = 0, that is, in the 
 zero'*^^ 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 
 
 «6 = '« • 51 (2.13) 
 
 and as the condition of the simultaneity of equations (2.1l) and (2.11'), 
 the equation of the surface of constant phase (® = constant) is 
 
 V@ 
 
 q2 
 
 c 
 
 2 (2.14) 
 
 For V = 0, as is seen from equation (2.9), q /c = Cq/c = M-^ 
 where p. is the refraction index for sound waves. The equation 
 
38 NACA TM 1399 
 
 Ivo] ^ = [i is called the "eikonal" equation. For v / 0, the ratio 
 q/c may likewise be considered the index of refraction of the medium, 
 but it nov depends also on the direction of propagation of the vaves. 
 
 The situation 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 | V0| = b@/6n, where Ss/Sn denotes dif- 
 ferentiation along the direction of the normal to the surface of con- 
 
 stant phase (® = constant), give 
 
 |v®| = 
 
 a _ ^0 
 
 5E 
 
 c + Vn 
 
 (2.15) 
 
 where Vj^ is the projection of the velocity of the wind on the normal 
 to the wave. With Sg/Sn known, the phase velocity of the waves Vf 
 can be determined. The equation of the moving phase surface is 
 $ = cot = ko® = constant. Differentiating this equation with respect to 
 time results in 
 
 On the basis of equation (2.15) there is then obtained 
 
 Vj = c + Vn (2.17) 
 
 that is, the pha:se 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 
 (2.17), 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 
 jtQ is obtained from the equations of geometrical 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. 
 The left sides of these equations agree with equations (2.1l). If the 
 notations ^q = (x-], Xg? ^■^) > ^n ~ ^4' ^^^ ^6 ~ ^5 ^^^ introduced 
 and equations (2,ll) are written in the form 
 
 k=l 
 
 ik 
 
 x]^ = i = 1, 2, 3, 4, 5 (2.18) 
 
NACA TM 1399 39 
 
 equations (2.12) can "be written in the form 
 
 y^ aik • x^ = ^i i = 1, 2, 3, 4, 5 (2.18) 
 
 k=l 
 
 By a known theorem of algebra, equations (2.18) will have solutions xj|. 
 only when the right sides are orthogonal to the solutions j-^ of the 
 adjoint system of equations: 
 
 5 
 ^^ aik • Yk = where a^^ = a^^ (2.19) 
 
 k=l 
 
 The condition of orthogonality is 
 
 5 
 
 b^y^ = (2.20) 
 
 z 
 
 k=l 
 
 With a^-^ determined from equations (2.1l), (2.11'), and (2.11") and 
 a^-^ transformed, yi, is obtained from equations (2.19) in the form 
 
 y = p • V©, y4 = q;, ^5 = "T 'i (2.21) 
 
 c 
 
 Substituting b, b,, and bg from equation (2.10) in equation (2.20) and 
 making use of equation (2.13) give the condition of orthogonality (eq. 
 (2.20)) in expanded form: 
 
 2 ^^ + 2jt^ div Vg + 2VVit(!) - (Vg, 7 log p qc^) • ^^ = (2.22) 
 
 -*■ 
 where the velocity Vg is given by (see fig. 3) 
 
 Vg = en + V (2.23) 
 
 -*■ 
 
 n being the unit vector along the normal to the surface of constant 
 
 phase • 
 
 Dropping the strokes of iIq and ^q, because the zero approxi- 
 mation is concerned in what follows, equation (2.22) is multiplied by 
 « and an equation for the square of the pressure amplitude is obtained: 
 
40 NACA TM 1399 
 
 2 
 
 ^ + div (Vgn''^) = (Vg, V log pqc^)n^ (2.24) 
 
 which together with equation (2.13) 
 
 rt 
 
 5 = 70 — (2.25) 
 
 pq 
 
 completely solves the problem of obtaining the sound pressure jt and the 
 velocity of the sound vibrations ^. 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 
 
 r2 
 
 T = I (p + 5) (v + 1)2 - PJ- = 1 p|2 + 5(^, t) (2.26) 
 
 where the remaining terms are rejected either as magnitudes of third- 
 order smallness or as magnitudes which within the framework of the lin- 
 ear theory should, on the average, give zero (for example, p(^f, X)) ' 
 Since 5 = n/c^ (compare eq. (2.4)), 
 
 T= i |V0|^ . ^+ (v, V0) . ^ (2.27) 
 
 pq pqc 
 
 Adding the mean potential energy of the second order U 
 
 U=|-4 (2.28) 
 
 pc 
 
 results, on the basis of equations (2.9) and (2.14), in 
 
 «2 . Co 
 
 e = T + U = ^ (2.29) 
 
 pqc 
 
 If equation (2.24) is divided by pqc /c^, then after simple reductions, 
 
 5I + div(eVg) = (2.30) 
 
 that is, the law of conservation of the average energy in geometrical 
 
 acoustics . This law, like the law for e^ and N-^ (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 geometrical acoustics are applicable. 
 
NACA TM 1399 41 
 
 -*■ 
 The magnitude eVg is evidently the mean energy flow 
 
 f = e\^g (2.31) 
 
 It follows immediately that the sound energy is propagated with the 
 
 -*■->■-*■ -*■ 
 
 velocity Vg = en + v, different from the phase velocity V-f. The veloc- 
 
 -♦ 
 ity V- is called the ray velocity . This velocity is equal to the 
 
 geometric sum of the local sound velocity en and the wind velocity v. 
 It coincides with the velocity of weak explosions according to Hadamard 
 (ref. 15). 
 
 On the hasis of equations (2.23) and (2.25)^ the energy flow may 
 also he represented in the form 
 
 N = K + — ^ ^) • ^ (2.31') 
 
 For V = 0, q = e^; and the previously derived (section 3) equation for 
 
 ->•-»• ->•->■ 
 
 the flow N = ir^ is obtained (the expression N-, = Tt-.^-, differs, how- 
 
 -> ->■ . 
 
 ever, from N = Tt^ since the latter vector represents the average value 
 
 in time of the energy flow while W-, is its instantaneous value) . If 
 
 the process is stationary, so that the mean energy of' the ^ound field 
 
 does not change (at least where the sound field has already filled the 
 
 space), from equation (2.30), 
 
 div (eVg) =0 (2.30') 
 
 From this equation it follows that, if tuhes are constructed the lateral 
 surfaces of which are formed hy lines along which the ray velocity is 
 directed ("ray tubes," fig. 4), the product e • Vg,s(s is the cross 
 section of the tube) is constant 
 
 eVgS = constant (2.32) 
 
 Substituting the value of e from equation (2.29) gives 
 
 where it-j^^ ^sl^ ^1 ' ^1 ' '^1 ' ^^^ "^1 ^^^ values of these magnitudes at 
 any 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 
 known at any section of it. To obtain the geometry of the ray tubes, 
 however, a solution of the problem of geometrical acoustics (equation 
 of the eikonal (2.14)) is required.' 
 
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 = cq = constant. The magnitude 
 q = CQ (since v = O) . Hence, from equation (2.33) for the conditions 
 considered, 
 
 rt^s = n^s 
 
 ^ • p/Pi (2.34) 
 
 In the special case of a plane wave, the cross section of a tube is 
 constant (s = s-^) and 
 
 n = 
 
 «1 . (p/p^)^/^ (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/p-. in an isothermal 
 atmosphere is determined by the barometric formula 
 
 P/Pl 
 
 e-»-H (2.36) 
 
 where x = M'g/PT, 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 is not plane but spherical, the cross section of the 
 tubes increases as the square of the distance from the source r^. Hence 
 for a spherical wave in place of equation (2.35), 
 
 Tt = «^ . ^ (p/Pl)^/^ (2.35-) 
 
 The velocity of the sound vibrations ^, in contrast to the pressure, 
 will increase. In fact, for a plane wave V0 = 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 
 
 J = ^ !1^ (p/p,)l/2 = ^ . Jli_ . (p,/p) 1/2 (2.37) 
 
 pc-|_ ^ p-^c-L -^ 
 
 The mean energy flow 
 
 N = it4 = n • — — (2.38) 
 
 P^c^ 
 
 remains constant. 
 
MCA TM 1399 43 
 
 In a similar manner, for the spherical wave, 
 
 t =^^^ . li (Pi/p)^'^' (2.37-) 
 
 Pl^l ^ 
 
 ^ ^ ^ ^1 ''l , . 
 
 r IX 
 
 -»• 
 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 = 
 
 AJr • £ = '\jrrT (2.39) 
 
 and the index of refraction of the sound wave \i will he 
 
 c 
 
 \ 
 
 LQ 
 
 (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 
 
 m ^ (I 
 
 ■^) = ^^ = Y- ^2.41) 
 
 (The X-axis is directed horizontally (fig. S) in the plane of the sound 
 ray and therefore it is assumed that 6 does not depend on z.) The 
 cosine of the angle cp between the x-axis and the normal to the wave 
 will be 
 
 Let 56 /Sx = cos cpQ, where (p„ is the value of cp for y = 0, that is, 
 on the ground surface, where T = Tq. From equations (2.4l) and (2.42), 
 
44 NACA TM 1399 
 
 cos cp = COS (Pq -Mm- (2.43) 
 
 i% 
 
 From this equation it is seen that, if, as is generally the case, 
 the temperature drops with the altitude, cos (p will decrease in absolute 
 magnitude and therefore the 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 x-axis) 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 = Cq - v(y) ^ (2.45) 
 
 and on the basis of equation (2.14), the equation of the eikonal will be 
 
 where r(y) = v(y)/cQ. 
 
 The velocity of the wind at the ground surface itself (y = 0) will 
 be assumed equal to zero (t(0) = O) . Assuming, also, as in (B) , that 
 the initial angle of the normal to the wave is equal to cpQ, bS/dx is 
 set equal to cos (pQ and from equation (2.46) is obtained 
 
 cos (pQ 
 •^^^ '^ = |1 - cos cpprl (2-47) 
 
 From this equation it follows that if the ray" is directed along the 
 wind {x ' cos <Pq > O), then as the velocity of the wind Increases with 
 
 In 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«<:l, this difference is not large. 
 
NACA TM 1399 
 
 45 
 
 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 
 impairment 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 becomes 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 
 
 P, and J are again 
 
 a. 
 
 direction cosines of the normal to the wave 
 
 determined from equation (2.14). Since Cq/c = -y^Tq/T and 
 
 q = Cq - (ve, v). 
 
 a = 
 
 50 
 
 ^ = 1 
 
 1^^ 
 
 T 
 
 5¥ 
 
 V8v> 
 
 (-^) 
 
 4^ 1 , 
 
 ^ 
 
 y 
 
 (2.48) 
 
 
 
 For their determination, it is thus necessary to know the function 
 from equation (2.14). 
 
4-fi 
 
 NACA TM 1399 
 
 As could have been seen from the equations of the preceding section 
 an essential part in the propagation 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 borrowed 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 6.2° 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 3 while the wind increases by 2.13 meters per second, and higher up 
 the temperature drops by 3.65° 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 
 sound-wave front in a nonhomogeneous atmosphere in the presence of wind 
 may be found by the reader in the work of S. V. Chibisov (ref. 16) in 
 which examples of zones of silence are likewise given. 
 
 The velocity of propagation of weak explosions (according to Hadamard) 
 which figures in the work of Chibisov agrees (ref. 12) with the ray 
 velocity Vg 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 Atmosphere 
 
 The propagation of sound in a medium the state of which changes little 
 over the distance of a sound-wave 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, the turbulence of the atmosphere. 
 These changes may be very rapid and their effect on the sound propagation 
 can by no means always be considered by the methods of geometrical 
 
NACA TM 1399 4:7 
 
 acoustics since the dimensions of the region in which an apprecialDle 
 change of state of the medium occurs may he 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 
 disturbances ("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 large-scale 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(x) = je^^^' ^). U.(dfi(t)) (2.49) 
 
48 NACA TM 1399 
 
 where Vj_(x) denotes a component of the velocity of the turbulent flow 
 (i = 1^2j3 are the numbers of the axes ox, oy, oz), q(q-j^, q^, q^) is 
 the wave vector belonging to the scale 1 =_^2jt/q, and d2(q) i_j an 
 element of volume in space of wave number q. Finally, U. (dS2(q)) is the 
 (infinitely small) Fourier amplitude defining the magnitude of the ve- 
 locity pulsations of scale I. It is an additive function of the volume 
 dS: 
 
 U^(!2^ + iSg) = Ui(%) + U^CSig) (2.49') 
 
 If Vj_(x) were_^a continuous function of the point x, there could 
 be written: U^(d2(q)) = Vj_Cq) • d2(Vj^(q)); the "density" of the velocity 
 in space q and the additive property would then be trivial since 
 
 U.(S2^ + 2^) = r v.d2 
 
 The density v^, 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, U^(dfi) 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)^. 
 
 The following assumptions are made relative to the statistical 
 properties of Uj^ : 
 
 (l) The velocity fluctuations associated with the different scales 
 are statistically independent so that the mean of Ui(2-i_) • U*(22) is 
 equal to zero 
 
 7 
 With regard to the mathematical basis of the expression of a 
 
 random function as an integral (2,49), see A. N. Kolmogorov (ref. 18). 
 
 In the following discussion, the presentation of A. M. Obukhov (ref. 19) 
 
 is followed (essentially). The same results, but by a somewhat different 
 
 method, were obtained also by Kolmogorov (ref. 20). 
 
NACA TM 1399 49 
 
 Ui(2i) • Ukfeg) = (2.50) 
 
 if the volumes Q-, and Qg '^.o not overlap (which means that the Uj[ 
 and Uj^ belong to different q) . The asterisk * denotes the conju- 
 gate complex magnitude. 
 
 (2) For coinciding volumes it is assumed that 
 
 Ui(a7)uf(Si) = *ikfei) (2.51) 
 
 is an additive function of the region Q . Physically this means that 
 the intensities associated with the different scales of turbulence are 
 combined. Since ^±]s_ is a certain mean magnitude^ it may be a smooth 
 function and may be expressed in terms of the "density" "^d^- 
 
 $ik(2) = r^i^(q)dQ (2.52) 
 
 The value -^^^^ shall be called the spectral tensor since it 
 determines^ as will be seen^ the distribution of the energy in a tur- 
 bulent flow over the different scales of the fluctuations I = 2jt/q. 
 If interest lies not in the complete velocity of the turbulent flow 
 but in only that part of it vP(x) which refers to the velocity fluc- 
 tuations having a scale less than I = 2tc/p; the expression for vP(x) 
 is obtained from equations (2.48) if the integration with respect to q 
 is extended over the range q > p: 
 
 v?(x) = r el(?' ^)Ui(d2C^)) (2.53) 
 
 U<i>P 
 
 The "moments of correlation" M?, (x'^ x") are determined by the 
 equation 
 
 Ul^d', 1") = via') ' vP(5") (2.54) 
 
 that is, as the mean of^ the product of two velocity con^ionents v^ and 
 v^ taken at two different points x' and x". The set- of magnitudes 
 
 Mj^-|^(x', x")(i, k = 1, i;, 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 
 
50 NACA TM 1399 
 
 ■points x' and x", that is^ on p = x' - x". Substituting v5(x) 
 from equation (2.53) into equation (2.54) gives 
 
 ^k(p) = r ^^^' ^+P)Ui(dS2(q')) • / e-i(^^ ^) • U^(d2(q")) 
 
 V^q^P »^q>P (2.55) 
 
 Use is made of the statistical independence of \J^ and Uv "belong- 
 ing to different q (condition (2.50)) and of the additivity (conditions 
 (2.49), (2. 51), and (2.52)) to obtain 
 
 mP^Cp) = / ei(^' P) ^^^a)da (2.56) 
 
 Upp 
 
 _^The motion of the fluid is considered incompressible so that 
 div V = 0. From equation (2.53) there then follows: 
 
 y~^sv^(^-) 
 
 1=1 
 
 r- = (2.57) 
 
 Applying this relation twice to equation (2.54) (differentiating once 
 with respect to x' and again with respect to x") results in 
 
 ^^ =0 (2.58) 
 
 /__, ax| . ax^ 
 
 i, k=i 
 
 From the preceding and from equation (2.56) it then follows that 
 the spectral tensor ij/j^j^(q) must have the form 
 
 ^.•v(q) = k-k - ^ )f(q) (2-59) 
 
 ik' 
 
 This tensor is now connected with the energy distribution in a turbu- 
 lent flow over the fluctuations of different scales Z. The energy 
 shall be considered as referred to unit mass so that the measure of 
 energy will be v^/z. The mean energy E(p), referring to the velocity 
 fluctuations the dimensions of which are less than I = 2n/p, will be 
 equal to 
 
NACA TM 1399 51 
 
 3 3 -5 p 
 
 E(p) = |(vP(x))2= ly^ (vf(x))2= iy"MP.(0) = 1 V / ^,,(q)cJ2 
 
 ^1 fe fc^^^^ (2.60), 
 or on the "basis of equation (2.59)^ 
 
 E(p) = 4jt / f(q)q''^dq (2.6l) 
 
 For determining the form of E(p)^ use is made of dimensional con- 
 siderations. The flow is assumed not only homogeneous but also isotropic 
 (of course again statistically in the mean). The turbulent motion of 
 such a flow must be maintained by a certain constant supply of energy 
 from outside^ for example^ by the energy of solar radiation giving rise 
 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 heat. The energy dissipated shall be denoted in 
 unit time (per unit mass of gas) by Dq. (it is equal to the supply 
 
 of energy from outside.) The dimensions of Dq are -1?'I1~^ {ctt? / sec^) . 
 
 In a developed homogeneous and isotropic turbulence its spectral state 
 must be determined by the supply of energy which maintains the turbu- 
 lence, that is, E(p) = f(Dq,p). Representing F in the form D^ • p™, 
 
 a dimensional equation for determining n and m is obtained in the 
 form 
 
 L^T-^ = (l2t-2)'^L"^ (2.62) 
 
 from which n = 2/3 and m = -2/3. The impossibility of forming any 
 nondimensional combinations from Dq and p leaves 
 
 E(p) = constant D^/^ p"^/^ (2.63) 
 
 A more detailed analysis by A. M. Obukhov (loc. cit.) shows that 
 
 2/- -2/3 
 constant = -^2 • x ' where x is a certain nondimensional number of 
 
 the order of 1; thus, in the notation of Obukhov, 
 
 E(p) = ^ .y-^ p-2/3 (2.63') 
 
 Since p = 2jt/2, E(p) = jS/S ^ 
 
52 NACA TM 1399 
 
 This law, established "by A. M. Obukhov (ref . 19) and A. N. 
 Kolmogorov (ref. 20) is usually briefly referred to as the "S/s" law. 
 From the law it follows that the energy of homogeneous and isotropic 
 turbulence is concentrated mainly in the region of large-scale fluc- 
 tuations of the velocity. The value of the energy E(Z) is restricted 
 by the maximum scale of the turbulence L determining the dimension 
 of the flow as a whole. For atmospheric turbulence L is the height 
 of observation above the earth's surface. 
 
 Differentiating equation (2,6l) with respect to p and using 
 equation (2.63) give 
 
 -ll/3 
 
 ^¥^(^) 
 
 Dn\2/3 
 
 f(p) =rp^""^" "^ =-f-Wj (2.64) 
 
 and therefore the spectral tensor is equal to 
 
 nkCq) = (Sii, -^Jrr^^/^ (2.65) 
 
 In concluding, the mean-square difference of the velocity component 
 taken at two different points of space is computed: 
 
 (vP(x') - vP(x"))^ = 2((vP(x'))^ - vP(x')vP(1^")) (2.66) 
 On the basis of equation (2.54), 
 
 (vP(?') - vP(^"))2= 2.^?i(0) - mP^(p)J (2.67) 
 
 from which, with the aid of equation (2.56), there is obtained 
 
 (v^(^') - vP(x"))2 =2 r \l- ei(q^ P^j^i^^^)^^ (2.67') 
 
 Introducing the new nondimensional variables a = q-,p, 3 = QoP^ 
 
 and r = qjP \dS2 = da dp dr/p^ and (q, pj = a^/p + P^l/p + X^/Q) where 
 
 5, r[, and ^ are the projections of pi and using equation (2.65) 
 result in 
 
 (v^(x') - v?(x"))2 = K^p^/S (2.68) 
 
 where the constant K is of the order of magnitude of j (sefe 
 eq. (2.64)). 
 
NACA TM 1399 
 
 53 
 
 A. M. Obukhov (ref . 19) gives an estimate of the value of x from 
 the fact that the energy of the atmospheric turbulence is derived from 
 the energy of the solax radiation. According to Brent (ref. 2l) 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 Dq = 5(erg/sec3), which leads to the value t = 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- 
 ing together with the mean wind. The value of the constant y or K 
 in equation (2.68) may then depend, however, on the absolute magnitude 
 
 of the mean wind velocity vq. This evidently has also been observed 
 in tests (see the following) . 
 
 11. Fluctuation in Phase of Sound Wave 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 accuracy of 
 operation of direction-finding acoustical apparatus. 
 
 The test of Krasilnikov consists essentially of the following: At 
 a point Q is placed a sound source (reproducer, fig. lO) at some 
 distance from two microphones Mj_ and lyu. The distance M-]_M2 = I 
 
 is the base of the directional-finding 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 Q^ different from 90°^ 
 
54 ^ NACA TM 1399 
 
 then on account of the different distances QJA-^ ^''^^ Q^ the sound 
 wave would arrive at the microphones M]_ and Mg with different phase. 
 By determining that position of the base M^^Mg (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 may be determined. On this 
 principle are based acoustical direction finders. Such difference in 
 phase may, however, also be obtained for the "correct" position of the 
 base MjM2 (at angle 90° to QB) if the physical conditions of the 
 sound propagation along the two rays QM^ and QMg are different. 
 Such difference in conditions is obtained as a result of the tiorbulence 
 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 QM-j^ and 
 QMg^ 'tbe difference in phase of the waves arriving at M-, and Mg is 
 
 likewise a random magnitude. This phase difference i|f was determined 
 in the tests of Krasilnikov; in particular, its mean-square value 
 
 \|f2 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 velocity 
 of sound and v^ is the projection of the wind velocity on the normal 
 
 to the wave. In this case the directions of the normals for the rays 
 QM-|_ and QMg differ little from the direction QB, which is taken 
 for the X-axis. The projection of the wind velocity on this axis is 
 denoted by v, and Vf = c + v is obtained. The phase of the wave 
 passing from Q to M]_ 
 
 L 
 *1 " "^ / c +^v^ " ^0 ~ "T / ^1^ (2.69) 
 
 (terms of the order of v-,/c and the differences between dx and 
 ds-[_ = dx/cos G are neglected; see fig. lO) where v-|_ denotes the 
 value of the velocity on the ray QM-, . A similar expression will be 
 obtained for the phase in the microphone Mg. For the difference in 
 phase, 
 
 i|r = <P2 - <Pi = -^ / (v^ - V )dx = iL / Av dx (2.70) 
 
NACA TM 1399 55 
 
 where vg is the value of the projection of the velocity in the second 
 ray (QMg) on the axis. The mean value of ^ is, of course, equal to 
 zero. The measure of t will be ^^ . From equation (2.70), 
 
 L L 
 
 >^^ = ^ / dx' / dx" • Av(x')Av(x") (2.7l) 
 
 ^ do Jq 
 
 The averaged magnitude under the integral sign is equal to 
 Av(x')Av(x") = [V3_(x') - vglx')] [v^(x") - VgCx")] 
 
 = v^(x')vi(x") + VgCxOvgCx") - V3_(x')v2(x") 
 
 v^(x")v2(x') (2.72) 
 
 On the basis of equations (2.57), (2.66), and (2.68), 
 
 ^1 - ^1 • ^2 = ^^^Iz' (2-'^^) 
 
 where r-,„ is the distance between the points 1 and 2. 
 
 Use is made of equation (2.73) to obtain 
 
 Av(x') . Av(x") = - ^2l2/3 ^ ^2/3 _ ^2/3 _ ^2/3 1 
 
 2 J^i'l" 2'2" 2'1" 2"1'J 
 
 (2.74) 
 from equation (2.72). 
 
 In figure 10 it is seen that 
 
 J.2 = r^ - r„ „ \^ f-, ■_ q2 
 
 I'l" - ^2-2" ^ W " ^2^ ^^■*" ^^ (e<<i) 
 
 r2,^„= r|„^, = (x^ - x^f + (x^ + Xg)^ 9^ 
 
 
 (2.75) 
 
 In this manner there is obtained from equations (2.7l), (2.74), and 
 (2.75) 
 
56 NACA TM 1399 
 
 (Xj^ - Xgj^/^d + e^jVsj (2.76) 
 
 Setting X = x-,/L and y = Xg/L gives equation (2.76) in the form 
 
 If in the preceding double integral are introduced the variables 
 
 X - y 
 
 ^ = — 2—^ and r] = X + Y, then for -»■ it does not depend on 9 
 
 and converges to a value of the order of 1. Hence 
 
 ^ = constant K^ f^^ L^/^6^/^ (2.77) 
 
 Denoting the length of the base M^^iyU by I and remembering that 
 6 = l/2L result in 
 
 N f2 ^ constant K -^ L^S^S^ (2.78) 
 
 Thus^ the mean-square fluctuation of phase of the direction finder is 
 proportional to the sound frequency <i>, to the square root of the 
 distance from the source, and approximately (exponent s/b) to the length 
 of the base. The test data of Krasilnikov (loc. cit.) very well con- 
 firm both the dependence on tjo (the tests were conducted in the range 
 
 s/b 
 
 from 1000 to 5000 hertz) and the dependence on Z ( ~Z ' ) . 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 . 23) 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 turbulence of 
 the atmosphere, if the observation times under consideration are not 
 too large, must not be considered isotropic (section lO) . 
 
NACA TM 1399 57 
 
 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 M-]_ and Mg 
 
 in the absence of turbulence will be 
 
 _ 2jtl 
 
 t = £±L cos a (2.79) 
 
 The error 6a in a due to the random fluctuations of i will be 
 
 Ba = ^ ., ^. • 5\lr (2.80) 
 
 2nl sm a 
 
 At large values of a, (a~n/2) for the mean-root deviation of &a 
 there is obtained 
 
 ^ ^ constant ^1/2^ _,/6 ^^^^^^ 
 
 Making use of the data of his tests, Krasilnikov determined the numerical 
 value of the constants entering equation (2.8l) as follows: 
 
 fP". 0.3 . J-Ve {h-f {^ (s.as) 
 
 where a is in degrees, 2 and L in meters, and the mean velocity of 
 the wind is in meters per second. For example, for Z = 1 meter, v = 2.7 
 
 meters per second, and L = 2000 meters, there is obtained 'v ba/- - 3 . 
 The value, if compared 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 (ibid.) 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 i|/ at 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 
 microphone M (fig. ll) . The computation was conducted on the basis of 
 the hypothesis (section lO) on the isotropic and homogeneous character 
 
58 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 
 
 A^^i/ = -^ / Av • dx (2.83) 
 
 ^S 
 
 where Av is the change in velocity during the same time. Hence 
 
 A^V'^ = (■^'\ f dx' I dx" • Av(x') . Av(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 At may be represented as the result of the displacement of the 
 turbulence by a small distance 6 = v -At. Then 
 
 Av(x')Av(x") = [v{x', 0) - v(xS 6)] [ v(x", O) - v(x", 6)] 
 
 = v(xS 0)v(x", 0) + v(x'; 5) • v(x", 6) - v(xS &)v(x", 0) 
 
 v(xS 0)v(x", 5) (2.85) 
 
 Making use of the "2/3" law gives 
 
 Av(x')Av(x") = - ^ 4(x' - x")^/^ - 2[(x' - x"f + b^l^^j 
 
 = k2 |(r2 + 62)1/3 _ ^^/^'j; r^ = (^. _ ^..)2 (2.86) 
 
 Substituting 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 
 
 /3 
 
 K^ = constant kSl^/s f^f (| f ^^ (2.87) 
 
 where the constant is found to be =3. Thus 
 
 /\Ja^= kVs lV3 i| (^ . At)5/6 (2.88) 
 
NACA TM 1399 59 
 
 Test data give the relation (v • At)*/^ rather than (v . At)^/^. It 
 is as yet difficult to explain the source of this divergence. Equation 
 
 (2.88), since A^^^^ L, v, At, and cu 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 
 
 (cm^/^/sec) 
 
 The curbulence measurements at the height of 2 meters above the earth 
 coriucted by A. M. Obukhov and N. E. Ershova give (for v ^ 3 m/sec) 
 
 the value K = 3.1 centimeters /^ per second. 
 
 -*■ 
 Gedicke (ref . 23) obtains for K (at v = 0.65 m/sec and height 
 
 1.15 m) the value 2.05 centimeters'^/^ 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 well-known 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 9; 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 were Dahl and Devick (ref. 25). The same phenomenon 
 of acoustical fading was investigated by Y. M. Sukharevskii in measure- 
 ments on mountains (Elbruz expedition of the USSR Academy 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 
 thoroughly 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 in a weak wind 
 (l to 2 m/sec or at an almost complete calm) considerable fluctuations 
 in the sound intensity (fading) are not observed, but the intensity of 
 
60 NACA TM 1399 
 
 the sound drops with increasing distance^ the damping 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 not 
 large; the directional characteristics of the source were not taken intc 
 account, and the conditions under which the points for the various 
 frequencies were taken were not identical. For this reason this result 
 does, not appear entirely reliable; it gives rather the order of magnitud': 
 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 (for 
 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 100 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 (ref s . 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 large-scale pulsations may be considered by the method 
 of geometrical acoustics. Hence the velocity of a turbulent flow must 
 be decomposed into two components v (macrocomponent) and u (micro- 
 component) : 
 
 = / e^Cq^^' U(d2(q)) 
 = I ei(^^^) U(d2(q)) 
 
 (2.89) 
 
 'q<qo 
 
 where v includes the mean velocity of the flow Vq. The magnitude 
 q^^ = k/i_i, where k = Zv./\, is the wave number of the sound wave and \x 
 is a nondimensional number »1. The dissipation of sound from a 
 
 parallelepiped L^ where L»X and L<27;/qQ will now be considered. 
 
 There is here subtracted the molecular absorption (Kneser effect 
 with account taken of the humidity of the air) . It has a considerable 
 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. 
 
NACA TM 1399 61 
 
 Under 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{l)/l, where Z is the scale of the 
 pulsations and u(Z) is the velocity of the pulsations associated with 
 
 this scale. According to the "2/3" law^ u^ = constant • 2^/3 «f (where 
 
 constant ~ 1 cnr/^/sec) ^ so that v = constantV^ . l~^/^ <^ f for all 
 values of f of practical applications.^ Hence in the propagation of 
 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 (v^if ) . 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 
 occior. 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 
 sound waves is denoted by \|; and the total velocity of the flow by 
 V = V + u to obtain from equation (2.84) (for yrTn = ^> V log c^ = 0, 
 
 v/c « l) 
 
 ^ ^ - ^^ + ^ (v> '^) - r (Vt. AV)dt = (2.90) 
 
 (v,'^)-/ 
 
 Passing over to a local system of coordinates in which v = 
 results in 
 
 9 
 It should be remarked that there exists a minimum scale of turbu- 
 
 lence I = Jjjj^^ = l/njx • ^1 [i^ /Bqp'^ (Dq 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 
 v« f may be violated only for f of the order of several hertz. 
 
62 NACA TM 1399 
 
 1 b^ 
 c 
 
 ^(«.'^)^/. 
 
 2 ^ - ^^ = - ~ l^' ^5t J+ I (Vt, Au)dt (2.91) 
 
 The right side of this equation will be considered as the disturbance. 
 By rejecting it con^jletely, the zero'^ii approximation i^q, representing 
 the fundamental wave, is obtained as 
 
 ^^ = A^^[o^t-),(^l,'^)] (2^32) 
 
 where n-, is the unit vector in the direction of propagation of the 
 fundamental wave k = co/c. The complete solution will be 
 
 ^ = ^0 "^ <P (2.93) 
 
 where (p is the dissipated wave. For large distances R from the 
 parallelepiped considered, tp is of the form 
 
 R i(aot-kR) , . 
 
 (P =1 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 cpQ in 
 
 the right side of equation (2.9l) in place of i. There is then obtained 
 
 
 dt = Q (2.95) 
 
 The solution of the wave equation (2.95) having the form of equation 
 (2.94), as is known, is equal to 
 
 "(^''J'-s J, ; ^-' 
 
 (2.96) 
 
 where dv' = dx' • dy' • dz ' and r is the distance between the points 
 X (point of observation) and x' (source of dissipated wave). Let n be 
 the unit vector in the direction of the dissipated ray (fig. 12), R 
 the distance from the center of the parallelepiped, and 9 the angle of 
 
 dissipation (angle between n-|_ and n) . Then, as follows from the 
 
 sketch, r = R - (x", n) (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 63 
 
 (p = 
 
 (2u'k2 + AuS ni)e^(^'^')dv' (2.97) 
 
 where the vector K is equal to 
 
 K = k(n - n-j_); K = 2k sin | (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 
 
 (2uk^ + Au, n-j_)e^^^J'^')dv' (2.99) 
 
 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 L^ is proportional to A L^ while the flow of 
 energy dissipated from the. parallelpiped is ohtained "by integration 
 
 over a distant sphere of radius R and is proportional to R^ 
 
 y |b| 2ds, 
 
 where dS2 denotes integration over all the directions of dissipation. 
 Since interest lies not in the instantaneous value of the dissipation 
 
 but in the mean value^ R^ / B^dS2 must be taken in place of the 
 
 previous expression^ where the bar over jBp denotes the averaging 
 over the velocity fluctuations of the turbulent flow. The mean decrease 
 of the energy flow in passing through the parallelepiped L will be 
 
 /SS^ = oNL (2.100) 
 
 from which a = Z^/WL, and since AN = P • R^ / |b| ^dS2 (p is the 
 factor of proportionality) and W = pA^L^^ 
 
 J bl^dfl 
 
 a = ^ !— ! (2.101) 
 
 a2l3 
 
54 NACA TM 1399 
 
 From equation (2.99) it follows that 
 
 2 = A^ / riv' / rlvVi(Kp); 
 
 Br = %. ^ I dv' / dv-'e^^'^f^^x 
 'l3 Jl3 
 
 16Jt2c2 
 
 r(2u'k2 + Au' ) (2u"k2 + Au'')l (2.102) 
 
 where p = x" - x' is the radius vector between the points x" and 
 x', and U-, is the projection of u on n-, . Introducing in place of 
 x' and X the relative coordinates p and the coordinates of the 
 
 center of gravity x = - — ^-^i- results in 
 
 |b|2 = _a!lL r^ 1(1 P). 
 •|4k%-L-L(p) + 4k^AM-j^3^(p) + A^M-^^-^^Cp) I (2.103) 
 
 where 
 
 M-l-l(p) = u{ . u^ = u^(^')u^(5") 
 
 is the moment of correlation. This moment is identical with the moment 
 M?, (p) introduced in section 10 (see eq. (2.54)) for i = k = 1 and 
 p = q„. Now equations (2.56) and (2.65) are used to find that 
 
 M^^(p) = / e-i("^'p) (l - -iJrq-lVSdq^dqgdq^ (2.104) 
 
 The multiplication of the expression under the integral in equation 
 (2.104) by -q2 and by q^^ respectively^ is obtained by simply applying 
 to M-,-,(p) the operators A and A . Substitution of the moment (2.104) 
 in equation (2.103) leads to integrals of the form 
 
MCA T.4 1399 65 
 
 "^ *Jq> 
 
 dq, . dq.^ . dq^ • e^^K-q^p) p (^) 
 
 
 1 "^^Z ^^3 
 
 = (2rt)3 . / dq-j_dq2dq2 • j (K-j_ - q-L)&(K2 " qg)^ 
 
 V-/q>qo 
 
 &(K3 - qj) . F(q) = (Zn)^ f(K) for K>qQ 
 
 = for K< q^ (2.105) 
 
 Here 6(x) is the symbol of the S-function (see section 6). Hence 
 
 , Br- is obtained as 
 
 '^jtA'-'L^k'^ 
 
 c" 
 
 {-$*^)iiy--''^' <--' 
 
 ;■ (for K = q^ otherwise Bp = O) . From this, on the basis of equation 
 (2.101), ^ ' ' 
 
 ^1 _ 4)rK-^^/^ dS-. (2.107) 
 
 where the integration over the angles is extended to the values K>q^. 
 
 Setting sin 0/2 = ^ and dS2 = sin d0 d(p = 4£ • d? • dp shows that 
 the integration over E = K/2k is extended from £ = l/2n to ^ = 1. 
 Carrying out this elementary integration yields 
 
 a = |j. 
 where 
 
 = ^t5/3 pf^^Z^J . 1 (2.108) 
 
 P = |(2Tt)V3 |i + 25(2n)-l/3 - 2l(2n)-V3 + 0(^-4)1. (2.109) 
 
 The iaagnitude ZriX^/'^X^/'^ is the velocity of the turbulent pulsations, 
 ; the scale of which is less than X. Thus the coefficient of damping 
 of the sound waves in a turbulent flow is proportional to the square of 
 the Mach number (Mg^ = u(\)/c) for the velocity of the turbulent pulsa- 
 tions of scale less thaia X and inversely proportional to the length of 
 
66 NACA TM 1399 
 
 the sound wave X. The magnitude Snyl/^^ 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 ll) and also of A. M. Obukhov and N. D. Ershov i 
 
 (section ll) give, for a moderate wind, 2jtr ' =6. As already pointed 
 out, the turbulence of the wind must not be considered isotropic so 
 
 that, in general, Znx ' is an increasing function of the wind velocity. 
 If use is made of the as yet not very reliable test data presented in 
 section 10, it is necessary to assume x 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 \~^i^ , 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 p., use is again made of 
 
 Sieg's data for a weak wind. In this casfe SnfV^ = 6. The coefficient 
 a is equal to 1.5 decibels in 100 meters, which in absolute units gives 
 a = 10~5centimeters"l. For f = 500 hertz (X = 68 cm) there is obtained 
 H = 10. This value of |i 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 com- 
 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 conjjosition the pressure p depends not only on 
 the density of the medium p and the entropy S but also on the con- 
 centrations Cjj of the individual components forming the medium (for 
 example, 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 
 
 l^Provided, of course, that the fundamental hypothesis of geomet- 
 rical acoustics on the smoothness of all changes in state of the medium 
 is not violated. 
 
NACA TM 1399 67 
 
 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"/p'^ where p" is the density of 
 the dissolved component, and p' is the density of the solvent 
 (p = p' + o" = p'(l + 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 j^he cor- 
 responding equations, the flow of the dissolved coniponent J" is noted 
 as 
 
 J" = vp'C + i (2.111) 
 
 i = - p'D^VC - p'DgVT + p'ugC (2.111') 
 
 where D-, is the coefficient of diffusion, D2 is the coefficient of 
 thermodif fusion, u is the mobility of the solvent in the field of 
 gravity, and g is the acceleration of gravity. The first term in 
 equation (2. Ill) 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, 
 
 ^i^+div J"= (2.112) 
 
 The density of the pure medium p' is subject, of course, to the 
 equation of continuity 
 
 2^+ div(p'v) = (2.113) 
 
 The required equation for C is obtained from equations (2.112) and 
 (2.113): 
 
 ^ + (vVC) = - i div i (2.114) 
 
 For the total density p = p'(l + C) there is obtained from 
 equations (2.112) and (2.113) 
 
68 NACA TM 1399 
 
 ^ + div(pv) = - div i (2.115) 
 
 The fundamental dynamic equation of hydrodynamics 
 
 5t + [^°^ ^^ ^] •" ^- 
 
 = --^+g+vAv + ^ Vdivv (2.116) 
 
 remains unchanged. The equation of entropy will be written in the 
 abbreviated form 
 
 ^ + (vVS) = ^ (2.117) 
 
 where i|f denotes the changes in entropy due to the irreversible processes 
 occurring in the motion of the fluid (\|/ contains terms proportional to 
 ^} ^} 'D-^' ^2-' ^'^^ ^) ^^'^ ^1^° ^hs possible supply of heat from without. 
 
 Equations (2.110), (2.114), (2.115), (2.116), and (2.117) form a 
 complete system of equations for a medium in which some component is 
 dissolved (water vapor in air, salt in water, and so forth) . 
 
 In the propagation of sound all the magnitudes characterizing the 
 
 -*■-*■ ■* 
 medium receive small increments so that v is replaced by v + C, p 
 
 by p + It, p by p + 5, S by S + a, and C by C + Z, where Z 
 
 denotes a small change in concentration of the dissolved component that 
 
 occurs in the medium on the passage of a sound wave. 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, D-^, D^, and u, that is, leaving aside the 
 
 irreversible processes accompanying the sound wave, give-'--'- 
 
 -► 
 
 3I + [rot v,1] + [rot I, V] + y(7, J) = - ^ + M (2. us) 
 
 — + (v, V5) + (5, vp) + p div £ + 5 div V = (2.119) 
 ot 
 
 ^ + (v, s/o) + (J, VS) = (2.120) 
 
 -^'he diffusion of the salt may give an absorption of sound in 
 addition to that due to the viscosity and heat conductivity. 
 
MCA TM 1399 69 
 
 If + (^. VZ) + (|,VC) = (2.121) 
 
 rt = c2 & + ha + gZ (2.122) 
 
 where 
 
 r> 
 
 c- 
 
 - dl.' - (iL- - M. (--3) 
 
 'S,C ^"^'p^C ^'^^'p^S 
 
 The square of the adiabatic velocity of sound for constant concentration 
 of the solution is c^. 
 
 These equations must be considered as the fundamental equations 
 for the propagation of sound in a nonhomogeneous and moving medium of 
 variable composition. If by C there is understood the concentration 
 of the water vapors in the air^ these will be the equations for the 
 propagation of sound in a humid 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 ^ O) , as in the presence of 
 gradients of the concentration of the dissolved component (VC / 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 = O) 
 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), 
 
 ft 
 
 z = Zo-e^*^ Zo = z6 + iiT"^ ■ • • {'^-^'2-^) 
 
 ^0 
 
 result in 
 
 aA = 
 
 Zo = (2.125) 
 
70 MCA TM 1399 
 
 that is, in the first approximation of geometric acoustics the sound is 
 propagated not only isentropically but leaves unchanged the composition 
 of the medium (Zq = O) . 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 velocity 
 
 and the pressure n will be subject to the law 
 
 p = constant (2.12?) 
 
 (compare section 1, 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), 
 
 g_^^g ^.-ha-gZ ^ (^--B-) 
 
 |_ (^ -ha -gZ ^^ p ^^^1 ^ (2.119-) 
 
 |^= - (i VS) (2.120-) 
 
 5^= - (^^^C) (2.121') 
 
 Setting rt/p = n and making use of equations (2.120') and (2.121- ) 
 give the equations for n and %: 
 
 ^\ „Sn vpl Sn _vp 
 2 ' 5t ■*■ „2_ 
 
 pc P c 
 
 ^ = - ^5t + 7^ • 5t + ::2^ ('P'^ (2.128) 
 
NACA TM 1399 71 
 
 1 ^^ai,|^I^j:zll.O (2.129) 
 
 where 
 
 Vp' = hvS + g . vC = Vp - c^vp (2.130) 
 
 Substituting Sn/St from equation (2.129) in equation (2.128) 
 gives the equation for the velocity of the sound vibrations 
 
 ^ = V (c2 . div I + iS2l^) . VP- • divt ^ 2£ . Mil (2.131) 
 
 This is the equation for the propagation of sound vibrations in a 
 medium at rest in which the density, tenrperatiire (entropy), and concen- 
 tration of the dissolved substance vary. It is seen from the equation 
 that for the computation of ^ 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 mediiim. 
 
 Equation (2.131) does not reduce to an equation for the potential 
 or the pressure. 
 
 After E has been found from equation (2.131), the sound pressure 
 is found from equation (2.129) as 
 
 1 . n . f J;^ div ? . ^^} at (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 yc^ is much greater than the terms 
 containing Vp ' is ass\imed. Then, rejecting the terms with Vp ' and 
 setting £ = - vcp (cp is the velocity potential of the sound vibrations), 
 the usual wave equation is obtained: 
 
 g^ = c^ A(P (2.133) 
 
72 
 
 MCA TM 1399 
 
 in which, however, c varies from point to point. 
 
 The term with Vc^ is Vc^ div ^ and in order of magnitude is 
 equal to yc • k^ ( k is the wave number) . The greatest term containing . 
 
 yp ' is yp ' • div ^Jp, in order of magnitude equal to yp ' • k^/p- 
 Hence the terms containing yp' may be rejected and the term containing 
 yc 
 
 '2 retained if 
 
 yc2 »^ 
 P 
 
 (2.134) 
 
 In order to obtain the condition satisfying this inequality, c^ 
 and p' are considered as functions of p, T, and C. Then 
 
 ■-' - ¥1,. • - ^ W,o " ^ fel. 
 
 vc 
 
 (2.135) 
 
 yp ' yp c£ 
 P "^ P " P 
 
 \^J • ^P + l^J • VT + (^f) 
 
 vc 
 
 * (2.136) 
 
 Here (Sp/^p) p = l/a^ (a^ is the square of the Isothermal velocity 
 of sound), (dp/hl)^ p = - pP (p is the coefficient of volume expansion), 
 
 and (Sp/^C)m = - px, where x = tt [■^j is the relative change of 
 
 volume of the fluid (gas) with change in the concentration of salt (or 
 vapor, respectively) . 
 
 Since 
 
 2 = £P . c2 
 
 a'" = 
 
 and 
 
 from equation (2.136) 
 
 7p' _ 
 P 
 
 a2p2T 
 
 Cv = a2p2T 
 
 yp + c2p . VT + c2xVC 
 
 (2.137) 
 
NACA TM 1399 73 
 
 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 
 satisfied. In fact, for water p = 2 • 10~* at 18° C, and at 4° C, 
 
 P = 0. The magnitude x= i (Sv/^C) rp for a solution of NaCl or KCl 
 
 at 15° is about 0.15 to 0.20. According to the measurements of A. Wood 
 (refs. 30 and 3l), the velocity of sound in sea water at t = 16.95° 
 
 and saltiness of 35.02 percent (that is, at C = 3.5 • 10" ) is equal 
 to 1526.3+0.3 meters per second and is governed by the equation 
 
 c = 1450 + 4.206t - 0.0366t2 + 1.137 • 10^(0 - 3.5 • lO'^) 
 
 whence 
 
 (Sc^/^C) = 2c • 1.137 • 103 = 1.42 • c^ 
 
 It is seen that ^e^/6c»xc^. Further, (bc'^/bl) ^ = 2c • 4.2 = 
 
 5.8 • 10-3 . c2 and Pc^ = 2 • 10"^ • c^, that is, (dc^/bl) q»P • c^. 
 
 2 
 Thus the magnitude Vc for salt sea water considerably exceeds 
 
 the magnitude ^^'/p. 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. 
 
74 NACA TM 1399 
 
 CHAPTER III 
 
 MOVING SOIMD SOURCE 
 
 14. \}a^re 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 cp is 
 
 1 
 
 A<P-4^= 0; A = ^+^ + ^ (3.1) 
 
 c2 at2 Sx2 Sy2 Sz2 
 
 It- is assumed that the position of a moving source of sound is determined 
 "by the coordinates 
 
 (3.2) 
 
 In this case it is convenient to introduce a system of coordinates 
 (S^iI^^a) connected with the sound source 
 
 ^ = X - X(t) n = y - Y(t) 
 
 ^ = z - Z(t) T = t (3.3) 
 
 In this system of coordinates the velocity of a wind Vq has the 
 components 
 
 Vqx = - dt - " ''^ 
 
 A 
 
 V - - dt - - ^y 
 
 
 V - - '^^ - -V 
 Oz dt z 
 
 J 
 
 (3.4) 
 
 Equation (3.l) is then transformed to the system of coordinates 
 (C^T^^^t)- For this purpose 
 
 (P(x,y,z,t) = cp(£: + X(t)/ Ti + Y(t), ^ + Z(t); t) (3.5) 
 
WACA TM 1399 "^^ 
 
 (3.6) 
 
 so that 
 
 S(p _ S(p S<p _ Sep 5<p _ S(p 
 
 that is, V^y(p = V^^f<P = V(p 
 
 Hence, the wave equation (3.1) in the system of coordinates ^, t], t^ 
 
 will be 
 
 2 
 1 S cp 2 ,-► , 5cp 
 
 c2 St2 c2 ^^^"'' ^ 
 i(v,v)(^,v)..i(|f.v).= 
 
 (3.7) 
 = 
 
 or , if in place of the velocity of the source v, the velocity of the 
 wind V^ is introduced, then 
 
 ife')-. ' "■'■' 
 
 This equation may be considered ^s the equation for the propagation of 
 sound in a medium moving with velocity VQ(t), depending on the time but 
 not depending on the coordinates. In fact, it almost agrees with the 
 previously (Chapter I, section 5) derived equation (1.85) governing the 
 propagation of sound in a medium in which the wind blows with constant 
 velocity Vq. The difference lies only in the presence of the last term 
 containing the acceleration dvo/dt. If it is assumed, however, that the 
 velocity 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 p^, are 
 
 -♦■ 
 5I + (V,V)V = - ^ + g; div V = (3.8) 
 
76 
 
 NACA TM 1399 
 
 With the assumption that V and p do not depend on the coordinates^ 
 there is obtained 
 
 dt 
 
 
 
 (3.9) 
 
 It follows that such motion i^ realized in a fictitious field of gravity 
 having an acceleration g = dYg/dt. Thus, in considering the sound field 
 of a moving source, the source is assumed as stationary j 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 
 dt 
 
 (3.10) 
 
 15. Sound Source Moving Uniformly With Subsonic Velocity 
 
 An arbitrary sound source moving with constant velocity v less 
 than the velocity of sound c will be considered. The velocity v is 
 directed along the x-axis . Changing to a. system of coordinates fixed 
 to the sound source 
 
 X 
 
 vt 
 
 C 
 
 ^ = y 
 T = t 
 
 (3.11) 
 
 yields a particular case of equation (3.7): 
 
 Acp - 
 
 1 S^y, . 2v S^cp v^ S^cp 
 
 c 3t c StS^ 
 
 ji_ ^ -r _ Q 
 
 c d^ 
 
 (3.12) 
 
 and introducing, as was done in section 5, a system of coordinates con- 
 tracted along the x-axis 
 
 e* = 
 
 vt 
 
 ^ = z 
 
 Ti = y 
 
 T = t 
 
 (3.13) 
 
 yields, in place of equation (3.12), 
 
 23 
 
 1 a% 
 
 c2 ^2 ■ VTTls c ^^rw 
 
 A (p - -5- — - + 
 
 A* = 
 
 ^4*2 5^2 
 
 P=F 
 
 ^V 
 
 (3.13') 
 
NACA TM 1399 
 
 77 
 
 I This equation agrees with equation (1.94:), ^ and the generalized theorem 
 
 ■ of Kirchhoff (see section 6) may be applied to it. It is evidently suf- 
 
 j ficient to restrict this report to the consideration of the sound of fre- 
 
 ' quency cd (in the system attached to the source); so that 
 
 (p = i|re 
 On the basis of equation (1.108), 
 
 -ikE 
 
 iODt 
 
 (3.14) 
 
 ^P = 
 
 4lt yj\^ 
 
 i 
 
 R' 
 
 ^(v)} 
 
 2 
 
 4jtVl 
 
 1 - 6^ <7 
 
 ^ 
 
 -ikE 
 
 dS + 
 
 > 
 
 dS 
 
 (3.15) 
 
 J 
 
 where i|fp is the value of the potential at the point of observation P, 
 and the surface S encloses the source. Further 
 
 R* = V^ + ^2 + ^2; R = 
 
 ■pg* + R» 
 Vl - b2 
 
 (3.16) 
 
 where R* signifies the distance (in the system g'*, r\, ^) from the 
 point of ovservation P to the point of the surface S(Q) : 
 
 
 Tl = T] 
 
 Q 
 
 ^ = ^ 
 
 Q 
 
 ^P 
 
 U 
 
 (3.17) 
 
 The wave field far from the surface S(R* ^ ") is now considered. For 
 large distances from the point P from the surface, as is seen from 
 figure 13, 
 
 R* = R* + R* • cos 
 
 'PQ 
 
 (3.18) 
 
 where R* is the distance OP, Rq is the distance OQ, and 0pQ is the 
 angle between OP and OQ. On the basis of equations (3.18) and (3.17), 
 
 R _ zM1_LR! = P^ +^P ^ -P€q + 5q • ^°s 0pQ ^ 
 
 V] 
 
 V] 
 
 Va 
 
 = Rp + A + 
 
 (3.19) 
 
 12it is necessary to bear in m.ind that p is now v/c, whereas in 
 section 6, (3 denotes Vq/c; thus 3 in section 6 and here differ in 
 sign (because Vq = - v) . 
 
78 
 
 NACA TM 1399 
 
 where 
 
 Rp = 
 
 P^P + ^P ^ 
 Vl - b2 
 
 A = 
 
 -3?| + R^ cos epQ 
 
 > 
 
 (3.20) 
 
 V5 
 
 J 
 
 Substituting the value of R (eq. (3.19)) in equation (3.15) and neg 
 lecting terms of the order 1/r*2 yields 
 
 -^ 
 
 4jtR| 
 
 ■ikA 
 
 
 dS 
 
 (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 vector 
 OP. These angles are different .depending on whether they are taken in 
 the contracted system ^ , t], ^ or in the initial system 4j t], ^ (they 
 differ by a magnitude of the order of P )• Let them be 9 , x in the 
 system %,, j\, t, (and 6*, X in the contracted system, respectively). 
 
 With the system C j\, t,, the following may be written: 
 
 -ikRp 
 
 n 
 
 (3.22) 
 
 where ^p in R and R? must be expressed in terms of ^p. 
 basis of equation (3.14), the following is obtained for <p : 
 
 On the 
 
 Rx 
 
 icDlt-- 
 
 (pi 
 
 (C^.Tlp,^p,T) = ^ 
 
 ,p^ 'Ip'bp 
 
 ^ 
 
 • QCe.x) 
 
 (3.23) 
 
 ■W>here Q(0,<P) is the integral 
 4j 
 
 .a(a,x) J"(|i - ..♦ ^)e-" 
 
 • dS +■ 
 
 2ii3k 
 
 VT 
 
 \|fe 
 
 -ikA 
 
 dS 
 
 (3.24) 
 
 The magnitude Q(0,X) determines the force of the sound source (it has 
 the dimensions of the volume velocity (cm /sec)) and its direction. If 
 
 Q(9,<P) is developed ia a series of spherical functions P2(cos 9)e a^ 
 where I = 0^1,2,3, " ' , and m = ±1, ±Z, ±3, • • • ±i ^ then 
 
 n +1 
 
 Z=0 m=-Z 
 
NACA TM 1399 79 
 
 When all the coefficients Q^^^ except Qq = Q, are equal to zero, then 
 a source of zero order results in 
 
 ^ y ' ~) 
 
 (p(Cp,Tip,^p,t) = ^ • Qq (3.26) 
 
 Kg 
 
 If, for example, only Q,-^q is different from- zero, then, since 
 P° = cos 0, 
 
 ( ""A 
 
 (p(C^,Ti^,Cp,t) = ^-—5 . Qio • cos e (3.27) 
 
 «p 
 
 that is, a dipole source where the dipole is oriented along the ^-axis. 
 The terms with I > 1 give multipole radiation. 
 
 Consideration will now he given to the dependence of <P on the dis- 
 tance. It is evident that the surfaces of constant amplitude i|; diverg- 
 ing in direction hy angles included in 0,(9 ,(f>) will be the surfaces 
 
 R^ = constant (3.28) 
 
 But Rp = W — + T] + ?^ that is, the surfaces of constant 
 
 Vl - 32 
 amplitude will be the ellipses (fig. 14) 
 
 5- + T]^ + f^ = constant (3.29) 
 
 1 - P'^ 
 
 The surfaces of constant phase will be 
 
 / Rp'\ 
 CL = a)(t •) = constant (3.30) 
 
 From this it is seen that the phase velocity along Rp is equal, to the 
 
 velocity of sound c. It is now assumed that the wave field (p 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 way than sim.ple 
 proportionality to t. Hence the observer will not perceive this sound 
 field as a field of harmonic vibrations (although in the system, attached 
 
80 NACA TM 1399 
 
 to the source harmonic vibrations were assumed). Nevertheless, if the 
 changes in the magnitude Rp are not too rapid, the frequency oj' can 
 be determined for the stationary observer as the derivative of the phase 
 a with respect to the time 
 
 The computation of the derivative dRp/dt, on the basis of equations 
 (3.20) and (3.18), yields 
 
 1 dRp 3^p/Rp 1 dCg P+SV^P 
 
 p . -; ^~|^ (3.32) 
 
 "" ^^ vr~^ "^ ^^ (1 - p2) 
 
 whence 
 
 CD' = OD 
 
 1+ P^ 
 ^ 
 
 1 - P2 
 
 (3.33) 
 
 This formula gives an 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): 
 
 CD 
 
 and, if behind the source, 
 
 "^' = 1-T1 (^?=-^p) (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 
 |3 are neglected, the following is obtained from formula (3.33): 
 
 O)' = aj(l + p cos 0) (3.34) 
 
 where 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 Velocity 
 
 The computation carried out in the preceding section shows that the 
 field at a great distance from a uniformly moving source has the form of 
 a field produced by a point source concentrated at the point (see 
 
MCA TM 1399 81 
 
 fig. 13)^ and the nature of the source is entirely concealed in the 
 function Q(0^^) determining the force and direction of the source. 
 On the "basis of this result the theorem of Kirchhoff may be avoided, 
 which, although it can he formulated also for a. nonuniformly moving 
 surface, obtains in this case a form which is very complicated and 
 unsuitable for applications. With the assumption that the source 
 moves along the trajectory 
 
 (3.35) 
 
 The true 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 : 
 
 2 
 A(P - -^ 2_| = _ 4rtQ(x,y,z,t) (3.36) 
 
 c'^ St"^ 
 
 In order to express the fact that the force Q is applied only at 
 the location of the source, use is made of the 5 functions introduced 
 in section 6 
 
 Q(x,y,z,t) = F(t) •5(x - X(t)) • 5(y - Y(t)) • B(z - Z(t)) 
 
 (3.37) 
 The magnitude F(t) gives the dependence of the force on the time in 
 the system attached to the source. Due to the introduction of the 6 
 functions, which are everywhere equal to zero except at the points 
 where their argument becomes zero, the force Q will 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 : 
 
 Q(C.ti,^,t) = F(t) . S(C) . S(ti) . 5(0 (3. 37') 
 
 that is, to the finding of a singular solution of equation (3.7'). The 
 solution of the wave equation (3.26) with the right side present, as is 
 known, reads (se.e section 6) 
 
 cp(x,y,z,t) J ^Q(xSy,z-,t -r/c) ^^ . ^3^^^^ 
 
82 
 
 NACA TM 1399 
 
 where r = V(x - x')*^ + (y - y')'^ + (2 - 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',y',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 at the point (x'^y',z') which was 
 present at the instant of time t - r/c. Substituting now the value 
 of equation (3.3?) in equation (3.38) yields 
 
 [F] 
 
 f(x,Y,z,t) 
 
 8(x' - [X]) • S(y' - [y])S(z' - [Z])dx' dy' dz ' 
 
 (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 are arguments of the 6 functions are introduced in place of 
 x',y',z': 
 
 (3.40) 
 
 and dx',<3y',dz' are transformed by the known formulas of integral 
 calculus 
 
 bx' Sy ' Sz ' 
 ^A~ "Sk Sa~ 
 
 dx'dv'dz' = 
 
 5x' Sy ' 5z ' 
 5x' hy' 5z ' 
 
 = I • dA • dB • dC 
 
 dA • dB • dC 
 
 (3.41) 
 
 The determinant I is readily computed from formulas (3.40), and 
 there is obtained 
 
 I = 
 
 ■^Ix] (x- - x) W] (y' - y) aizj (z- - z)" 
 
 "^ 
 
 Tr 
 
 1 
 
 It - ^)1 
 
 (3.42) 
 
MCA TM 1399 83 
 
 where [v-p] is the projection of the velocity of the source ^ in the 
 direction of r taken at the instant of time t - r/c . 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 
 6 functions, the result of the integration should simply be equal to the 
 value of the function under the integral at the point A = B = C = 
 (see section 6), that is, 
 
 <P(x,y,t) =^ (ifL. l)^^3^^^Q (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 = the 
 following results : 
 
 (x' - x) = [X] - x^ 
 
 (y' - y) = [Y] - y > (3.44) 
 
 (z' - z) = [Z] - 
 
 By taking the square of these equations and combining term by term, 
 an equation for obtaining the value of r at the point A = B = C = 
 is obtained. This value is denoted by E. By the method indicated the 
 following equation results from equation (3.44): 
 
 R^ = <x - Xlt - - 
 
 or 
 
 where 
 
 (* - ?)] ^ f - i^ - ?)]' ^ [ - <- 1 (--' 
 
 f(R) = (3.46) 
 
 f(R)=j^x-x(t-|jj .|^y-Y(t-f)[ .[^z-z(t 
 
 r2 
 
 (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 
 over the points A=B=C=0 in equation (3.43) goes over into the 
 sum over 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, w here R is the distance Q'P; the 
 instantaneous distance, however, r = V(x-X(t))2 + (y-Y(t))2 + (z-Z(t))2 
 is equal to QP. Substituting in equation (3.33) the value r = R yields 
 
84 NACA TM 1399 
 
 ^i^-^/c) (3.48) 
 
 <p(x,y,z,t) = ) 
 
 RVl - P^ 
 where, as is easily verified by equa.tions (3.42) and (3.47), 
 
 Vl - p2 . R* = R 
 
 [-r] 
 
 df 
 
 dR 
 
 (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) = have a second positive root, f(R) must pass 
 through an extreme value, that is^ df/dR must become zero. From 
 equation (3.49) it is seen that in this ca.se [v-d] must be equal to 
 c, which is im.possible. Hence, for v < c. 
 
 ■tl 
 
 ^(x,y,z,t) = ^^ ^^ (3.50) 
 
 * 
 
 13 
 where R is the only positive root of equation (3.46). 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 x-axis 
 
 f(R) = fx - v(t - l)]^ + y2 + z^ - R^ (3.51) 
 
 13ln section 5 the solution has the form F(t + r/c)/r . The dif- 
 ference between them and e quation (3.40) is only an apparent one. In the 
 first place, the factor -yjl - p^ ^±g, not enter for the reason that in • 
 section 5 there was no interest in the absolute strength of the source. 
 Further, equation (3.3l) has also a formal leading solution. Thus, in 
 equation (3.40), Q(x',y',z',t + r/c) can be taken. Th e chose n sign + 
 
 yields, in pl ace of equation (3.40), cp = F(t + E')/l^l - p^^ 
 R* • Vl - P^ ~ I - "*" t^R^'AL where [vj^] ' is the value of v-p 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 R-, = R, it also has the 
 solution Rg = - R". Hence, in order to obtain a lagging solution of 
 
 equation (3.46), it is necessary to take R > if starting from 
 Q(x',ySz',t - r/c) while it is necessary to take R < if start- 
 ing from Q(x',y'^z'jt + r/c). But this root is precisely equal, to -R-^. 
 
NACA TM 1399 
 
 85 
 
 From equation (3.46) the already familiar result is obtained 
 
 _ pg* + R 
 
 ■«■ 
 
 "N 
 
 R^ 
 
 V5*2 ^^2+^2 
 
 vt 
 
 vrr 
 
 (3.52) 
 
 J 
 
 The solution obtained (eq. (3.50)) represents the field of a 
 zero source. By combining such sources^ however^ with the proper phases 
 and disposing them according to a known method, a wave field having any 
 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 insta.nt of time, for 
 example, t = (that is, if F(t) = for t < O), there would be present 
 a wave front, that is, of a surface which would be reached by a distur- 
 bance starting out from the source. 
 
 From each position of the source a wave starts out at time t at 
 the distance R = ct. Substituting this value of R in equation (3.46), 
 the equation of the wave front is obtained: 
 
 jx - X(0)j + fy - Y(0)} + jz - Z(0)j = c^t^ (3.53) 
 
 that is, a sphere of radius ct with center at the point where the 
 source began to function (that is, at x = X(o), y = Y(o) , z = Z(o)). 
 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 a.re, in 
 many respects, in agreement with the known results of Lenard-Wichert 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 
 system the frequency cd, the form of q) (eq. (3.47)) is restricted: 
 
 cp(x,y,z,t) 
 
 iijolt 
 
 R^ 
 
 Q 
 
 R' 
 
 '^ 
 
 Q 
 
 la 
 
 R*^^^T72 
 
 (3.54) 
 
86 NACA TM 1399 
 
 From the instantaneous frequency cjd' perceived by a certain observer 
 not moving together with the source, the derivative of the phase a 
 with respect to the time is understood 
 
 O)' 
 
 da /, 1 dR\ ,„ ^^, 
 
 = dt = ^1;^ - c dtj ^2-^^) 
 
 This formula must be considered as the most general formula for express- 
 ing the Doppler effect. It was nresented 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 formula was not 
 considered. For an observer not attached to the source, the spectrum 
 of the wave field (p(x,y,z,t), notwithstanding the harmonics of the 
 source, will appear as continuous and the intensities of the individual 
 frequencies will be determined by the amplitudes f(x,y,z,a:)) in the 
 expression 
 
 (p(x,y,z,t) = / ¥(x,y,z,cD')e^"^''^d(JD' (3.56) 
 
 It may be asked under what conditions the action of this entire ' 
 frequency spectrum is equivalent to the action of a single one cd' which 
 depends on the time according to equation (3.55). The answer to this 
 question is simple and is connected with an analysis of the work of the 
 sound receiver used by the observer. Let this receiver be a certain 
 resonator with a tiirie constant equal to T. In such a resonator the 
 frequencies will be established in time T. If the time dependence 
 of the force acting on the receiver is written in the form 
 
 cp(x,y,z,t) = -^ . e = Ae (3.57) 
 
 R 
 
 where cd' is the "instantaneous" frequency (eq. (3.55)) and A is 
 the "instantaneous" amplitude (A = Q/r (t), the dependence of A and 
 cjd' on the time may be neglected under the conditions that 
 
 (l) A varies slowly by comparison with the changes of phase co't, 
 that is , 
 
 1 
 A 
 
 dA 
 
 dt 
 
 oa' 
 
 (3.58) 
 
 (2) The frequency cd' changes little in the time T during which 
 the frequencies are being established 
 
 ^ • T « CD' (3.59) 
 
 dt 
 
NACA TM 1399 
 
 87 
 
 From the preceding it can be seen that the Doppler effect may be observed 
 only for sources with sufficiently large damping (small T) . These con- 
 ditions will be analyzed in more detail; but now, if they are assumed 
 satisfied, the Doppler effect will be considered for the case of an 
 observer and a sound source moving uniformly and rectilinearly but at 
 a certain angle to each other. On figure 17 is shown a source Q mov- 
 ing with velocity "^ and an observer P moving with velocity V. The 
 velocity of the observer relative to the source will be if = V - v. In 
 order to compute R, equation (3.30) is used. Substituting in R the 
 value C and passing from motion along the x-axis to motion along any 
 direction (which is done by simple rotation of the system of coordinates) 
 yield 
 
 (?. v/c) 
 
 R = 
 
 d 
 
 r2(l - v2/c2) 
 
 M 
 
 (1 
 
 .2/^2 
 
 /c2) 
 
 (3.60) 
 
 where r 
 
 -»■ -y / 
 
 is the instantaneous distance QP = rp - rQ. Now, dR/dt can 
 
 be computed^ taking into account the fact that both the source and the 
 
 observer are moving, so that 
 
 Vt + ?P 
 
 = vt + f> 
 
 (3.61) 
 
 A somewhat long but simple computa.tion lea.ds to the following result for 
 oj' : 
 
 CD' = to <1 
 
 (I^f. (%!)]. 
 
 (1 - v2/c2) -^ 
 
 2/ 2 
 
 1 - v7e 
 
 (3.62) 
 
 where the vector r? is equal to 
 
 V 
 
 r'^d - vVc") + 
 
 (^, v)' 
 
 (3.63) 
 
 This is the most general formula, for the Doppler effect for a uniformly 
 moving source and observer. From this formula it is seen that^ if they 
 are relatively motionless (u = O), cd' = cd. For a motionless observer 
 (V = O) there is obtained 
 
 Oj' 
 
 ca 
 
 1 + ("i^v) 
 
 1 - v2/c2 
 
 (3.62') 
 
88 
 
 NACA TM 1399 
 
 an 
 
 d for a motionless source (v = O) 
 
 CD 
 
 ' = Jl - ^V^ ; n° = 1 
 
 (3.62") 
 
 For an estimate of da:,' /it and dA/dt let A ^ l/i". Condition (3.58) 
 then reduces to 
 
 f il<- 
 
 or r » 
 
 CJD' 
 
 (3.58') 
 
 If the observation is made in the wave zone, then r» 2jtc/a)'; hence 
 equation (3.58) is satisfied in all those conditions where, in general, 
 the initial formulas derived for the wave zone are applicable. The case 
 is otherwise with condition (3.59). If doo'/dt = - d%/cdt^ is computed, 
 with use of equations (3.60) and (3.6l), then with an accuracy up to a 
 magnitude of the first order with respect to v/c and v/c there is 
 obtained 
 
 T «r c 
 
 (-^) 
 
 (3.59') 
 
 where u is the projection of the relative velocity on the direction 
 of source to receiver and u^ is the projection on the direction per- 
 pendicular to this line. For a relative velocity u of the order of 
 c for certain positions (small u ), the magnitude of the time constant 
 T should ^e much less than r/c and condition (3.59') may be very re- 
 strictive. When this condition is violated, the sound of the harmonic 
 of the source itself will be received as an impulse containing different 
 frequencies continuously distributed. 
 
 18. Sound of an Airplane Propeller 
 
 The sound of an airplane originates fundamentally from two sources : 
 the propeller and the engine exhaust. The sound of the propeller like- 
 wise has a dual character. In the first place, a rotating body, such 
 as the propeller of a motor, gives rise to periodic changes in pressure 
 and velocity of the air near the plane and swept by it. These periodic 
 changes of the air are accompanied by small compressions and rarefactions 
 which are propagated in the form of a sound wave. The sounds of this 
 origin are called rotational sounds. 1*^ In the second place, from the 
 propeller blade, as from any moving body in the air, vortices are shed 
 which likewise impart periodic impulses to the medium surrounding the 
 propeller . 
 
 These periodic impulses are the cause of the second sound, the so- 
 called vortical sound. In section 25 the origin of this sound and its 
 
 14 
 
 This term was introduced by E. Nepomnyashchli. 
 
NACA TM 1399 
 
 fundamental properties will be considered in detail. For the present^ 
 however, the discussion will be restricted to pointing out the fact that 
 the frequencies of this sound are very high and are strongly absorbed in 
 the air so that in observing the sound of a distant airplane only the 
 rotational sound, and at that its lowest harmonics (and also the lowest 
 harmonics of the exhaust), are heard. Hence, it will be entirely rea- 
 sonable to consider in this section only the rotational sound. In fig- 
 ure 18 is shown the propeller of an airplane and its enclosing surface 
 S on which the disturbances brought about by the motion of the propeller 
 will be studied. The faces S' and S" of this surface (fig. 18) will 
 be considered so far removed from the surface of rotation of the pro- 
 peller that the motion of the gas on this surface may be assumed as 
 linear (with the exception, of course, of the general forward motion 
 of the air) . 
 
 The possible frequencies of the rotational sound will be considered 
 first. Let the propeller have n blades and make N rotations per 
 second. It is then evident that at each point on the surface S, due 
 to the rotation of the propeller, the state will be periodically repeated 
 nN times per second so that the fundamental frequency (cyclic) of the 
 rotational sound will be 
 
 cDq = 2nnN (3.64) 
 
 and its harmonic will be en = ODQm, where 
 
 m = 2,3,4,... 
 
 The computation of the intensity of the sound and its direction char- 
 acteristic for these frequencies for a given shape of propeller and 
 for a given speed presents exceptional difficulties .15 Hereinafter 
 the discussion will be limited to the investigation of the most general 
 features of this sound and to qualitative estimates. 
 
 After the control surface S is shifted to the region where the 
 periodic disturbances have become linear, the properties of the potential 
 and its derivatives on the faces S' and S" of the surface S will 
 be considered. A cylindrical system £, , d,X} rigidly attached to the 
 airplane so that <P = (p (4* P^X;'"t) , will be taken as a system of 
 coordinates . 
 
 Since the propeller rotates uniformly in the same plane in which 
 the angle x is measured, X and t should enter (p only in the com- 
 bination t - x/'^} where ao = 2jtN = cdq/u is the angular velocity of 
 rotation of the propeller. 
 
 See note, p. 93. 
 
90 
 
 MCA TM 1399 
 
 Expanding <P in a Fourier series with respect to the time t with 
 period T = 2jt /cdq yields : 
 
 
 >k 
 
 m=- 
 
 t 
 
 111= -» 
 
 / (3.65) 
 
 --/ 
 
 In the following it is sufficient to consider separately each of 
 the harmonics 
 
 (p = i(f 
 m m 
 
 (C*,p)e^Kt-n>-n-X) 
 
 (3.66) 
 
 The theorem of Kirchhoff (section 6) is now applied to the potential of 
 any of these harmonics and the wave field f at a point P is considered 
 at some distance from the airplane. According to equation (3.33), | 
 
 ^i((%it-kniRp) 
 
 ^m(^v''^v>^v'^) = 
 
 f-m^^p^'ip^bp 
 
 Rr 
 
 (3.67) 
 
 where ^ , t]^, ^p are the coordinates of the point of observation P, 
 
 an 
 
 d 0, on the basis of equation (3.24), is equal to 
 
 ^«Qm = 
 
 
 e-"^*^ . dS . 
 
 > 
 
 ^ (3.68) 
 
 dS 
 
 -/ 
 
 where (3 = v/c , v is the velocity of the airplane, 1^ = (J^/c , and, 
 according to equation (3.20), the magnitude A is 
 
 A 
 
 -P£q + R^ • cos 
 
 PQ 
 
 Vi _ p^ 
 
 (3.69) 
 
 The symbol Q is a point on the surface S (fig. 18). From the same 
 figure it follows that 
 
 cos 0pQ = cos 9^ ' cosq + cos(^p-^Q)sin 0* • sin 
 
 ^Q 
 
 i^ = V^^T^ 
 
 ^ = h* 
 
 (3.70) 
 (3.70') 
 
NACA TM 1399 
 
 91 
 
 where h* is the distance to the control surface, p = -\/ r^ + t2 is the 
 distance from the axis of the propeller, 0p and (pp are the angles in 
 
 the polar system determining the position of the point of observation P, 
 and 
 
 S' (or S") 
 
 0g and ^Pg are the same angles for the point Q of the surface 
 
 It is evident that cos 0? = h*/Vp^ + ^*^ ^^^ 
 
 = p/Vp 
 
 Q 
 
 .*2 
 
 sin 0;T = D/-i/p^ + h'^^. Substituting this value of A in equation (3.68) 
 
 and i|fjjj from equation (3.66), the integration with respect to 
 be carried out. It is here necessary to bear in mind that 
 
 iz cos(x-X')-im-n-x' -im-nX 
 
 2 dx' = 2jtie • I^(z) 
 
 •^Q 
 
 can 
 
 (3.71) 
 
 where T^^{7.) is a Bessel function of the first order (m • n). With 
 use of equation (3.7l) the following is obtained from equation (3.68) 
 
 for the surface S ' (4q = h^) 
 
 -im-nXi 
 
 4jt(; 
 
 = 2jtie 
 
 / 
 
 
 
 P dp 
 
 k h' 
 
 ml 
 
 v^ 
 
 .(p-cos Q^) 
 
 ikjn % 
 
 m 
 
 cos 0p + 
 
 > (3.72) 
 
 where r_ is the radius of the control surface, which may be equated 
 to the radius of the propeller, and the magnitude ^Rgp/Sn = SI^q/S C is 
 replaced for large Rpg by cos 0p. For the surface Sg a similar 
 expression Q" is obtained which differs from equation (3.72) in the 
 substitution of -hg for ?*. Combining (^ and Q^ yields 
 
 -im-nXp 
 
 Qn. = 2 ^ 
 
 ^'m 
 
 sin 9^ 
 
 ^ 
 
 dp 
 
 m-n 
 
 X 
 
 ^ 
 
 iXn 
 
 e^2 
 
 
 ik_ 
 
 m 
 
 cos 
 
 (V)i 
 
 + ikjjj cos 0|; 
 
 (t ) 
 
 ^3.73) 
 
 ]} 
 
 J 
 
32 
 
 NACA TM 1399 
 
 On account of the smallness of the magnitude kj^h , the phase multipliers 
 e 1 and e 2 jn^y be expanded in a power series in k^.h'* 
 
 •1 - i^ ^1 ,„ ^^, 
 
 iX 
 
 e " = 1 + 
 
 V 
 
 (P - cos 0p) + 
 
 1 - p^ 
 
 iXp ik h* 
 e = 1 
 
 1 - 3 
 The following is then obtained: 
 
 ^ -im-nXp 
 
 h* 
 / - (3 - cos ©p) + 
 
 Vl _ B^ 
 
 X' 
 
 
 
 p do ^ 
 
 -m 
 
 p sin 0p 
 
 > 
 
 ^^ 
 
 m-n 
 
 V 
 
 1 - p^ 
 
 [Ajjj(p) + B^(d) cos 0* + 
 
 C (p) cos^e* + •••] 
 
 rm 
 
 (3.74) 
 
 ^ 
 
 (3.75) 
 
 J 
 
 where A^(p) ms-y tie considered as the strength density of zero-order 
 sources distributed in the plane of rotation of the propeller 
 
 ^<^'Ka-©/:S^ 
 
 [(^m)l - (O2] + 
 
 >v 
 
 ^VP 
 
 Vl - 32 
 
 [(a ^ ^ -m ^ ^-i 
 
 > (3.76) 
 
 2v2 
 
 2p"'k 
 
 m 
 
 Vl - B^ 
 
 [(V)i t^ + (>^m)2 hJJ 
 
 >' 
 
 the magnitude B (0), as the strength density of dipole sources 
 
 \(o) = - ik^ [(V)i - i^^)^] 
 
 -\ 
 
 ik 
 
 ra 
 
 Vl - p2 
 
 M^f^e 
 
 id. 
 
 1 > 
 
 ^ 
 
 (3.77) 
 
 y 
 
Cjo) = - 
 
 P^ 
 
 , 2 
 km 
 
 i(%\ 
 
 hj + (^^2 
 
 •hj] 
 
 
 
 
 
 
 Vl 
 
 -p2 
 
 
 Therefore 
 
 , the 
 
 unkr 
 
 lown fui 
 
 actions "if 
 m 
 
 and 
 
 ^ijH' 
 
 for ET = 
 
 hf 
 
 and 
 
 C*= - 
 
 hf. 
 
 These 
 
 funct 
 
 ions are 
 
 NACA TM 1399 93 
 
 the magnitude C (p), as the density of quadripole sources 
 
 „2,_2 
 
 (3.78) 
 Vl~T~p2 
 
 and so forth 
 
 calculated now for 
 
 independent of each other beca.use the value of any one of them on the 
 
 control surface S determines uniquely the solution of the wave equation. 
 
 They can "be given only in those cases where it may be assumed from some 
 
 preliminary considerations that the assumed values of -^^ and S\|/jjj/SC 
 
 approximate the true values and are thus in agreement with each other. 
 The computation of these magnitudes presents the fundamental problem for 
 the com.putation of the sound of an airplane. ° It is necessary to call 
 attention to the following circumstance. In the integral (3.75)^ the 
 magnitudes P^, B^, Cj^,'-' must not change their signs as functions of 
 
 D, at least in the region of most effective values of p (in the working 
 part of the propeller blade). It is easily seen that the same refers 
 
 also to the magnitude Iiiin(kmP ^■'"" ^p/VT - (B^)- In fact; 
 kjjjp = 2rtnm]\[p/c = nmv(D)/c; where v(d) is the rotational speed of an 
 arc of the propeller. The roots Xj^^^ of the equation Ijjjj^(x) = 
 possess the property that x-^^ > mn^ but v/c < 1. Hence, in the range 
 of integration < p < rQ, the argument Ij^^^ is less than xj^^ . 
 Because of this I can be moved outside the integral sign^ replacing 
 p by a certain effective value p = R . There is then obtained 
 
 1 -iin-ncPp f\Ro sin 0* 
 
 < 4 + B^ • ^°s ej + Cjj, • cos^0» - 
 
 P^ / > (3.79) 
 
 } 
 
 l^Attempts to compute these magnitudes have been frequently made 
 (see the references at the end of the chapter^ in particular^ the book by 
 E. Wepomnyashchii , "Investigation and computation of the sound of an air- 
 plane propeller"). These computations are not, however, entirely reliable 
 because they make use of the relations of linear acoustics in the nonlinear 
 region. In particular, no account is taken of the presence of a constant 
 air flow; the magnitude oBi|r/(3t (where p is the air density) is equated 
 to the pressure p on the blade of the propeller, whereas 
 
 p = pS\|r/St - p • (Vi|f)2/2 
 
 and so forth. It is therefore difficult in this way to attain anything 
 more than agreement in the most general features . 
 
94 NACA TM 1399 
 
 where A^, Bj^, '^m'"' ^^^ ^^^^ values of these magnitudes over the 
 length of the propeller blade. Since the magnitudes Am, Bm, Cm, • • • 
 represent the coefficients of expansion in the small parameter k h*, 
 the value of A^^^ among the terms in braces in equation (3.79) should 
 be predominant; that is, there is a source of zero order. Hence, the 
 directional characteristic of the sound of the airplane propeller will 
 be determined essentially by the factor I^^ while the remaining terms 
 in equation (3.79) will only deform somewhat and displace the directional 
 curve given by this factor. Since not only do the roots of the equation 
 
 -"-TTir,^^^ ~ ^ exceed mn but also those values of x'' which correspond 
 mil mn ^ 
 
 to the maximum l^U) , the expression Imn (kjj^Q ^In 0*/Vl - pS) will . 
 monotonically increase with increase in 0* to n/z and then drop to I 
 for e = n. Thus the maximum of the radiation will lie at 0* = n/z, 
 that is, in a plane perpendicular to the line of flight of the airplane 
 (in the plane of rotation of the propeller). ^^ 
 
 This curve is given in figure 19 (curve a). In fact, there is 
 generally observed an assymetry of the directional curve (curve b of fig. 
 19) which indicates that the part played by the dipole radiation can not 
 be entirely ignored in comparison with the part played by the radiation 
 of zero order. Both curves refer to a system of coordinates which are 
 at rest relative to the airplane. Now the intensity in the sound spec- 
 trum of the propeller will be determined. For this the magnitude Q)^ 
 in equation (3.79) has the sense of a volume velocity. Its fundamental 
 term contains the magnitude Aj^ equal approximately to the sum of the 
 velocity components of the air normal to the surface S. These velocities 
 are produced by the compression of the air in the motion of propeller 
 blades and may be represented in direct dependence on the velocity of 
 motion of these blades. 
 
 Consider the velocity component u(t - (p/a),o,?*) normal to the 
 surface S-]_ . The same expansion (eq. (3.65)) in a Fourier series i 
 applicable to it that applied for <P, namely. 
 
 s 
 
 m 
 
 im(a)Qt-n(p) 
 
 u (p,?*) e , (3.80) 
 
 m 
 
 17 * 
 
 The difference between 6 and 6 is ignored since these angles 
 
 differ by a magnitude of the order of p . 
 
NACA TM 1399 95 
 
 ; whence 
 
 P2n 
 
 / ^*\ . -inmX 
 
 1 
 T 
 
 
 
 ■{ 
 
 A-,o,r e . dt (3.81) 
 
 CD / 
 
 ■ If the width of the blade at p is equal to Z(o) ? it may be assumed 
 
 > that u as a function of time has the form of an impulse lasting over 
 
 the time T = l/v = l/o^p, so that u = Uq for < 2 < t and u = 
 
 1 outside this interval. Carrying out the proposed integration in equa- 
 i tion (3.81) yields 
 
 I f ,^\ . ^-imnX 
 
 I- . I ^ 
 
 i „ / -im- \ 
 
 = - — • 1S__ ±l=^[e - ij 3.82) 
 
 T imjDQ 2ran \ J ^ ' 
 
 From this it is seen that the amplitude of u very slowly decreases 
 with increasing m so that the spectrum of the sound of the airplane 
 should be very rich in harmonics, as is actually observed to be the 
 case. ° 
 
 19. Characteristics of Motion at Supersonic Velocity. 
 
 Density Jumps (Shock Waves) 
 
 Before the problem of immediate interest, that of sound radiation 
 from a source moving with supersonic velocity, is discussed, considera- 
 tion will be given to those special phenomena which arise in the flow 
 about a body with velocity of motion exceeding the velocity of sound in 
 the medium c . 
 
 The essential difference between a flow with v > c and a flow with 
 V < c may be considered from the equation for the velocity potential ^ 
 describing the flow of a compressible fluid. According to the general- 
 ized equation of Bernoulli (eq. (1.27')), 
 
 w 
 
 jr?=^-|(V.)^ (-B3) 
 
 1 8 
 
 The assumptions herein were too simplified, of course, to expect 
 
 anything more than a qualitative conclusion. The computation of the form 
 of the Impulses is carried out in the book by E. Nepomnyashchii. As pre- 
 viously pointed out, however, it would be necessary to choose values of 
 the impulse on a suitable control surface, whereas generally their values 
 are computed in the plane of the propeller. 
 
96 NACA TM 1399 
 
 On the other hand, the equation of continuity reads 
 
 ^ - div(oV<i>) = (3.84) 
 
 (since v=-v^). Noting that Sw/^t = i |£ ^^ = ^ || and Vv = ^vP 
 
 and in equation (3.84) expressing Sp/St, yp in terms of Sw/Bt, yw, and 
 w in terms of $ with the aid of equation (3.83) yields 
 
 W, v[^ - I (V$)^JU (3.85) 
 
 If a local system of coordinates x, y, and z is introduced such that 
 the axis ox is directed along the normal to the surface $ = constant 
 (i.e., along the direction of the velocity v at the point considered) 
 and the axes oy and oz lie in the tangent plane, equation (3.85) 
 
 assumes the form 
 
 {'-S) 
 
 ^N S^* s% ^h 2v S^* 1 a^^ 
 
 + 
 
 Sx^ ay2 Sz^ c2 ' htbx ' c2 St 
 
 - -p — p = (3.86) 
 
 If at a point of the flow the velocity v exceeds the local velocity of 
 sound c, the coefficient before b^^/bx^ becomes negative so that the 
 coordinate x assum.es, as it were, the same status as the time; the 
 equation of elliptical type relative to the coordinates turns into an 
 equation of the hyperbolic type. These two types of equations fundamen- 
 tally differ from one another. The hyperbolic equation has discontinuous 
 solutions which are not uniquely determined by the boundary conditions. 
 A simple example illustrating this fact will subsequently be given. In 
 fact, in the motion of a body at supersonic velocity, there arise in the 
 medium the so-called density jumps or shock waves. These jumps are 
 propagated over a great distance from the moving body along surfaces 
 which for a small magnitude of the jump approximately coincide with the 
 characteristics of equation (3.86). In the density jump the state of the 
 medium changes discontinuously. Such discontinuous change is undergone 
 simultaneously by all the magnitudes characteristic of the medium: the 
 velocity, the density, the pressure, the temperature, and the entropy. 
 By studying the propagation of the sound from a source moving with super- 
 sonic velocity, it would be systematic to start from that state of the 
 medium which is produced by the motion of the source and to consider the 
 sound as a small disturbance. However, at this time general methods of 
 
 I 
 
NACA TM 1399 
 
 97 
 
 solution of the problem of the supersonic flow about a body are not 
 available} and, therefore, no theory is available -which permits finding, 
 in this case, the fields of velocity and pressure and determining the 
 magnitude and position of the density jumps which arise with supersonic 
 flows. For this reason the discussion will be restricted to the consid- 
 eration of certain partial problems B,nd to a qualitative analysis of the 
 phenomena. Consideration will now be given to the simplest cases of 
 supersonic motion which permit an uncomplicated mathematical analysis. 
 The profiles of a thin infinitely long wing are shown in figure 20. The 
 flow in this case is two dimensional and its velocity will be assumed as 
 V > c. If it is assumed that the wing is thin (and the angle of attack 
 small), the disturbance imparted by it to the flow may also be assumed 
 small. Corresponding to this assumption, the potential ^(x,y) is 
 represented in the form 
 
 $ = - vx + (P(x,y) 
 
 (3.87) 
 
 where (p is a small correction and the higher powers of it may be 
 neglected. Substituting equation (3.87) in equation (3.86) and neglect- 
 ing terms containing higher powers and derivatives of (p yield 
 
 1 _ i!\ Si ^ a^ ^ 
 
 (-S) 
 
 ^r 
 
 
 
 (3.88) 
 
 where c is the value of the velocity of sound in the undisturbed flow. 
 Setting 
 
 X = T 
 
 ^ 
 
 ,.!>. 
 
 (3.89) 
 
 gives, in place of equation (3.79), 
 
 :^2 >2 
 
 o cp o (p 
 
 By2 " St2 
 
 = 
 
 (3.90) 
 
 As also follows from the general theory, an equation of the hyperbolic 
 type is obtained. If t is considered as the time, it coincides with 
 the equation for the propagation of waves in one dimension (y) with a 
 velocity equal to 1. 
 
 The general solution of this equation has the form 
 
 (P = f^ (t - y) + fg (t + y) 
 
 (3.91) 
 
98 MCA TM 1399 
 
 The disturbances, giving rise t o cp, are disposed (along the wing profile' 
 
 from. T = to T^ = Z/yP^ - 1 and are propagated according to equation 
 
 (3. 91) without change of their intensity along the lines j - '^ and 
 
 y = - T (for example, PQ and P'Q' on fig. 20(a)). The assumption that 
 
 f 2 7^ for y > would mean that the disturbance would travel ahead 
 
 of the wing at any large distance. This contradicts causality and, 
 therefore, it is assumed that f 9 = for y > and for the same 
 
 reasons f-[_ = for y < 0. " Then 
 
 and 
 
 tp = fi (x - y) y > 
 
 (P = ^2 (f + y) y < (3.92) 
 
 With this choice of solutions the disturbances concentrate in the strips 
 OAEO' and OA'B'O'. The inclination of these strips ^'3 determined by the 
 equation 
 
 y = ^T = ^ - ^ (3.93) 
 
 Vp2 _ 1 
 
 so that the angle e = AOO', called the Mach angle, is equal to 
 
 sin e = - = - (3.94) 
 
 P V ^ ' 
 
 The form of the functions f-]_ and fg can now be connected with the 
 
 form of the wing profile. Denoting the normal to the surface of the 
 wing by it, the following condition exists on the surface of the wing: 
 
 |i = - V . cos(x,n) + ^ cos(y,n) + ^ cos(x,n) = (3.95) 
 on dy ^ ^ ' ' 
 
 which expresses the fact that the components of the velocity normal to 
 the wing surface are equal to zero. If the wing profile is thin and the 
 angle of attack of the elements of its surface is everywhere small, 
 cos(x,n) = and cos(y,n) = 1. Hence the condition of equation (3.95) 
 can be approximately written as 
 
 \^/v=0 
 
 V cos(x,n) (3.96) 
 
 y=o 
 
 19 
 
 In this supplementary requirement there is also expressed the 
 
 property referred to above of equations of the hyperbolic type. 
 
NACA TM 1399 99 
 
 The sign + holds for the upper surface; the sign - for the lower 
 surface. Substituting cp from equation (3.92) yields for the upper 
 surface 
 
 (; 
 
 
 cos(x,n) (3.97) 
 
 and since cos(x;n) is given on the wing profile as a function of x, 
 and therefore also as a function of T, there is thereby determined the 
 potential f]_(T^) with an accuracy up to an unknown constant. In the 
 same manner there is also found f2('^)- From equations (3.92) and (3.97) 
 an additional velocity on the x-axis is obtained 
 
 Sep df-|_(T: - y) ^ 
 Avy = - 3— = - =— -— : = — rrm^iiz cos(x,n) (3.98) 
 
 where cos(x^n) is considered as a. function of (t - y) • 
 
 With the aid of equation (3.83) the change in pressure Ap = p - p 
 
 as compared with the pressure in the undisturbed flow p^ can also be 
 obtained. Thus, for small Ap, from equation (3.83) 
 
 ^=^.iV|£^^ (3.99) 
 
 Pq ut c, c 
 
 The constant v /2 is so chosen that in the undisturbed flow, where 
 (V^)2 " 
 and n 
 yield 
 
 (V'*')^ = "V and ^/Bt = 0, p = Pq. Substituting -^ from equation (3.77) 
 and neglecting higher powers of cp and powers of the derivatives of ^ 
 
 Ap = o^ V ^ (3.100) 
 
 '-' ox 
 
 whence on the basis of equation (3.78) 
 
 2 
 Ap = P^ • cos(x,n) . (3.101) 
 
 VpS - 1 
 
 At the point x = (the point of meeting of the flow with the 
 profile) cos(x,n) $ 0, and at the point of departure (x = Z) cos(x,rj) ^ 0. 
 Outside the interval < x < 2 , cos(x,n) = 0. Hence the pressure Ap 
 and the velocity Av have the form shown in figure 20. At the point of 
 approach a discontinuity of the motion occurs. The resistance of a thin 
 wing computed in this manner agrees well with test results (ref. 34). 
 Both the pressure Ap and the velocity Av^^ maintain their values 
 constant along the line y = ±'^, that is, along lines inclined to the 
 flow by the Mach angle £ (sin e = c/v) . 
 
100 NACA TM 1399 
 
 The solution presented above, demonstrating the presence of dis- 
 continuities in the supersonic flow about a body, is suitable essentially 
 for infinitely small density jumps. The theory of density jumps of 
 finite magnitude can not be obtained from a consideration of only the 
 differential equations of hydrodynamics since these equations lose their 
 validity precisely in the region of discontinuity and must be replaced 
 by suitable boundary conditions. In order to find these, a density jump 
 of the form represented in figure 21 will be considered; equation (3.83) 
 is the region of the undisturbed medium and equation (3.84) the region 
 of the jump. Let the jump move with the velocity V in the positive 
 direction of the x-axis. It is natural to take a system of coordinates 
 in which the jump is at rest. In this system the velocities of the gas 
 along the x-axis in rerions of equations (3.P3) and (3.6l) will be 
 
 u-L = - V 
 
 Ug - V 
 
 (3.102) 
 
 where Ug is the absolute velocity of the gas in the region of the jump. 
 To derive the conditions at the jump it would be necessary to rewrite the 
 fundamental equations of hydrodynamics in Integral form. As was explained, 
 however, in chapter I, these equations represent no other than the three 
 laws of conservation and this fact may be utilized by applying these laws 
 directly to the region of the density jump. The matter, momentum, and 
 energy flows on both sides of the density jump must be the same. Making 
 use of the expressions for these magnitudes (eqs. (l.9), (l.lO), and (l.ll) 
 a.nd neglecting for the present the viscosity and the heat conductivity, 
 the law of conservation of matter is obtained 
 
 0]_u-|_ = pgUg (3.103) 
 
 where o-]_ and pg ^■'^^ 'the density of the gas before and after the jump. 
 Further, the law of the conservation of momentum is obtained 
 
 P-^uS + p^ = pgul + pg (3.104) 
 
 where p-j^ and pg are the pressure before and after the jump, and 
 finally the law of the conservation of energy is obtained 
 
 13 13 
 
 2 '^1^1 "^ '^l^l^l ■*■ Pl^l ~ p f'2^2 "•" P2^2^2 ''" P2^2 (3.105) 
 
 where E-|_ and Eg are the energy of unit mass of the gas before and 
 after the jump. 
 
NACA TM 1399 101 
 
 With the use of equation (3.103) ^ equation (3.105) may be written in 
 the following form: 
 
 12 12 
 
 •e u-]_ + w-i_ = ■o ^2 '^ ^2 (3.105') 
 
 where w = E + p/p is the heat function. From these three equations are 
 ohtained 
 
 u = _ ./ f2 . P2 - Pi ^ _ ^ (3.106) 
 
 1 II P;l P2 - Pi 
 
 Pi P2 - Pi , . ^^ . 
 
 "2 = - A (3.107) 
 
 ^ "' P2 P2 - Pi 
 
 and also with the use of the equation for an ideal gas 
 
 E = -i^ . ^ ^=.-J—2 (3.108) 
 
 r-ip r-ip 
 
 the relation of Hugoniot (ref . 35) is obtained 
 
 1 /P2 Pl^ 
 
 (r 
 
 hTi|-9 = i<^.-3)g-i) M 
 
 Equations (3.106) ^ (3.107), and (3.109) permit computing all the data 
 referring to the density jump as soon as the pressure p-, and the den- 
 sity of the gas p]_ ahead of the jump are given, and also one of the 
 magnitudes characterizing the jump, for example, po . 
 
 In conclusion, the cha.nge in entropy occurring in a density jump 
 will be computed. From equation (1.34) it follows tha.t the entropy of 
 unit mass of the gas is equal to 
 
 S = So + c^ln ^(^) (3.110) 
 
 From this equation the change in entropy is obtained 
 
 'P-p /poXT Pp Pp 
 
 So - S. = c In — I — ] = c In -^ + c In — (3.111) 
 
 ^his relation was earlier established by Rankin (ref. 36); see alsc 
 reference 37. 
 
 I 
 
102 MCA TM 1399 
 
 If use is made of the relations of Hugoniot, it is not difficult to show 
 that for a density jump this magnitude is greater than zero so that the 
 processes in the jump have an irreversible character. It is precisely 
 for this reason that it is impossible to restrict oneself to the dif- 
 ferential equations of hydrodynamics which do not take into account such 
 processes. The motion of the jump, as is seen, proceeds in the direction 
 of increasing entropy since the gas has less entropy before the jump 
 (eq. (3.87)) than after it (eq. (3.87)), and the jump moves in the dir- 
 ection from (2) to (l) . The velocity of this motion V = - u-|^ is readily 
 found from the preceding equations if pg and p-|^ are eliminated from 
 
 equations (3.106) and (3.109). There is then obtained 
 
 ^1 = ^^ = ^ • |(r + 1) ^ + (r - 1)1 (3.112) 
 
 where c is the adiabatic velocity of sound in the gas at rest (eq. 
 (3.84)). Since p_ > p-, , therefore V^ > c^; that is, the jump always 
 
 moves with a velocity greater than the velocity of sound in the medium in 
 which it originates. The relations herein derived will be used in 
 analyzing the work of a sound receiver moving with a velocity greater 
 than the velocity of sound in the medium. 
 
 20. Sound Source Moving with Supersonic Velocity and Having 
 
 Small Head Resistance 
 
 In this section consideration will be given to the radiation of 
 sound by a source moving with velocity v > c and having a small head 
 resistance. The theory of such a sound source is, to a considerable 
 degree, analogous to the theory presented in section 19 of an infinitely 
 thin wing. The sound source will be imagined as located on the body 
 (fig. 22). The profile of the body will be given by a curve in a 
 cylindrical system of coordinates (o,K)X) 
 
 P = Oq{£) (3.113) 
 
 The cross section of the body ttdq will be considered inf initesimally 
 small. 
 
 It is assumed further that the surface of this body or a part of it 
 performs small vibrations of frequency go. This vibration will be the 
 sound source. The potential of the flow ^ will be given in the form 
 
 $=-v • x + (Pq + (P (3.114) 
 
 where v • x is the potential of the undisturbed flow, cp^ is the 
 potential produced by the motion of the body, and (p is the potential 
 
NACA TM 1399 103 
 
 produced ty the vibrations of the surface of the body (it is proportional 
 to e ) . The potential cp is of no interest since in a system of 
 
 coordinates connected with the body it does not depend on the time.'^-'- 
 The assumption of the small cross section of the body permits restriction 
 to the linear theory. In virtue of this the solution will be a super- 
 position of the steady solution and the unsteady sound field. The prob- 
 lem thus reduces to the determination of fp. For solving this problem 
 the method of sources will be used. The field of a point source of 
 sound m.oving with supersonic velocity will first be determined and then 
 a suitable distribution of these sources over the surface of the body of 
 revolution will be taken. In a system of coordinates attached to the 
 body let there be a point source at the points ^^, r\^, ^„ lying on the 
 
 surface of the body under consideration. In a stationary system of 
 coordinates, the coordinates of this source will be 
 
 X = vt + £q 
 
 ^ = ^0 
 
 Z = ^0 (3.115) 
 
 The strength of this source dQ will be assumed as infinites imally small 
 and proportional to an element dOQ = 2rtpQ ■ d?Q of the surface of the 
 
 body on which it is located 
 
 dQ = q{t,KQ,T\Q,^Q) • da^ • S(x - vt - ?q) x 
 
 S(y - Ti^)(z - ^^) (3.116) 
 
 In this formula the small ma.gnitude 
 
 dF = q(t,?o^TiO'^o)^^0 (3.117) 
 
 has the same meaning as F in equation (3.37). In correspondence with 
 equation (3.43), the solution of the point source will be written in the 
 form 
 
 Xo(x.y.z,t) = y_M^^9^ (3.118) 
 
 where [q ] = q(t - R/c,Cq,tiq,^q) and R and R* are as previously 
 detern-ined from equations (3.46) and (3.49). However, in the case v >c 
 
 21 
 
 The potential (p^ may be determined by a method similar to thai 
 
 presented in section 19 for a thin wing. See T. Karma' n , reference 34, 
 
 page 81. 
 
104 NACA TM 1399 
 
 the previous assertion on the uniqueness of the positive root of equation 
 (3.46) is not true. Solving equation (3.46) 
 
 f(R)-(x-v(t-f)-.,J, 
 
 yields 
 
 (y - r^y + (z - U^ - R^^ = (3.119) 
 
 ±R* - PC* 
 
 R = ^ R* = Vf*2 _ p2 (3.120) 
 
 V32 - 1 
 
 where 
 
 X - vt - &^ 
 
 V3 2 _ 1 Vr2 _ 
 
 3^-1 
 
 11 = y - ^0 
 
 ^ = z - ^0 (3.121) 
 
 where, as will soon "be shown, both roots of equation (3.120) are greater 
 than zero. From expression (3.120) for R* it is seen that ^ must 
 be greater than c^ so that the entire solution lies within the cone 
 
 ^*2 - p^ ^ 
 that is, 
 
 -^ = p^ (3.122) 
 
 p'^ - 1 
 
 The generators of this cone start from the point vt + ^q, tiq, ^q, at 
 which the source is located, and, as is seen from equation (3.122), are 
 inclined to the velocity v (to the axis E,) by the Mach angle e 
 
 sin e = £ (3.123) 
 
 V 
 
 With the possibility of a disturbance ahead of the excluded source, 
 a restriction to the region e < (fig. 22) is necessary. But -P4* 
 for C < is always greater than R . Therefore, R is positive and 
 both solutions (eq. (3.120)) are lagging ones. The physical meaning of 
 this double solution lies in the fact that at each point P (fig. 23) 
 enclosed within the Mach cone two sounds arrive. If at the instant con- 
 sidered the source occupies the position Q, then Q ' and Q" are two 
 
 00 
 
 A similar assumption was made in the theory of a thin wing when 
 
 fg for y > and fi for y < were neglected. 
 
NACA TM 1399 
 
 105 
 
 effective positions of the source from which the sound arrives at the 
 point P at the instant t. At suhsonic velocity there is only one 
 effective position. 
 
 The solution for a point source does not have significance in the 
 immediate neighborhood of the source (where it becomes infinite). From 
 the computations it is seen that at supersonic velocity of the source 
 there exists not only a. singular point but an entire surface (the Mach 
 cone) at which the solution becomes infinite. It follows that with 
 restriction to a point source, it is impossible to assign a mea,ning to 
 the solution (eq. (3.118)) not only near the source itself but also near 
 the Mach cone. However, use may be made of this solution for construct- 
 ing the field of a distributed source and also for a. qualitative analysis 
 of the phenomena for supersonic velocities. 'Assuming that q depends 
 harmonic a.lly on the time t, 
 
 q = qo(€o'iln'^n)^ 
 
 iCDt 
 
 (3.124) 
 
 a solution representing the field of an element of surface of the body 
 is obtained from equation (3.118) 
 
 - ' Ri^ ( r: 
 
 tiix>\t-—- 
 
 dP = 
 
 qodOQ 
 
 R 
 
 ^ 
 
 tou- 
 
 + e 
 
 (3.125) 
 
 where R-[_ and Rg are the two roots of equation (3.119). This solution 
 
 is valid within the Mach cone having its vertex at the point E^,^^,t,^- 
 
 The total field due to all the elements of the surface carrying out a 
 vibration with frequency oo will be 
 
 cp 
 
 qodaQ 
 
 iia\t 
 
 na 
 
 where the integral is extended over the region 
 
 + e 
 
 (3.126) 
 
 >P 
 
 ? = X 
 
 vt 
 
 ?o<o 
 
 2 2 
 
 p = (y - tiq) + (5 
 
 ^n) 
 
 0' 
 
 (3.127) 
 
 It is assumed that the radiating elements are disposed along rings from 
 ^ = -I to ?Q = so that in the cylindrical system of coordinates 
 
106 
 
 NACA TM 1399 
 
 and setting 
 yields 
 
 depend on (p. Bearing in mind that da.Q = Dq ' ^"^ ' dC 
 
 QqPq = aQ(CQ)/2jt, for oq ->■ 0, yield from equation (3.116) 
 
 (p = 
 
 "o(«o)«o 
 
 '^ 
 
 iojlt 
 
 + e 
 
 (3.128) 
 
 If the length of the wave is much greater than the dimensions of the 
 radiator (kZ = ooZ/c «l), the phase factors may be taken outside the 
 integral sign and this yields 
 
 v^ 
 
 1 
 
 i^(t-^j ixo(t--^j 
 
 + e 
 
 (3.129) 
 
 The last integral agrees with the integral considered by Karman in his 
 theory of the resistance of thin bodies at supersonic velocity and has 
 everywhere a finite value (ref. 34). It may be noted that at a distance 
 from the Mach cone where R* » 2 , the m^agnitude R* may be taken outside 
 the integral sign and there is then obtained a quite simple result 
 
 (p = 
 
 R 
 
 'V^ 
 
 + e 
 
 where 
 
 ^O^^O^^^O 
 
 (3.130) 
 
 (3.131) 
 
 The magnitude aQ(5Q) must be determined from the condition that the 
 
 derivative - Stp/Sp for o "*■ should be equal to the velocity of a 
 surface element carrying out vibrations with frequency oj. The method 
 for determining a (E, ) was given by Karman in the preceding mentioned 
 
 theory of the resistance of a thin body of revolution (ref. 34). 
 
 From the solution of equation (3.130) ^ it follows that surfaces of 
 constant amplitude will be the hyperboloids R* = constant, that is, 
 
 ?' 
 
 - 1 
 
 - = constant > 
 
 (3.132) 
 
 These hyperboloids are represented in figure 24. They asymptotically 
 touch the Mach cone. For subsonic velocity the surfaces of constant 
 amplitude are ellipsoids (see fig. 14) and for a stationary source they 
 
MCA TM 1399 10"^ 
 
 are spheres. At a point of space P', lying outside the Mach cone^ there 
 will in general "be no sound field and at each point P, lying within this 
 cone^ there will "be two fields originating from the two effective posi- 
 tions of the sound source Q' and Q" . With the assumption that the 
 conditions at which a Doppler effect occurs (see section 17) are sat- 
 isfied, the conclusion is drawn that at the point P there will "be 
 received two 'instantaneous' frequencies simultaneously 
 
 00' = 
 
 ^ =^ 1 - c dT (3.133) 
 
 In this case there thus occurs what might more properly be called not a 
 Doppler displacement of the frequency but a Doppler splitting. The fre- 
 quencies 00 ' and cd" are easily computed on the basis of formulas for 
 R-|_ and R^ (eq. (3.20)) 
 
 00' = 00 
 
 <^" = "^ — ^ (3.134) 
 
 P - 1 
 
 In particular, on the x-axis (p = O) is obtained 
 
 
 - 1 
 
 r - 
 
 ■ 1 
 + 1 
 
 /- 
 
 1 
 
 CD' 
 
 OJ 
 
 00 
 
 a:i 
 
 P + 1 
 
 (3.135) 
 
 From this it is seen that if 1 < p < 2, then |oo'| > oo and |a3"| < ca; 
 but if p > 2, then both frequencies are less than oo, that is, in this 
 case lowered tones (as compared with the initial (oj) ) are heard. 
 
 21. Sound Field of a Sound Source for Supersonic "Velocity of Motion 
 
 I 
 
 ■ In the preceding section a sound source of infinitely small cross 
 
 section moving uniformly with supersonic velocity was considered. With 
 the assumption of a source of this shape, the entire problem was con- 
 sidered linearly; the state of the medium in this extreme idealized 
 case was represented as a simple superposition of states, one of which 
 
108 NACA TM 1399 
 
 was determined by the motion of the body (solution of Karman) and the 
 other by the vibrations of its surface (radiation of sound). For finite 
 dimensions of the cross section of the body, such simple superposition 
 does not take place. The trans lational motion of a body of finite sec- 
 tion produces in the medium considerable changes in the density, pressure 
 and temperature and leads to the formation of density jumps (shock waves) 
 of finite magnitude. 
 
 Because of the difference in the compression of the stream about a 
 body at various points of the body, the velocity of sound c is not the 
 same at all points. As a result the Mach angle, too, e = arc sin c/v, 
 is different for different points of space about the body. The surfaces 
 of discontinuity (of the density jumps) do not, for this reason, have 
 the shape of a cone and only at a distance from the body do they possess 
 this simple shape. In figures 25 and 26 are shown shock waves arising 
 during the motion of artillery projectiles of various shapes obtained by 
 the schlieren method. ^^ Thus, the state of the medium near the body 
 itself is very complicated, and, as has already been mentioned, the 
 solutions of the hydrodynamic equations for this case are at present 
 unknown.^"* At a. great distance from the body, however, the situation 
 is simpler. It may be imagined, at least for explaining the geometric 
 and kinematic aspects of the matter, that the disturbance at some dis- 
 tance from the body is the result of the compounding of disturbances 
 propagated with the velocity of sound from each point of the surface of 
 the body. In this the differences in the velocity of sound propagation 
 near the body must be unavoidably ignored, and therefore the assumed 
 point of view essentially ignores the finite dimensions of the body so 
 that a point source of sound would have to be considered. This, however, 
 leads to an infinitely large magnitude of the shock wave on the Mach cone. 
 Hence, the theory of a point source may be applied to the problem con- 
 sidered, restricting it, however, to a consideration of the kinematic 
 side of the phenomenon. Such problems, for example, as the magnitude of 
 the shock wave and its change with distance from the body cannot be con- 
 sidered from such a point of view. With these reservations the preceding 
 theory (section 20) of the point source may be applied to the problem of 
 the radiation of a sound source moving in an arbitrary manner . According 
 to equation (3.48) the field of a, harmonic point source is determined by 
 the potential 
 
 V k 
 
 23 
 
 On photographing by the schlieren method see reference 38. 
 
 Except for the case of the flow about a cone (see Ackeret, ref. 2) 
 
NACA TM 1399 109 
 
 where the magnitudes R (in the general case their number also may be 
 greater than two) are determined as the positive roots of the equation 
 
 .(H).{x-x(t-f)}%[-.(t-|)} 
 
 2 
 
 + 
 
 R^ = (3.137) 
 
 25 
 and the magnitudes RT are determined by the equation 
 
 
 I* 
 
 ^=\ 
 
 (3.138) 
 
 It is evident that the surfaces 
 
 R* = (3.139) 
 
 'k 
 
 are no other than the surfaces representing the envelopes of the elemen- 
 tary disturbances propagated from the sound source. This makes possible 
 a simple geometrical construction of the surfaces R^ = 0^ which are the 
 surfaces of the potential discontinuity^ that is, of the shock-wave sur- 
 faces (strictly speaking, there is no justification in considering the 
 state in the immediate neighborhood of these surfaces since on these 
 surfaces (p = » ) . 
 
 For this purpose it is necessary to construct a family of spheres 
 representing the fronts of the spherical waves issuing from the source 
 at different instants of time and to draw the envelope of these spheres. 
 In figure 27 this construction is shown for a. uniformly moving sound 
 source: (a) for subsonic velocity (in which case there is no envelope) 
 and (b) for supersonic velocity. In this case the envelope is a Mach 
 cone with vertex at the location of the source. From the latter con- 
 struction there is clearly seen also the twofold character of the field 
 for V > c: at the point P, at the instant of time assumed on the 
 drawing, a disturbance arrives from the two points Q' and Q" (behind 
 and ahead of P) . 
 
 Density jumps are also frequently called shock waves or ballistic 
 waves . 
 
 The suitability of these terms will be understood if the density 
 jump is considered relative to a stationary observer or, in general, a, 
 sound (pressure) receiver. The density jump, moving together with the 
 
 Here R* differs from the preceding factor. 
 
110 
 
 NACA TM 1399 
 
 sound source, on passing by the sound receiver leads to a discontinuous 
 increase of the pressure in the receiver with smoother pressure changes 
 behind the jump. In figure 28 is shown the pressure in the ballistic B 
 and nozzle wave N from a 305 millimeter shell according to a recording 
 by E. Eksklagon. It is thus seen that the pressure in the receiver will 
 have the character of an impulse or shock. 
 
 The problem of finding the envelope of the elementary disturbances 
 may thus be considered as the problem of finding the front of the bal- 
 listic or shock wave. A rational analytical solution of this problem will 
 also be presented. Equation (3.137) in the general case is transcendental 
 with respect to R so that its direct solution may be very difficult. 
 It is expedient to try to obtain the curves of the intersections of the 
 wave-front surfaces with some coordinate plane in parametric form with 
 R as a parameter. Equation (3.137) is quadratic with respect to x, y 
 and z while equation (3.138) is always linear with respect to the same 
 variables. By taking any section of the required surface with a plane, 
 
 for example 
 
 z = z ' 
 
 y 
 
 can be expressed by equation (3.138) as a func- 
 
 tion of X, z', and R; substituting in equation (3.137) yields a quad- 
 ratic equation for x. Its roots will be 
 
 X = Xi(R,z') 
 X = X2(R,z') 
 Substituting these values in equation (3.137) gives 
 
 y=Yi(z',R) 
 y=Y2(z',R) 
 
 (3.140) 
 
 (3.140') 
 
 The equations x = X-j^, y = Y-, , and x = Xg , y = Yg give the curve of 
 
 intersection of the shock-wave front by the plane z = z ' in parametric 
 form. 
 
 If the source originated at the instant of time t = 0, the values 
 of the parameter R for the instant t lie in the interval 
 
 ^ R <: c • t 
 
 The instant of occurrence is the origin of a special disturbance source 
 which on being propagated from the point of origin (X(0),.Y(0), Z(0)) in 
 the form of a spherical wave gives an additional sperical wave front 
 which, generally speaking, is not a surface of discontinuity. At sub- 
 sonic velocity of motion of the source, the wave front is entirely 
 formed by this sphere having its center at the point of origin of the 
 source (fig. 27(a)). An example of such a wave is that occurring at 
 the instant of discharge of a missile from the barrel of a gun (discharge 
 sound) . 
 
NACA TM 1399 111 
 
 This wave is called a "muzzle wave" in contrast with the ballistic 
 wave which moves together with the missile. Figure 29 shows a sketch of 
 the muzzle and ballistic waves for a uniformly accelerated motion of a 
 disturbance source starting at the' point at instant t = 0. From 
 the diagrams shown for t = 1, 3^ 4, and 8, the reader can follow the 
 developm.ent of the ballistic wave which; in this example, overtakes the 
 m.uzzle wave. The cross-hatched regions are those in which there are two 
 effective positions of the sound source and at which, therefore, there 
 should be heard the superposition of two sound fields, one starting from 
 the side of the sound source, the other from the opposite side (see 
 positions Q." , Q', Q, and P .in fig. 23). 
 
 In figures 30 and 31 are shown the development of a, ballistic wave 
 of an artillery missile for a, uniformly retarded (fig. 30) and uniformly 
 accelerated (fig. 3l) motion. ^° Figure 32 shews a ballistic (G) and a 
 muzzle (m) wave for an accelerated curvilinear motion of a disturbance 
 source . 
 
 Of? 
 
 Figures 30, 31, and 32 were taken from the paper of L. Prandtl 
 (ref. 39); figure 28 from the book by E. Eksklagon (ref. 40). 
 
11? 
 
 NACA TM 1399 
 
 IS 
 
 CHAPTER IV 
 
 EXCITATION OF SOUND BY A FLOW 
 
 22. General Data on Vortical Sound and Vortex Formation 
 
 The most common reason for the occurrence of sound in a medium x^ 
 the periodic motion of bodies immersed in the medium and having a suf- 
 ficiently high frequency, for example, the vibration of the prongs of a 
 tuning fork, the rotational motion of the blades of an airplane or ship 
 propeller, and so forth. The occurrence of sound is not restricted, 
 however, to only such cases as these. Sound also arises when there is 
 a constant flow about stationary solid bodies (or, what amounts to the 
 same thing, in the motion of bodies with constant velocity) when it 
 would appear that there were no causes that give rise to periodic 
 phenomena. An example of this type of sound formation is provided by 
 the whistle of the tension rods of airplanes, the rigging of ships, the 
 sound of wires and strings ("aeolian harp"), and the whistling in the flow 
 about angles, slots, and so on. It is important in this connection to 
 note that the capacity of the string, for example, for vibrating plays 
 a secondary part because these sounds occur also in the flow about non- 
 yielding solid bodies. The initial causes for the occurrence of sound 
 in these cases are not connected with the vibrations of bodies but with 
 the phenomena of vortex formations in the flow of a fluid about bodies. 
 The corresponding sound is therefore called a vortical sound. The two- 
 fold origin of the sound of an airplane propeller 'has already been 
 pointed out. On the one hand, the sound of the propeller is caused by 
 the periodic motion of the blades (rotational sound); and, on the other 
 hand, a flow takes place about the propeller blades which leads to vor- 
 tex formation and also to the occurrence of a particular vortical sound. 
 The fundamental laws of vortex formation in the flow about bodies are 
 now considered in more detail. However small the viscosity of the medi- 
 um about the body may be, the very existence of the frictional forces 
 produced by it leads to the formation of vortices in the initially po- 
 tential flow. In order to clarify this aspect of the problem, use will 
 be made of the equation of motion of a viscous, imcompressible fluid 
 (the_^equation of Navier-Stokes) . According to equation (l.l3), setting 
 div V = gives 
 
 Bv ,-> x-*- ^ 
 
 ft "- ("'^)^J 
 
 (v,v)v = - VP + mAv div V = (4.1) 
 
 The flow about a body of characteristic dimension d is now considered, 
 and the velocity of the approaching flow is denoted by v. In place of' 
 X, y, z, and t, the nondimensional variables x' = x/d, y' = y/d. 
 
NACA TM 1399 113 
 
 z' = z/d, and x' = tv/d are used; and p = p'pv and v = v •~v' are 
 set up. Equation (4.l) then reduces to the nondimensional form 
 
 TT; + (^'> V') V' = - V'P' + ^ A'v' (4.1') 
 
 ot Re 
 
 where Re is the Reynolds number 
 
 Re = ^ (4.2) 
 
 u 
 
 (u = li/p^ the kinematic viscosity). 
 
 From the hydrodynamic equation thus reduced to the nondimensional 
 form, it is seen that at large Reynolds numbers Re the last term in 
 equation (4.1) maybe rejected; and, therefore, in this case the viscous 
 stresses play a vanishingly small part in comparison with the effects 
 arising from the inertia of the fluid. The equations of motion of an 
 ideal fluid are thus obtained. Hence, if the approaching flow were 
 potential, it would have to remain so. This conclusion, however, is 
 true only at a large distance from the body and is not true in the im- 
 mediate neighborhood of and behind the body. The velocity v on the 
 surface of the body itself is, because of the adherence of the fluid, 
 equal to zero. Far from the surface it assumes a value close to that 
 of the approaching flow (v' = l). This change of velocity occurs in a 
 thin layer which is called the boundary layer^"^. The thickness of this 
 layer 5 may be estimated from the fact that in this layer the action 
 of the viscous stresses is comparable with the effect produced by the 
 inertia. This means that in this layer the last term in equation (4.l) 
 is comparable with the remaining terms . These latter terms are of the 
 order of 1. Since in the boundary layer the velocity over its thickness 
 5 varies from to 1, the magnitude of the derivative S^v/c^n^, where 
 n is the normal to the surface of the body, will be of the order of 
 v/b^; and in nondimensional form the magnitude Av = ci^v/^n + S^v/Bs 
 (s is the tangential length) will be 
 
 A'v' = (Z 2/52)^2^1/^^^.2 ^ ^2^,/^g.2 ^ ^2/52 (instead of A'v' = I'^/l'^ = 1 
 outside the boundary layer). From this it is concluded that in the 
 boundary layer l/Re • I'^/b'^ ^ 1; that is, 
 
 7. 
 
 S = 
 
 Re (4.3) 
 
 In this thin layer the flow may be considered as corresponding to the 
 potential flow of an ideal fluid. The existence of a boundary layer, 
 however thin it may be (large Re, small viscosity), leads to essential 
 changes in the flow behind the body. In figure 33(a) is represented the 
 potential flow about a cylinder and in figure 33(b) the flow as it is 
 
 27 
 
 For details on the boundary layer and vorticity, see ref- 
 erences 41 and 42. 
 
1 11 NACA 'I'M 1399 
 
 av'tually obtained. The tliin boundary layer b'b" becomes unstable at ' 
 the point b" and gives rise to vortices shed from the body. The ' 
 formation of these vortices may be explained in the following manner: 
 Tlie stream line a'a"a'" of the potential flow near the surface of the 
 body is considered. In the region a 'a" the stream moves with accelera- 
 tion, and the pressure at a" drops, as follows directly from the law 
 of Bernoulli 
 
 p +T = constant (^_4) 
 
 because of the narrowing of the stream in the region a" . On the other 
 hand, in the region a"a"' the stream moves against an increasing pres- 
 sure and is consequently retarded. In the ideal case of an absolutely 
 nonviscous fluid, the particles of the fluid successfully overcome this 
 rise in pressure, converting the stored-up kinetic energy into potential. 
 In the presence of friction, however, part of the kinetic energy is 
 spent in overcoming the forces of friction, and the store of kinetic 
 energy of the particles is now insufficient for overcoming the increas- 
 ing pressure. As a result, a reverse flow arises in this region. The 
 point of occurrence of this flow b" is called the point of separation 
 of the boundary layer. The picture of the flows that arise here is 
 represented in more detail in figure 34. This reverse flow forms a 
 vortex which gradually increases, approximately up to the dimensions 
 of the body, and which finally breaks away from the body (fig. 33). The 
 same also takes place at the lower point of separation. The development 
 of the vortex on one side, however, hinders the development on the other. 
 Hence, the development of the vortices and their separation occurs al- 
 ternately, now one side, now on the other side of the body. The separat- 
 ing vortices form behind the body a double chain of vortices which are 
 gradually dissipated. This double chain of vortices is termed a Karman 
 vortex street. The theory of this concept will be discussed subsequently. 
 For the present, it is merely pointed out that so far no mathematical 
 computation of the periodic separation of the vortices has been obtained. 
 By numerical methods, Boltze (ref . 43) has succeeded in showing mathe- 
 matically the development of a vortex behind the point of separation. 
 Figures 35 and 36 are photographs of a developing vortex in the flow 
 about a cylinder and also a Karman street formed behind the cylinder at 
 Re = 250. Although the frequency of the separation of vortices cannot 
 as yet be computed mathematically, important conclusions can nevertheless 
 be derived from dim.ensional considerations. From the magnitudes charac- 
 terizing the flow about the body, v, the flow velocity; d, the dimen- 
 sion of the body; and u , the kinematic viscosity, two magnitudes f 
 and f ' can be formed having the dimensions of frequency 
 
 f= x(Re) I (4.5) 
 
 f = x'(Re) ^ (4.5') 
 
 d2 
 
NACA TM 1399 115 
 
 where x and x' are nondimensional coefficients depending on the 
 Reynolds number. The first of the frequencies is a characteristic fre- 
 quency of the possible periodic motions of the fluid at large values of 
 the Reynolds number when the effect of the inertia of the fluid predom- 
 inates; the second, on the contrary, is of significance in the case of 
 predominant viscosity (small Re). Vortices arise only at large Reynolds 
 numbers and therefore it may be expected that the frequency of separation 
 of the vortices should be determined by equation (4.5). It may appear 
 strange that the frequency of the vortices arising exclusively from the 
 viscosity of the fluid is determined iDy equation (4.5) and not by equation 
 (4.5'). This paradoxical character is, however, only an apparent one. 
 If it is desired to make use of equation (4.5') for determining the fre- 
 quency of separation of the vortices, then the magnitude d would have 
 to denote not the dimension of the body but the thickness of the boundary 
 layer 5. When 8 is substituted from equation (4.3) into equation 
 (4.5'), a result agreeing with equation (4.5) is obtained for f ' . It 
 may be remarked that 'equations (4.5) and (4.5') differ, of course, only 
 in that x and x' in both cases depend little on Re. This phenome- 
 non is, in fact, observed in actual cases. The periodic separation of 
 the vortices with frequency (equation (4.5)) gives rise to periodic im- 
 pulses of small compressions and rarefactions which are propagated at a 
 distance from the body in the form of a sound wave the frequency of 
 which agrees with f . This is the wave which is denoted as the vortical 
 sound. The frequency of the vortical sound was first investigated by 
 Strouhal for a vibrating string in an air flow (aeolian harp) . From 
 his tests, Strouhal derived precisely equation (4.5) with x(Re) = 0.185.^^ 
 The value of the Strouhal coefficient depends on the shape of the body, 
 on the choice of the characteristic dimension d, and not much (in a 
 certain interval of Reynolds numbers) on the Reynolds number. For a 
 sphere or cylinder, d denotes the diameter. For a plate having width 
 I and thickness b at angle of attack a to the flow, 
 d = Z sin a + b cos a. 
 
 For such determination of d, from test data (refs. 33 and 45) 
 For a cylinder: x= 0.20 in the range 10^^ Re < 3- 10"^ 
 
 For a plate: x = 0.165 to 0.180 for 10^ < R"e < iTaxio^ (at angles of attack 
 20° to 90°) . 
 
 The values of x obtained by different authors differ little from 
 one another. A more detailed investigation of the spectrum of vortical 
 sound (ref. 45) shows that the equation of Strouhal (equation (4.5)) must 
 be generalized in order to take into account the upper harmonics of the 
 
 ^%his derivation of Strouhal was disputed on the basis that the in- 
 vestigated string is itself capable of vibrations. However, later inves- 
 tigations (see, for example, refs. 45 and 46) confirmed the equations of 
 Strouhal for rigid bodies where the vibrations are due exclusively to the 
 vortices . 
 
116 NACA TM 1399 
 
 fundamental frequency 
 
 fn ^^(Re) I n n = 1, 2, 3, • • • (4.6) | 
 
 These harmonics, although weakly expressed, are nevertheless observed^^ 
 ■(fig. 37) . The intensity of the Strouhal sound has been investigated 
 in considerably less detail. According to the observations of W. Holle 
 (ref. 46) for the flow about thin cylinders (diameter d, length I), the 
 intensity of the vortex sound at the distance r from the radiator is 
 equal to 
 
 I=a-^,db (4.7) 
 
 r'^ 
 
 where Holle assumes for n the value 7 ( in general, according to his 
 tests, 6 < n < 8) and a = S-IO"^"^ in centimeter-gram- second units. 
 According to the observations of Yudin (ref. 47) on the Intensity of the 
 sound of a blower, n = 6 is obtained. The same result for n has been 
 arrived at by Y. M. Sukharevskii from observations of the vortex sound 
 In a wind tunnel (unpublished computation of the Physics Institute of 
 the Soviet Academy of Sciences). Nepomnyashchii (ref. 33) from measure- 
 ments on the vortical sound of a propeller (fig. 38) arrived at the 
 value n = 4, which corresponds to the theoretical curve 40 log v (fig. 
 38). In fact, at least for large angles of attack (a > 20°), the curve 
 60 log V corresponding to n = 6 better corresponds with the results 
 of his measurements. With regard to the directional characteristic of 
 the vortex sound, the observations of Yudin (ref. 47) show that it agrees 
 with the direction characteristic of a dipole the axis of which Is per- 
 pendicular to the direction of flow about the body (e.g., a propeller 
 radiates a vortical sound primarily in the direction of its axis sym- 
 metrical in front and behind) . In figure 39 is shown the directional 
 characteristic for the vortical sound of a propeller. The theoretical 
 explanation of these laws will be given subsequently. For the present, 
 the fact is noted that the high degree of dependence of the intensity 
 of the vortex sound on the velocity of the flow (n $, 6) has often ap- 
 peared paradoxical because from the dimensions of the magnitudes it was 
 assumed that, since the intensity of the sound is proportional to the 
 square of the pressure and pv /Z is a measure of the pressure, n 
 should be equal to 4. The error of this reasoning is based on the fact 
 that the magnitude pv /2 is a measure of the pressure only in an 
 
 For large Reynolds numbers (Re > 10 ) , the expressed vortical fre- 
 quency evidently does not, in general, exist. The spectrum of the vorti- 
 cal sound becomes practically continuous and the Strouhal frequency 
 (equation (4.5)) becomes only a suitable measure for the frequencies rep- 
 resented in such a spectrum. 
 
 Figure 37 is taken from the article by Holle (ref. 46). 
 
NACA TM 1399 
 
 117 
 
 incompressible fluid. In the wave region far from the body, this pres- 
 sure, which decreases in inverse proportion to the square of the dis- 
 tance, is practically equal to zero; but the important part of the pres- 
 sure, which decreases in inverse proportion to the first power of the 
 distance, is entirely connected with the presence of compressibility of 
 the gas or liquid. In general, from considerations of dimensionality, 
 it can be concluded that n is greater than 4. In fact, the density 
 of the flow of sound energy is equal to 
 
 (4.8) 
 
 1 = ^ 
 
 pc 
 
 where Jt is the pressure in the wave, p 
 c the velocity of sound. 
 
 the density of the medium, and 
 
 If n is measured in units of pv^/2, then 
 
 ..A 
 
 pv^ 
 
 (4.9) 
 
 where the nondimensional coefficient x ma-y depend on the Reynolds num- 
 ber; the Mach number v/c; the ratio l/r, where I is some dimension 
 of the body; and on the observation angles 0, (p. Since at large r, 
 on account of the law of conservation of energy, I must be inversely 
 proportional to the square of the distance, the following expression 
 applies for r->- «: 
 
 2 
 In place of Z , it is possible, of course, to take the product Id. 
 
 Further, in the absence of compressibility (c-^^), x' = (since the 
 
 sound in the absence of compressibility is not radiated) . Hence, x' 
 
 must be proportional to a certain positive power of the Mach number 
 
 (v/c) . In this way there is obtained 
 
 I = a . M . PI^ • fiX (*-ll) 
 
 •©' 
 
 where a depends on Re and on the angles 9, ijr, and s > 0. The depend- 
 ence of a. on the Reynolds number in the range where the resistance 
 of the body depends little on Re must be weak so that physically a 
 depends only on the angles and determines the directional characteristics 
 of the sound. 
 
 Additional considerations permit determining also the least value 
 of s. The sound source may be assumed as a zero-order source (a does 
 not depend on 9, cp ) , a dipole (a = a' cos^Q, where the direction of the 
 dipole axis cannot be indicated, and 0=0), and so forth. It will 
 now be shown that the zero- order source should be excluded. In fact. 
 
118 KACA TM 1399 
 
 the strength of a. zero-order source Q is equal to the volume velocity 
 
 vdS (4.12) 
 
 ■I 
 
 n 
 
 where the integral is taken over the surface S about the source near 
 the latter itself, and v^^ is the normal component of the velocity of 
 the fluid to the chosen surface S. In this region, the fluid may be 
 considered as imcompressible since the wave length X of the vortex 
 sound 
 
 \ c 1 c J , , 
 
 X = f=--ci (4.13) 
 
 (for v< c,x< l) is always much greater than the body. But for an in- 
 compressible fluid the flow through a closed surface is equal to zero. 
 Hence, Q = O.^l 
 
 A diople radiation may thus be expected.^ Since its strength is 
 is also proportional to the velocity of flow v while the intensity of 
 the radiation of a dipole source is proportional to the square of its 
 strength (i.e., v^) and to the fourth power of the frequency (f^ ~ v ), 
 the result that the exponent s in equation (4.1l) must be equal to 2 
 
 is obtained; that is. 
 
 I=a'(Re) • cos^a "^ '^ (,.ll) 
 
 The magnitude of the coefficient a' cannot, of course, be determined 
 from dimensional considerations. With regard to the direction of the 
 dipole axis (S = O), it should, at least for symmetrical bodies, be sur- 
 mised, on the basis of the symmetrical succession of the separation of 
 the vortices from the upper line of separation and from the lower, that 
 the axis of the dipole is directed along a line perpendicular to the 
 flow (see section 25). 
 
 31 
 
 A possible objection to this conclusion is that for the radiation of 
 
 sound the compressibility is essential, and that taking it into account 
 
 gives Q ^ 0. But taking the compressibility into account means exuan- 
 
 sion in powers of v^/c ; hence, Q would be proportional to v . v^/c^ 
 
 and the square of Q proportional to v^v'^/c'*. Since the intensity of 
 
 a zero-order source is further proportional to f^ ~v^, there would be 
 
 obtained I~ v^, i.e., the succeeding term after equation (4.1l) in the 
 
 expansion in powers of v/c. 
 
 32 
 
 The same conclusion was arrived at by E. Y. Yudin on the basis of his 
 
 measurements of the directional characteristics of the sound of blowers. 
 
NACA TM 1399 119 
 
 23. Theory of the Karman Vortex Street - Computation of 
 
 the Frequency of Vortex Formation 
 
 Karman and Rubach (refs. 48 and 49) succeeded in constructing a 
 theory of a double chain of vortices representing an idealization of the 
 vortex street which actually arises behind bodies moving in a fluid 
 (see fig. 36) . 
 
 The Karman-Rubach theory refers to the flow about infinite cylinders 
 and plates so that only two-dimensional flow is considered in a plane 
 (x^y) coinciding with the plane of the cross section of the body. 
 Along an axis parallel to the generator of the cylinder or plate, the 
 flow is assumed unchanged. 
 
 The two-dimensional flow of an ideal incompressible fluid may, as 
 is known, be described by a complex velocity potential $(z), z = x + iy. 
 The components of the velocity v along the ox and oy axes, v 
 and V , are computed from this potential by the formula 
 
 ^x-i^y=-i (4-12) 
 
 The component along the third axis, however, in view of the assumption 
 of the two-dimensional character of the flow, is equal to zero. If $ 
 is known, the pressure p can also be found. If § depends on the 
 time, p is computed from the generalized Bernoulli equation 
 
 
 ^o-pH^^M-ltl 
 
 d*|2 2 2 2 
 
 — i = V + V = V 
 
 dz X y 
 
 (4.13) 
 
 In this expression, R(S'i'/St) denotes the real part of ^^/^t. 
 The complex potential of one vortex filament at the point z = z, will be 
 
 'Z'Z 
 
 If #(z,t) = (p(x,y,t) + it(x,y,t) is an analytic function of z, then 
 <p and ^ satisfy the equations of Laplace, Acp = 0, and Ai|r = 0, where 
 the derivatives of (p and ^ are subject to the Cauchy-Riemann conditions: 
 Scp/^x = S\l;/By, and S<p/Sy = - Sxjr/Sx. If <p is taken as the velocity poten- 
 tial, the equation \|f = constant will give the streamlines orthogonal to 
 the surfaces <P = constant. The velocities vx, Vy are -bcp/bx -and -S(p/Sy, 
 Because of the conditions of Cauct;y-Riemann, 
 
 -d#/dz = -S<p/Sx -iB\|//Sx = -S(p/Sx + iB(p/Sy = v^^. - iv 
 
 which gives equation (4.12). Equation (4.13), if $ is expressed in 
 terms of cp^ reads: p = Pq + pBcp/St -pv2/2, which agrees with equation 
 (1.29). For details on the complex potential, see any textbook on 
 hydrodynamics . 
 
i::o 
 
 *k(^) = 2k • ^ 
 
 n{: 
 
 \) 
 
 NACA TM 1399 
 
 (4.14) 
 
 where T is the circulation of the velocity and I any length. The 
 simple computation of the velocities v^ and Vy from this potential 
 gives the flo w about the point z = zj^ shown in figure 40. The velocity 
 
 V2 2 
 v^ + V changes with the distance r from the axis of the cylin- 
 drical vortex according to the law 
 
 = r/2 
 
 itr 
 
 '^' 
 
 similar to the change in the magnetic field about an infinitely long 
 cylindrical wire. If there are several vortices located at different 
 points z-i_(x-L_,y-|_), zgCxg^yg), • ' -, Zj^(xk^yk)^ "the total potential # is 
 obtained by summing over-all $j^. The potential <i>' of an infinite seriea 
 of vortices having the circulation r' and located at the distance I 
 from each other will now be considered. In figure 41 are shown two such 
 series of vortices. The vortices of the upper series are located at 
 the points z'y^ = x'j^ + iy'j^, where x'k = Ik, (k = 0, +1, ±2, • • •) and 
 y'k = h/2. Since the potential $ is determined only with an accuracy 
 up to a constant, it is possible in equation (4.14) under the logarithmic 
 sign to divide by any number so that the sum ^ may be written in the 
 form 
 
 r J (2; - zn)rt 
 §'(z) = ^<ln ° 
 
 2jri 
 
 In 
 
 In 
 
 (^) 
 
 W)]) - h- 
 
 k=l 
 
 (z 
 
 ZAjIT 
 
 (4.15) 
 
 and, because of the known representation of the function sin(Trz) 
 
 sm rtz = JTZ 
 
 n^*) 
 
 (4.16) 
 
 k=l 
 
NACA TM 1399 
 
 121 
 
 there is obtained 
 
 <!>' = r In sin — (z - zA) 
 
 2jti I ^ 0' 
 
 (4.17) 
 
 In the same way^ the following expression is obtained for the second 
 series of vortices: 
 
 *" = il ^ =" f (^ - ''o' 
 
 (4.18) 
 
 The velocity of motion of the vortex chains will now be determined. It 
 is readily seen that one chain cannot move. In fact, all the vortices 
 are under the same conditions and for computing the velocity of motion 
 of the chain it is sufficient to compute the velocity of any of the 
 vortices. The latter is equal to the velocity resulting from all vor- 
 tices except the one considered since one vortex filament does not by 
 itself give a forward velocity. In view of the symmetry, however, it 
 is evident that the vortices situated on the right and left sides of 
 the vortex under consideration impart to it equal and opposite velocities 
 (this is easily verified by the equation v = -d#'/dz, if the potential 
 of the vortex under consideration is subtracted from equation (4.17)). 
 In the presence of two chains, conditions are different. In this case 
 the total potential is given by 
 
 $ = #' +<!." = ^ Jji sin T (z - Zfi) + 
 
 27ri 
 
 isi sin I (z - z;;) 
 
 (4.19) 
 
 and the complex velocity will be 
 
 VX - iVy = 
 
 d* V ^ rt / 
 
 XT = " ■=377 cot y v: 
 dz 2ni i 
 
 ^6) 
 
 -I- cotil (z - z") 
 2rti I 
 
 (4.20) 
 
 The vortices of each of the chains will move in the identical 
 fashion; but since the chain itself does not move, the first chain will 
 be displaced only under the effect of the second, while the second only 
 under the effect of the first. The velocity of the first chain is 
 therefore obtained if the velocity produced by the second chain is com- 
 puted at the point where some vortex of the first is located (for example, 
 z = Zq). The velocity of the first chain will thus be 
 
 r" 
 
 V 
 
 X 
 
 iV = 
 
 y 
 
 2jTi 
 
 cot 
 
 (: 
 
 '0 
 
 ^o) 
 
 and of the second 
 
 (4.21) 
 
 V" - iV" = + r^ cot ^ (z' 
 X y 2jti I ^ 
 
 ^5' 
 
 (4.22) 
 
""■^^ NACA TM 1399 
 
 In order that the chains move without change of the relative configura- 
 tion, it is necessary that 
 
 v- - iv;= v;;- iv; (4.23) 
 
 that is, r" = -r'. If it is desired that the direction of the circula- 
 tion correspond with that shown in figure 41, it is necessary to take 
 r 7 r > 0. Assuming further that the chains move parallel to themselves 
 It IS required that V^ = V^ = 0. This condition permits determining 
 the magnitude of the shift of the vortices of one chain relative to the 
 vortices of the other. When this shift is denoted by b and the dis- 
 tance between the chains by h (see fig. 41 ), the expression 
 z5 - Zq = b + ih is obtained. In order to determine b, it is necessary 
 to equate V' to zero, that is, to the imaginary part of equation (4.21) 
 or equation (4.22). For this it is necessary to make use of the equati 
 
 on 
 
 cot (X + iY) = ^^" 2X sinh 2Y 
 
 cosh 2Y - cos 2X " cosh 2Y - cos 2X ^^'24) 
 where 
 
 cosh C = (e? . e-^)/2, sinh C = (e^ - e-?)/2, tanh ? = sinh ?/cosh ? 
 After si:iiple reductions there is obtained from the condition V^ = V" = 
 
 . 2itb 
 "^" -y- = (4.25) 
 
 that is, b = or b = 2/2. In the first case the vortices of the t^o 
 series are one above the other; in the second case they are arranged In 
 
 It .T f .f '^°™ '" ^'^^"^ "I- K^'^^^- ^nd Rubach (re?! Is) hfve 
 shown that the first arrangement (b = 0) is not stable whin Z^^v, . 
 one fb - T /?) TQ c-t-a-Ki ^ -^ ^y lb not staoie, while the second 
 one ^D - l/^) IS stable for a wide class of disturbances if 
 
 cosh /l^^ - /^ 
 
 [ij - "^^ (4.25') 
 
 that is. 
 
 h 
 
 = 0.28 
 
 2 
 
 ^ ^J"^.^^^^ "^^ ^^^ ''^^^° '"/^ ^^ determined. The obtained picture 
 of the disposition and motion of the vortices very nearly corresponds 
 with what IS observed in tests on the flow about cylinders and plates 
 [see fig 36). In particular, experiment confirms the value of the 
 ratio h/2 given here. 
 
 When the real part V^ of the complex velocity v; - iV" (eq. (4 2l)) 
 IS computed for b = 2/2, the velocity of motion of the vortex street 
 
 is obtained 
 
 x 22 ^^"^ [^ — (4.26 
 
NACA TM 1399 
 
 123 
 
 With the aid of the law of the conservation of momentum applied to the 
 approaching flowj the body, and the vortex street behind the body, Karman 
 and Rubach (ref . 48; see also ref . 49^ the hydromechanics book by Kochin, 
 Kibel, and Roze) succeeded in establishing a relation between the coef- 
 ficient of head resistance of a body C^ and the ratios l/d and u/v, 
 where d is the diameter of the cylinder or the width of the plate and 
 V is the velocity of the body. At the same time, they identify the 
 street arising behind the body with the infinite vortex street just con- 
 sidered (fig. 42). Very good agreement is obtained with test results 
 as illustrated in the following table : 
 
 Body 
 
 u/v 
 
 Z/d 
 
 h/z 
 
 Cw 
 
 Theory 
 
 Experiment 
 
 Theory 
 
 Experiment 
 
 Cylinder 
 Plate 
 
 0.14 
 .20 
 
 4.3 
 5.6 
 
 0.28 
 .28 
 
 0.28 
 .30 
 
 0.91 
 
 1.61 
 
 0.90 
 1.56 
 
 The determination from this table of the ratio u/v permits also com- 
 puting the Strouhal coefficient x in equation (4.5) for the frequency 
 of the vortex sound for the cylinder and the plate. Thus, in a system 
 of coordinates in which the body is at rest, the vortex street moves 
 with a velocity, equal in absolute value to (v - u), in a direction 
 opposite to the motion of the body (fig. 42) . When the vortex street 
 is displaced by I, the entire picture of the vortex motion repeats 
 itself. Hence the period of the motion is T = Z/(v - u) and the 
 (fundamental) frequency will be f = (v - u)/Z. After each time inter- 
 val T, there occurs behind the body a new completely developed vortex 
 pair. Since it is by these vortices that the vortex sound is generated, 
 the frequency of the vortex sound should be equal to 
 
 /i - ^U ^ 
 
 whence 
 
 -(-^v)f 
 
 (4.27) 
 
 (4.28) 
 
 When the values u/v and Z/d are substituted from the table given 
 previously, x= 0.20 (for the cylinder) and x= 0.14 (for the plate) 
 are obtained, which are in very good agreement with the experimental 
 data previously given. In this way the theory of Karman and Rubach 
 connects the computation of the head resistance of a body with the com- 
 putation of the formation of vortices arising behind the body. 
 
 24. Pseudosound. Conditions of Radiation of Sound by a Flow 
 
 In practice it is often necessary to deal with a sound receiver 
 under conditions where the receiver is immersed in an unsteady flow. 
 
124 ^ jjACA TM 1399 
 
 that is, in a flow the pressure and velocity of which vary not only in 
 space but in time. Examples of such flows are a wind which constitutes 
 a turbulent flow possessing a certain mean velocity, the stream of water 
 in a ship's wake separating behind a ship or from some projecting part 
 of its body, and so forth. An idealization of such wake is the Karman 
 vortex street which moves with the velocity u = F /S -yjzi so that the 
 pressure and velocity of the flow at each point vary in time with the 
 period T = Z/u. The changes in pressure and velocity of the pressure 
 fluctuations in the sound receiver are generally considered as acoustic 
 interferences. From this point of view the subject will later be con- 
 sidered in the section on the wind shielding of receivers. For the 
 present, however, the problem will be presented with special interest 
 placed upon those sounds which are produced by this flow in the re- 
 ceiver. If the frequency of these impulses is sufficiently large, the 
 receiver in such unsteady flow will "hear" a sound (or noise, depending 
 on the spectral composition of these pulsations) . Here those additional 
 sounds which may arise from the vortex formation in the sound receiver 
 itself are neglected with the receiver being assumed ideal in this respect. 
 The effect of the pulsations existing in the flow (on the receiver) may 
 be inseparable from the effect of a sound of similar spectral composi- 
 tion. In both cases, the receiver will receive a sonic effect. The 
 sonic vibrations of the medium, however, and the pulsations of the un- 
 steady flow are physically widely different. In the first case it is 
 a question of the small changes of state of the medium associated with 
 its compressibility. The sonic vibrations are propagated with the veloc- 
 ity of sound, and this velocity is determined by the elasticity of the 
 medium (c^ = dp/dp). In the case of pulsations in an unsteady flow, the 
 compressibility (if the velocities in the flow are much less than the 
 velocity of sound) plays an entirely secondary role. The motion of the 
 fluid may be assumed as entirely incompressible and still the pulsations 
 of pressure and velocity can take place and will be received by the 
 receiver as a changed pressure. The velocity of propagation of these 
 pulsations bears no relation to the velocity of sound and is equal to 
 the mean velocity of the flow. The second difference lies in the fact 
 that the sound waves are subject to the principle of superposition (be- 
 cause they may be assumed as linear vibrations of the medium), whereas 
 the pulsations of velocity and pressure in an unsteady flow represent 
 a twofold nonlinear phenomenon and are not, of course, subject to the 
 superposition principle. These physical differences make it necessary 
 to term the sound received by a receiver immersed in an unsteady flow 
 a "pseudosound." It should be borne in mind that an unsteady flow may 
 be the cause of the occurrence of the usual sound propagated with veloc- 
 ity c . An example of this may be provided by the same vortex sound 
 which arises in the flow about bodies. The conditions under which sound 
 is produced by a flow will be considered subsequently. 
 
NACA TM 1399 ■ 125 
 
 What has herein been said with regard to the sound of an unsteady 
 flow may easily be illustrated by the example of the Karman vortex street, 
 which may be considered as one of the simplest schemes of nonsteady 
 flow. For this purpose, the velocity and pressure in a Karman vortex 
 street will be computed. The pressure receiver will be assumed at rest, 
 so that the computation will be carried out in a system of coordinates 
 in which the vortex street moves with the velocity u = F /2 -y/2Z. In 
 this system of coordinates, the coordinates zq and Zq (of eqs. (4.17), 
 (4.18), and (4.19)), distinguishing the positions of the vortices of the 
 first and second street, will be functions of the time 
 
 (4.29) 
 
 Zj^) = ut + i I 
 
 z;; = ut - i I 
 
 From equations (4.19) and (4.20) there follows: 
 
 ^ _ Bf a# u ^ - n fv iv ^ (^-2°) 
 
 The pressure p, on the basis of the Bernoulli's equation (4.13), is 
 equal to 
 
 2 
 P = Pq + puv^ " P ~ (4:. 31) 
 
 Further, from equation (4.20) for T' = -T" = -T we have 
 
 ^x - i^y = -^ ^^^ f (^ - ^0^ - ^ ^°* f (^ - ^;) (4-32) 
 
 When equation (4.24) is used for the determination of the real and 
 imaginary parts, the following expressions are obtained from equations 
 (4.32) and (4.29): 
 
 v^ = 
 
 sinh 2 
 
 -(,.|) 
 
 2jt / h\ 
 
 TV -2/ 
 
 sinh 
 
 cosh 
 
 2jt/,^ h\ 23T / ^s / (4.33) 
 
 TV -2)- =°^T ^^ " ^"^M 
 
126 NACA TM 1399 
 
 r . 2jt , ^s 
 
 ~2n~7 h\ 2n , T 
 
 cosh 
 
 f(y-^)- coslll (x- ut)J (4.33') 
 
 In view of the complexity of the equations, the pressure p will be de- 
 
 I 
 
 termined for two extreme cases: (a) on the axis of the street (y = O), 
 (b) outside the street, for y > 2 > h. In the first case from equations 
 (4.31)^ ( 4 . 5 5 ) , and (4.33'), the following expression is obtained when 
 r/22 = -yfz -u and cosh(rth/Z) = -y/Z are taken into account: 
 
 ani 
 
 2 2jt 
 
 l_ 
 
 ° ' cos2 ^ (x - ut) K cos2 ^ (x - ut) 
 
 ,2 I 4+2 sin2 ^ (x 
 
 P. :74^. r < 1 '— 
 
 1 - 
 
 nh I ^ " , ? nh J (4.34) 
 
 cosh^ -r- ^^ cosh' - 
 
 From this equation it is seen that the amplitude of the variable pres- 
 sure on the axis of the chain is of order of magnitude equal to pu , 
 and the fundamental frequency of vibration of this pressure is 
 CD = 4jtu/2 = 2a3Q (coq = 2rtu/z is the fundamental frequency of the chain) 
 For large y, the terms of the order (e"2'^y/^)2 and higher being neg- 
 lected, there is obtained 
 
 p = Pq - 4pu2e ^ -y/?. cos — (x - ut) + • • • (4.55) 
 
 As is seen from equation (4.55), the pressure gradually approaches pQ, 
 its amplitude is now equal to 4-^/2 pu^e'^'^y/^, and the fundamental fre- 
 quency CD = ai„. The spectrum of the vibrations depends essentially on 
 the position of the receiver in the street; at the depth of the street 
 the predominating frequency of the pseudosound is 2(jdq and outside it, 
 
 CDq. 
 
 It should be observed that this computation of the pressure refers 
 to an ideal receiver which does not introduce a distortion in the flow. 
 A real receiver situated in a flow unavoidably changes it near the body 
 of the receiver. The pressure received by the receiver will depend not 
 only on what occurs in the flow itself but also on the character of the 
 flow about the receiver. For this reason it is necessary to take into 
 account the precise manner in which the pressure distribution of the 
 flow changes when the receiver is introduced. 
 
NACA TM 1399 127 
 
 Since the computation of such distortions is practically nonrealiz- 
 able, the discussion must be limited to approximations. The equations 
 of hydrodynamics 
 
 { 
 
 |l+ (^ V v)>= - VP (4.36) 
 
 permit writing down for the pressure the following equation, which is 
 valid for the order of magnitude considered: 
 
 p = op - — I + 3pv5v + constant (4.37) 
 
 T 
 
 where 5v is the magnitude of the velocity pulsations in the flow, T 
 the period of these pulsations, a and p numerical coefficients, and 
 I a length determining the gradients (for example, yp ~ p/j ) . If the 
 linear dimension of the pulsations is A, T = a/v. In the flow itself, 
 evidently, Z = A, so that both sides in equation (4.37) are of the same 
 order (provided the flow is not near the steady condition) Near the 
 body of the receiver, the characteristic length determining the gradients 
 will be either A or a dimension of the receiver d, if d < A. In 
 the first case (d>A ), there is obtained from equation (4.37) 
 
 p = (a + |3)pv5v (4.38) 
 
 in the second (d « A) 
 
 p = (a • I + (3)pv6v = ppv6v (4.39) 
 
 The coefficients a and p depend on the character of the flow and on 
 the shape of the receiver body because it is a function of the pressure 
 near the receiver. An essential conclusion, important for the wind 
 shielding of the receiver, is that in the case where the dimensions of 
 the receiver are very much less than the dimensions of the pulsations 
 the receiver will register the pressure changes which are produced not 
 by the local acceleration Sv/St but by the change in the aerodynamic 
 pressure pv^/S; that is, the situation would be that which would be 
 obtained for a slow change of velocity of a constant flow. In this 
 case, therefore, it is permissible to consider the flow about the re- 
 ceiver as a constant flow and, since the pressure distribution on the 
 receiver is known for such flow 
 
 p 
 
 OV / ^ 
 
 p = X • ^ (4.40) 
 
 (where x depends on the point of the surface of the receiver), the 
 changed pressure may be computed by the equation 
 
 5p = X pv5v ( 4 . 41 ) 
 
 ( 5v is the pulsation of velocity). In the case d > A, such simplified 
 
128 MCA TM 1399 
 
 "quasistationary" consideration is no longer possible. This case will 
 be considered in more detail in sections 29 and 30. 
 
 The pressure pulsations computed previously, which have been termed '' 
 pseudosound, are due to the motion of an incompressible fluid. The com- 
 pressibility could have been taken into account as a further small cor- 
 rection of the order of xx^/c"^. The question may be raised, however, 
 whether it is possible that behind these small corrections there is 
 nevertheless hidden a true sound propagated with its characteristic 
 velocity c. The answer to this question must be given in the negative. 
 If the receiver moves together with the street, that is, with velocity 
 u, all the magnitudes will become constant. In such a system of co- 
 ordinates, the flow of the vortex street becomes stationary. It is now 
 shown that if there exists a system of coordinates in which the flow is 
 stationary, such flow cannot radiate sound. The possibility of reducing 
 the flow to a stationary form means that all the magnitudes characteri- 
 zing the flow depend on the time only through the combination x - ut, 
 so that by taking the new system moving with velocity u (x' = x - ut) 
 a stationary flow is obtained. Hence, the potential of the velocities 
 $ will likewise be a function of (x - ut) (even if the compressibility 
 of the fluid is taken into account); that is, 
 
 § = ^(x - ut,y,z) (4.42) 
 
 This potential is expanded into a system of cylindrical waves passing off 
 from the flow 
 
 (x - ut,y,z) = J C(a,p)e^°'^''"''*^Ho(Pp)|3d(3da (4.43) 
 
 where p = -/x^ + z is the distance from the axis of the flow, Hq is 
 a Hankel function, and a and (3 are parameters of the expansion. 
 Each of the individual cylindrical waves 
 
 *^(x,p) = C • e^^-^^-^^) . HQ(fip) (4.44) 
 
 for large p assumes the asymptotic form: 
 
 ia(x-ut)+iPp -, 
 
 #^(x,p) = C • ^ _ = -— . ei(ax+3p)-iajt (4.45) 
 
 ^ yp -y/p 
 
 where the frequency go = au. But for sound waves the phase velocity is 
 c; hence for these waves 
 
 2 ^2 ^ 2 U"^ 
 a^ + B"^ = — = a^ -g (4.46) 
 
 c ^ 
 
 # 
 
\Ai'A IM 1590 i;:y 
 
 or 
 
 a-') 
 
 p2 ^ a2(iil - ij (4.47) 
 
 It follows that if u < c, then (3 is imaginary and therefore the vi- 
 brations $Q,p damp out exponentially with increasing distance from the 
 flow; in other words ^ for a flow velocity less than the velocity of 
 sound, the sonic field in the wave region (|(3|p»l) is equal to zero. 
 
 For supersonic velocity (u > c), equation (4.47) is possible also for 
 real p. Since p/a = tan e , where e is the angle between the normal 
 to the wave and the flow velocity, the following expression is obtained 
 for u > c (eq. 4.47)) : 
 
 sin e = -^ (4.48) 
 
 that is, radiation is possible only under the Mach angle. This result 
 has already been obtained by a different method (cf. section 20). From 
 this it is seen that a flow moving with subsonic velocity may radiate 
 only in the case where it cannot be made stationary in any system of 
 coordinates. As a particular case, it then follows that the infinite 
 Karman vortex street cannot radiate sound. Its entire field, even in 
 the case of large frequencies u/l, will be pseudosonic. 
 
 25 . Vortex Sound in the Flow about a Long Cylinder or Plate 
 
 The occurrence of vortex sound in the flow about a body of simple 
 shape, such as a cylinder or plate, is now considered in more detail. 
 Figure 42 shows a section of the cylinder under consideration and the 
 vortex street obtained behind it. A system of coordinates ^,^,K will 
 be taken in which the body is at rest while the air moves with velocity 
 V in the direction of the ^ axis. If in the stationary system of 
 coordinates the velocity of the vortices in the street is u, this 
 velocity in the chosen system of coordinates will be V = v - u (cf . 
 section 23) . The continued existence of the Karman street is maintained 
 by the periodical separation of vortex filaments from the edges of the 
 body in the flow. If the period of the street is denoted as previously 
 by Z, the frequency of the vortex separations, as has already been 
 explained in section 23 (cf. eq. (4.27)), is equal to 
 
 -- ■- V , . 
 
 X • - (4.49) 
 
 I d 
 
 At the distance, for example, l/2, from the body the formed vortices 
 turn into a regular Karman street and move on uniformly with velocity 
 V = V - u. Hence the state of the flow in this region (C> j/2) will 
 depend on the time through the combination 4 - Vt; and in agreement 
 
130 NACA TM 1399 
 
 with what was said in section 24, the occurrence of vortex sound will 
 be entirely due to the period ■ c ~eneration of vortices about the body 
 (the region to the left of O'O" in fig. 42). It has already been shownj 
 
 (section 22) that the Vdve length of a vortex sound X = d is much 
 
 XVI 
 
 greater than the dimensions of the body d. Because of this circumstance, 
 a region can be drawn about the body in which the motion of the fluid 
 may be considered as incompressible, the more so as the distance from 
 it is increased, so that the linear-equation for the potential cp, con- 
 sidered in section 6 (see eq. (1.94)) may be considered as valid 
 
 A<p - i ^ 23_ 1 ai(£_ ^ (^_5Q) 
 
 ^2 :nv+2 
 
 ^* Vi - 02 ^ ^t • ac' 
 
 where 
 
 C'' = 4 /Vi _ 2 ^= X - vt, p = v/c 
 
 The linearity of the phenomenon is attained in that region where the 
 second term of the Bernoulli equation p = pScp/St - p(V<p) /2 may be 
 neglected. If, for the purpose of an approximation, the relations 
 existing in the Karman street are considered, it may be assumed on the 
 basis of equation (4.35) that the second term p(V(p)2/'2 is small in 
 comparison with the first for the distances from the axis of the street 
 H satisfying the inequality l/Z-yJz » e'^''^"-/'' , which is the case for 
 H = 1/2. Actually the vortices do not break away from the body in an 
 entirely regular manner so that in the spectrum of the vortex sound 
 there are present, in addition to the fundamental Strouhal frequency 
 f, also other frequencies (upper harmonics of f and sound noises). 
 Consideration is restricted to the fundamental frequency f which 
 dominates in intensity and corresponds to the formation of the ideal 
 Karman street. For this reason, only that part of the potential cp 
 will be considered which depends harmonically on the ti.ne with frequency 
 f = xv/d. The corresponding wave number 2jrf/c will be denoted by 
 k(k = 2jrf/c = cu/c) . A certain control surface S is now passed about 
 the body and the Karman street so that near the body it goes through the 
 previously mentioned region where, on the one hand, equation (4.49) 
 holds and, on the other hand, the motion of the fluid may be considered 
 as incompressible . Ahead of the body, this surface will be considered 
 a plane AB continued by the planes AC, BD (see fig. 42) covering the 
 vortex street. Further, it is evidently sufficient to consider a seg- 
 ment of a cylinder of length (-L/2< ^ < L/2) since the state along the 
 cylinder does not vary and the end effects will be neglected. If the 
 values of the potential and its derivatives on this surface are known^ 
 applying the theorem of Kirchhoff (eq. (1.103)) generalized for equation 
 (4.49) gives the value of the potential at any point of space. The 
 integral over the chosen surface breaks up into two essentially different 
 parts: the integral over the surface 0"BAO' lying in the region 
 
NACA TM 1399 l^-L 
 
 without waves, and the Integral over the planes O'C and 0"D, enclosing 
 the Karman street and lying in a considerable part of the wave region. 
 The values of the potential cp and its derivatives on the first of the 
 aforementioned surfaces may be replaced by the values ^ representing 
 the motion of an incompressible fluid. The integral over AB then 
 drops out since this surface is drawn through the undistributed flow 
 where $0=0, and there remain the integrations over AO ' and BO". 
 On the planes O'C and 0"D^ passing to infinity^ the potential <p may 
 be represented in the form of the sum of the potentials ^ A and (p . 
 The first represents the potential of the Karman street, and its integral 
 vanishes in the wave region (the street does not radiate). The second, 
 cp ' , represents the part of the wave field due to the shed vortices. The 
 integral of this part will give at a point of observation P a certain, 
 in general nonvanishing, result which will be denoted by <pp. If the 
 part of the field at the point P due to the integration over AO ' 
 and BO"' is denoted by (pp^ the following is obtained for the wave field 
 at P: cpp =(Pp +(Pp. Since the surfaces of integration AO' and BO" 
 pass near the source, the integration over them should give the princi- 
 pal part of the field (pp. The field cpp, however, having the same 
 physical cause as tpp, cannot possess symmetry other than cpp (they 
 are both produced by the same incompressible motion of the fluid near 
 the cylinder). Hence, the magnitude cpp is at least of the same order 
 as (pp and has the same symmetry. Therefore, it is sufficient to 
 compute (pp for estimating the order of magnitudes and determining the 
 symmetry of the field (zero source or dipole, etc.). This field is ob- 
 tained by the application of theorem (1.108) to the surface AO' and 
 BO"; that is, 
 
 --E 
 
 
 where £-]_ and fo are the coordinates of point A and point 0_, respec- 
 tively, so that ^2 ~ ^1 ~ ^ ^^^ E = 1/2. It will now be assumed that 
 the symmetry in that region where the vortices are generated for the 
 part of the flow having the principal frequency is the same as the sym- 
 metry of the flow in the developed Karman street; that is, it is assumed 
 that the vortices are developed in alternation, first on the upper edge 
 of the body then on the lower, with a phase shift n (this more descrip- 
 tive requirement is somewhat more rigid than the requirement just formu- 
 lated for the component of the motion having the frequency f, but from 
 
^^'' NACA TM 1393 
 
 the second the first necessarily follows). This symmetry is characteri- 
 zed by the relations 
 
 (4.53) 
 
 and leads- to a dipole radiation with a dipole the axis of which is per- 
 pendicular to the flow (plus higher multiple radiations the intensities 
 of which will decrease (kZ)^ times with increase in the multiplicity of 
 
 the poles). In fact^ if in equation (4.5l) e-i'^/^and o(e"^^^/E*)STi 
 are expanded in a Taylor series in powers of kH «1 and r -»■ "^ the 
 following expression is obtained from equations (4.52) and (4.53): 
 
 ^.i(a^t-kR) ^j^ 
 
 (4.54; 
 
 The magnitudes 5 'J' /Sri and ^n/^ are proportional to v^ and their 
 mean values differ from v only by a factor. The rragnitude H is ap- 
 proximately equal to l/S, Cp - ?-] -I. Finality, Sr/St] differs from 
 cos 9 only by quantities of the order of y'^/c , where 9 is the angle 
 between the r^-axis (axis of the dipole) and the direction to the point 
 of observation P. In place of equation (4.54), the following may 
 therefore be written: 
 
 ika • V e^^^^^-^^ ^. ^ (4.55) 
 
 ^ P = 4^f— • • LZ^ cos 
 
 where a is a numerical coefficient « 1 (representing essentially 
 the mean value of the nondimensional velocity l/vd^Q/^Ti on the 
 
 34 
 
 The symmetries of the Karman street corresponding to these rela- 
 tions are easily verified if it is borne in mind that the potential of 
 one vortex chain extending over the length y ' = h/2 is symmetrical 
 with respect to the substitution of -(y - h/2) for (y - h/2)^ and the 
 other, extending over the length y" = - h/2, is syraruetrical with re- 
 spect to the substitution of -(y + h/2) for (y + h/2), and that the 
 phases of the potentials along the x-axis for the fundamental frequency 
 are displaced by a half period. 
 
NACA 'I'M 1399 
 
 133 
 
 r)-axis in the plane AO) . The energy flow W in a system of coordinates 
 in which the medium is at rest and the body moves with velocity v is 
 now computed. According to equation (I.6O), this flow is equal to 
 
 where cd' is the frequency which has been changed because of the Doppler 
 effect: 
 
 CD' = 27rf 
 
 Since the quadrupole radiation was neglected, there is no point in re- 
 taining higher powers of v/c in substituting (p from equation (4.49) 
 into equation (4.50). Neglecting these terms yields: 
 
 ., g^cos^e pGD^v^ -r2-,4 (4.58) 
 
 w - 7^ — ?r -2 J-' i 
 
 ^ 32n^r'^ c'^ 
 
 and the total energy radiated per second will be 
 
 Noting that gd = Snf = ^^^ (l - u/v), where u is the velocity of the 
 
 vortices, gives 
 
 I 
 
 N 
 
 jtfa£cos^ pv^ J 2 /'i _ iA^ (4.58') 
 
 2r 
 
 
 If? .== £f5 L^l - ^) (4.59') 
 
 where, according to the table given previously, for the cylinder 
 (1 - u/v) =0.86 and for the plate = 0.80. The equation obtained 
 earlier from dimensional considerations (see section 22)- is again ob- 
 tained. The direction of the dipole axis is now fixed; however, the 
 axis extends perpendicular to the direction of the flow. It is seen 
 further that the intensity is proportional to the square of the length 
 of the segment of the cylinder (or plate). According to the observations 
 of W. Holle (ref. 46) for small aspect ratios (L/d <; 15), the intensity 
 of the sound is proportional to a power of L, near 2. For L/d > 30, 
 according to reference 46, I is proportional not to L^ but to Ld. 
 The essential point evidently is the fact that for large L/d the co- 
 herence of the radiation by the individual parts of the cylinder is dis- 
 rupted. This consideration is very likely if it is remembered that the 
 long vortex filaments, as they are considered in the Karman theory, are 
 not very stable and break up into certain segments of length AT..'^^ 
 
 35 
 
 This assumption sould te verified bj experimental check. 
 
^^^ NACA TM 1399 
 
 The intensity will then be proportional not to L^ but to 
 
 E 
 
 AL^ = ^ Z!1.2 = L^ 
 
 where AL does not now depend on L so that AL = 3d, where p is a 
 certain numerical coefficient depending in general on d/L. For medium 
 values of the aspect ratio L/d, p = L/d; and for large values of L/d, 
 P = constant. 
 
 Thus in place of equations (4.52) and (4.53), the following will 
 apply for long bodies; 
 
 N = ^ g-' • P cos^d Pv^ ^^ / u\* (4.58") 
 
 ^ = ¥»'^' !r"(i-/ (-5^") 
 
 If the results of the previously mentioned tests of Holle are used, it 
 is to be expected that p = L/d for L/d ~ 10 and p = constant for 
 L/d > 20. 
 
 Both from the earlier derived equations and from those now obtained, 
 it follows that the intensity of the vortex sound is proportional to 
 the density of the medium p and inversely proportional to the cube of 
 the sound velocity. Hence the intensity of the vortex sound, for other- 
 wise equ.-l conditions, is in water 10 times as great as in air. When 
 the Intensity in decibels is expressed by the ratio to the threshold 
 pressure 2x10-4- bar, there is obtained 
 
 N(db) =80+10 log IP£ (4.60) 
 
 4 ^ ' 
 
 According to the results of W. Holle (ref. 46), the intensity of the 
 vortex sound N is 80 decibels for a cylinder of length L = 22 5 centi- 
 meters and diameter d = 1.2 centimeters, for v = 35 meters per second at 
 the distance r = 1 meter (and cos e = l) . From these data and equation 
 (4.58 ), the value «%^p/2 = 10" -^ is obtained, whence for 3 = 10 
 there is obtained m = 1.4-10-2. This value is in good agreement with 
 the initial assumptions of the theory, in fact, a essentially reduces 
 to the value of the ratio v /v at the distance y = z/2 from the street. 
 According to equation (4.33'), at this distance 
 Vy/v = ue-^/v = 0.2e-" = 10-2. 
 
 26. Remarks on the Vortex Noise of Propellers 
 
 Tests show that the vortex noise of propellers has a spectrum in 
 which one of the frequencies stands out relatively strongly, so that the 
 
NACA TM 1399 I35 
 
 spectrum consists of a sharp peak on a diffused background (fig. 43). 
 This characteristic of the vortex noise becomes understandable if account 
 is taken of the fact that its intensity increases very rapidly with the 
 velocity (as v^) . This noise may, in fact, be considered as generated 
 by the vortices shed from the different parts of the blade. A concep- 
 tion of the spectrum of this sound can be obtained if the individual 
 parts of the blade are assumed to give rise to independent vortex for- 
 mations and if to each part of the blade is applied the equation for the 
 intensity of the vortex sound derived previously from considerations of 
 dimensionality and considered foT the special case of a cylinder or 
 plate. The length of a segment of the blade over which the profile and 
 its angle of attack changes little will be denoted by AR. The width of 
 the profile at the same segment will be denoted by 2(R). The intensity 
 of the vortex sound generated by this segment will then be 
 
 AI=rUR)ARv^R) T = "'"'^ r""' (4-61) 
 
 2r" 
 
 where v = SitRN is the peripheral velocity of the segment, R is the 
 distance from the axis of the propeller, and W is the number of rota- 
 tions of the propeller. The frequency which is predominantly radiated 
 by this segment will be 
 
 f(R) = X 4^ = 2«xW • -4-7 (4.62) 
 
 ^ ^ dXRj d(R) ^ 
 
 where d(R) is the width of a plate equivalent to the blade element. 
 The following expression may be set up: 
 
 d(R) = I sin a + b cos a (4.63) 
 
 where o. is the angle of attack of the segment, I the width, and b 
 the thickness {l , b, and a are functions of R). 
 
 From equations (4.62) and (4.63), the following terms can be found; 
 R = R(f), and also d(f), Z(f), AR(f) = (dR/df)Af. Substituting in 
 equation (4.6l) yields 
 
 AI = rUf)(i6(f)f^Af IP (4.64) 
 
 which gives the spectral distribution of the vortex sound radiated by 
 the propeller. It has a sharp maximum about a certain frequency f . 
 This is evident from the fact that f and R are approximately linearly 
 connected (R ~ f), and for f->- " , R -* Rq, where Rg is the radius of 
 the propeller (in fact, d(RQ) = 0; then from equation (4.62) there fol- 
 lows f = '») . Hence in equation (4.58)^ the factor f^ rapidly increases 
 while the factors dR/df and d6(f) approach as f-^". 
 
 Equation (4.64) can, of course, give only a very rough idea of the 
 spectral composition and the intensity of the vortex so md of a propel- 
 ler, since the assumptions made in its derivation do not pretend to 
 
136 NACA TM 1399 
 
 great accuracy. The angle entering the coefficient r, as is known 
 from section 25, is the angle between the ray at the point of observa- 
 tion and the dipole axis which is perpendicular to the flow. Since the 
 blades move perpendicularly to the propeller axis, this is the angle 
 between a ray at the point of observation and the propeller axis. The 
 maximum intensity of the vortex noise will therefore be radiated ahead 
 of and behind the propeller axis, as, in fact, observed (see ref. 47). 
 It should be recalled that the s-und of the propeller rotation (cf. 
 section 18) is, on the contrary, radiated in directions almost perpen- 
 dicular to the propeller axis. The frequencies of these two sounds, 
 as has already been remarked, are likewise different. The frequency of 
 the rotation sound is fg = Nn (n, the niomber of blades, cf . section 
 18), while the frequency of the vortex sound is equal to 
 
 f = ■>^ • — T" (4.65) 
 
 d 
 
 where R and d are the values of R and d for the most intense 
 frequency. The ratio of the frequencies of these two sounds will be 
 
 il = 2^x . I . 1 I 
 
 fo ^ n (4.66) ' 
 
 Since n = 2 or 5, x s 0.2, and R is generally several times (about 
 
 6 to 10) times as large as d, the frequency of the vortex sound exceeds . 
 
 the frequency of the sound of rotation by several times. 
 
 < 
 
 27. Excitation of Resonators by a Flow 
 
 In the preceding sections, the origin of the sound in the flow of 
 air about bodies was considered. This theory cannot, however, be applied 
 directly to bodies of any shape. It was tacitly assumed that the body 
 has a relatively simple geometrical shape capable of being characterized 
 with sufficient completeness by a single length d, which also determines 
 the frequency of the radiated sound by the equation of Strouhal f = xv/d. 
 For bodies of more complicated shape, the case is otherwise. It is 
 clear, for example, that if on a body of simple shape with characteristic 
 dimension d-^ there is a projection with characteristic dimension do, 
 there will be two vortex frequencies for the same velocity of the flow v: 
 
 2 2 dg 
 
NACA TM 1399 137 
 
 The presence of any projections, sharp angles, discontinuities in the 
 profile, roughnesses, and so forth, may essentially change the sound 
 spectrum. Entirely different characteristic phenomena arise in those 
 cases where the body possesses not convexities but concavities. The 
 latter are acoustic resonators possessing proper vibrations with fre- 
 quencies Ug and damping coefficients hg . The proper frequencies of 
 such a resonator are determined by its dimensions d and the velocity 
 of sound c : 
 
 a • ^=) 
 
 0^=1 %(^ ■ hj (4.68) 
 
 where \|; is a certain numerical coefficient. The value of the damp- 
 ing coefficient depends further on the viscosity of the air |j. and on 
 its thermal conductivity x (if the thermal conductivity of the walls 
 of the resonator is much greater than the thermal conductivity of the 
 air, then hg does not depend on it) . It may be said that, in the 
 presence of cavities in the body which are capable of resonance, the fre- 
 quencies that can be associated with the body depend not only on the 
 ratios v/d but also on the ratios c/d. The simplest examples of such 
 resonators will be, for example, pipes open at one or both ends, Helmholtz 
 resonators (in the form of bottles), and so on. All resonators of such 
 kind may easily be made to emit a sound in an air flow by blowing at 
 their mouths. This phenomenon may be on the most diverse scales, from 
 the whistling in the wind of a small cavity of a receiver microphone 
 (wind static) to the catastrophic excitation of the vibrations of an 
 open wind tunnel that may lead to the destruction of the tunnel and 
 buildings . The same phenomenon in the last war was applied by the 
 enemy in the so-called whistling bombs designed for psychological effect. 
 It finds application to other more suitable purposes in military matters. 
 Also, all musical wind instruments and sirens are essentially based on 
 the phenomenon of the excitation of vibrations by an air stream. 
 
 In all these cases there may be distinguished two mutually inter- 
 acting systems: the vortices arising in the flow about the body on the 
 one hand and the resonator on the other. The vortices do not, of course^ 
 represent a rigid system and, strictly speaking, their action on the 
 resonator cannot be considered as the action of an external given force. 
 On the contrary, it is to be expected that the vibrations of the resona- 
 tor have themselves an effect on the formation of the vortices and on 
 their frequency and intensity so that the entire system must be considered 
 as self- vibrating nonlinear system, the state of which is described by 
 
 An open wind tunnel represents a resonator pipe with open ends and 
 curved like a torus. The flow which excites the vibrations is the flow 
 within the timnel itself, and vortex formation is obtained at the exit 
 of this stream in the working section. Interesting investigations of 
 the vibrations arising in such system have been conducted by S. P. 
 Strelkov (ref . 50 ). 
 
138 NACA TM 1399 
 
 the velocity v and the damping coefficient of the resonator hg . From 
 considerations of dimensionality, the following formula may be written 
 for the amplitude of the pressure fluctuations in the resonator: 
 
 P = P — <P -J-' — (4.69) 
 
 In the region of maximum excitation of the resonator (autoresonance) , 
 this amplitude should be inversely proportional to the damping coeffi- 
 cient h_ : 
 
 Os f^'^^ 
 
 P3= P — ^ CP3 ^-^J (4.70) 
 
 where the stroke on the velocity v indicates that the equation holds 
 only for a certain value v = v' . The nonlinear phenomena occurring in 
 the systems under consideration cannot, at the present stage of the 
 theory of vortex formation, be considered in more detail mathematically. 
 The computation of the vortices that arise in the flow about a body even 
 in the absence of a resonator is as yet an unsolved problem. It is all " 
 the more reason to expect little success in the computation in the pres- 
 ence of a resonator when, for example, there may occur an interaction 
 of the frequencies of vibration of the vortices with the frequencies of 
 the resonator, phenomena which are characteristic for autovibrating ( 
 systems. It is therefore of interest to know to what extent it may be I 
 useful, for practical purposes, to employ a more primitive point of | 
 view in which the nonlinear character of the relations between the vor- 
 tices and the resonator is ignored and the pressure pulsations produced 
 by the vortex formation are considered as a given external force applied 
 to the resonator. It is evident that such a simplified approach to the 
 phenomenon is possible only in the case where the system of vortices has 
 a considerable degree of independence so that the amplitudes and fre- 
 quencies of this system are essentially determined by the velocity and 
 geometry of the flow and not by the vibrations of the resonator. If 
 such is the case, the nonlinear phenomena, such as the interaction of 
 the frequencies, could be considered relatively unimportant, and it 
 would not be absolutely required to account for such phenomena in ap- 
 proximations intended for obtaining only the most essential information. 
 It is possible also to assime a priori that the case is otherwise, namely 
 that in the presence of a resonator the vibrations of the vortices as a 
 whole are determined by the vibrations of the resonator interacting with 
 the flow. The problem proposed could be solved only by an experimental 
 method. Experiments on the excitation of resonators by air streams are 
 reported in reference 51. As a resonator there was taken a four- sided 
 tube closed at one end and placed in an air flow the velocity of which 
 could be brought up to 35 meters per second. At the bottom of the tube 
 was a measuring microphone, with the aid of which the pressures of the 
 vibrations arising in the pipe were transmitted. This pipe was readily 
 excited at definite velocities of the stream, emitting a sound with its 
 natural vibration frequencies Og = c(2s + l)4=l, s = 1, 2, 3, • • •. 
 
NACA TM 1399 139 
 
 A picture of the flows about such a pipe and within it is shown in figure 
 44. The circulation arising in the pipe, characteristic in general for 
 all concavities in a stream, is very slow and has no essential value for 
 the phenomena of interest here. On the other hand, of extreme importance 
 is the region about the mouth of the resonator where, as in the case of 
 the flow about solid bodies, an unstable dividing boundary (ABC) is 
 formed between the stream and the stagnant region. It is in this bound- 
 ary that the vortex formation is obtained, which must, therefore, essen- 
 tially depend on the geometry of the mouth of the resonator. In order 
 to explain the character of the vortex formation aside from the depend- 
 ence on the presence of the resonator, the resonator was damped by a 
 damper of cotton and netting placed on the bottom of the resonator . The 
 flow around the mouth was thereby practically unchanged and the resonator 
 was, in effect, eliminated. The damping was chosen such that the fre- 
 quency characteristic of the measuring microphone located at the bottom 
 of the resonator coincided with the frequency characteristic of the 
 microphone itself. In this way it was possible to determine the spec- 
 tral composition of the pressure pulsations due to the vortex formation 
 at the mouth of the radiator. It was found that the frequencies of the 
 vortices were in accordance with equation (4.67): 
 
 f^ =x I n n = 1, 2, 3, • • • x = 0,65 ' (4.7l) 
 
 where d is the length of the side of the mouth of the resonator. The 
 value of the coefficient x is given for the angle of attack a of 70° 
 (fig. 44) in the neighborhood of which there was observed the excitation 
 of the resonators. In this way the existence of two overtones of the 
 Stroiohal frequency was confirmed, which led in section 22 to the general- 
 ized formula (4.6). The amplitude of the pressure of these overtones, 
 as was to be expected, is proportional to the square of the flow velocity: 
 
 where a = 70°, P-]_ = 0.055, Pg = 0.020, and Pg = 0.010. Figure 45 shows 
 the frequency of the vortices as a function of the flow velocity v. 
 The same figure shows also the natural frequencies of the resonator u 
 indicated by the horizontal lines. At the points of intersection of 
 these lines, that is, for 
 
 indicated on the figure by small circles, which are the points of reso- 
 nance, the excitation of the resonator was to be expected. This was 
 actually confirmed. On removal of the damper, the resonator was excited 
 at the stream velocities v', determined from equations (4.7l) and (4.73): 
 
l^*^ . KACA TM 1399 
 
 v' = ^ (4.74) 
 
 xn 
 
 The existence of the vortices permits, at least as an approximation, 
 computing the pressures arising in the resonator as a result of the 
 action of the vortices. If q(a)) denotes the coefficient of amplifica- 
 tion of the resonator for the frequency cu(ao = 2nf), the amplitude of the 
 pressure p on the bottom of the resonator will be as follows, if a 
 pressure of frequency co and with amplitude P is applied at its 
 mouth: 
 
 p = q(cD)P (4.75) 
 
 This equation assumes that the vibrations are linear; q(cD) depends on 
 the shape of the resonator, but for all resonators in the region of 
 resonance frequencies (cjd = 2jtu ), q(cjD) is inversely proportional to the 
 damping coefficient h . Comparison of the results of computation by 
 this equation with the measured values of p shows (ref . 5l) agreement 
 in the order of magnitude. The observed difference attains 6 decibels 
 (2 times), which already serves as an indication of the fact that the 
 divergences are due not to the errors of measurement but to the fact 
 that the assumption of a rigid vortex system fails to correspond; such 
 a system is actually subject to the inverse effect of the vibrations of 
 the resonator (autovibrating character of the phenomena). In figure 46 
 are given the excitation curves of a resonator (l as a function of v). 
 
 The maximums of the excitation correpsond to the resonances of the 
 vortex frequencies and the natural frequencies of the resonator. They 
 are indicated by the same letters as the circles in figure 45 . The last 
 maximum c corresponds simultaneously to two resonances (fig. 46), when 
 the second overtone of the vortices coincides with the first overtone 
 of the resonator and simultaneously the fundamental tone of the resonator 
 coincides with the fundamental tone of the vortices^"^. The vibrations 
 that arise in this case are biharmonic. 
 
 The height of the maximums is inversely proportional to the damp- 
 ing coefficient, as was confirmed by a change in the damping of the res- 
 onator. In the same manner, the dependence of the frequency of the vor- 
 tices on the dimensions of the mouth of the resonator (eq. (4.71)) is 
 also confirmed. The computation of the amplification coefficient of 
 the resonator for the different resonators is found in many texts on 
 acoustics . 
 
 37 
 'This circumstance is incidental and is caused by a characteristic 
 
 feature of the given resonator. 
 
NACA TM 1399 141 
 
 For the simple harmonic Helmholtz oscillator (fig. 47) the equation 
 of vibrations reads : 
 
 'i + 2\i + a?^K = ^ (4.76) 
 
 where k, is the velocity of motion of the air mass in the resonator 
 throat, h the coefficient of damping, odq the natural frequency, s the 
 throat area, P the variable pressure applied from without, and M the 
 mass of air moving in the throat of the resonator. Also, M - pLs, where 
 L is the effective length of the throat, p the density of the air; 
 L = 2 + aa, where 2 is the length of the throat, a its radius, a a 
 numerical coefficient equal, for the case of a circular opening, to it/2 . 
 The natural frequency gdq is equal to: 
 
 0)0 = c ^ (4.77) 
 
 where V is the volume of the resonator. When equation (4.77) is 
 solved for the external force, having frequency od, the amplitude of 
 the displacement C„ is obtained: 
 
 ^ _SP 1 (4.78) 
 
 (if cD^ » h) . The changes in pressure within the resonator with adia- 
 batic change of the volume of air enclosed in it will have the amplitude: 
 
 p.pc^f^PC^^^ (4.79) 
 
 Substituting this in equation (4.72) gives 
 
 p = P 
 
 2 
 "^0 (4.80) 
 
 ■/i 
 
 {J - mif + 4h=^a)g 
 
 whence 
 
 q(a)) 
 
 
 2 
 
 O) 
 
 (4.81) 
 
 -Ji 
 
 f 2 2. ,J2 2 
 (cD - cUq) + 4h cOq 
 
 At resonance (od = co„) , q(cD) = cDQ/2h, so that the amplitude of vibration 
 of the resonator excited by the flow of air will be 
 
 P=Pp^^ (4.82) 
 
 2 2h \ ' 
 
 The numerical coefficient 3 depends on the shape of the throat 
 
 and on the angle of approach of the flow (as was mentioned for the rec- 
 tangular throaty for a = 70°, p = 0.055). 
 
142 NACA TM I399 
 
 A second simple case of a resonator is a pipe closed at one end, 
 such as that considered in the previously described tests (fig. 44). 
 
 In this case it is a question of the vibrations of a distributed 
 system. The displacement of the air along the axis of the pipe, which 
 will be chosen in the direction of the x-axis, is here subject to the 
 wave equation: 
 
 c3t (3x 
 
 where 5 is the friction coefficient that takes into account the losses 
 in the heat conduction and viscosity of the air^^^ 
 
 The pressure p at each point is equal to: 
 
 p= - pc2M (4.84) 
 
 ox 
 
 Equation (l7) must be solved for the boundary conditions: 
 
 (^)x=0=° 
 
 -pc^(^) =P (4.85) 
 
 \^x/x=Z 
 
 expressing that fact that at the closed end of the pipe (x = O) the air 
 is at rest while at the open end the pressure is equal to the external 
 applied pressure P. If the fact that the air near the mouth of the 
 pipe takes part in the vibrations is taken into account, the last bound- 
 ary condition must be satisfied by the pressure P', representing the 
 reaction of the associated air mass. If the impedance of this mass, 
 generally termed mouth impedance, is Z = X + iY, then P' = pc(X + lY) . 
 In place of equation (4.85), the following expressions hold: 
 
 (^).= n = 
 
 'x=0 
 
 .pc2 (2i) = pc(x+ iY)(4)^^^ +P (4.86) 
 
 \^^/ x=l 
 
 The active part of this impedance X is due to the losses in radiation, 
 while the reactive part Y is determined by the mass of the air vibrating 
 along with the resonator. These magnitudes, for an orifice of area s, 
 are equal to : 
 
 A simple method of computing this coefficient is given in 
 
 reference 4 . 
 
NACA TM 1399 143 
 
 X = A 
 
 ^=0.7^^ 
 
 (4.87) 
 
 and details on them may be found, for example, in the work of Y. L. Gutin 
 (ref. 52). Assuming that the external pressure depends harmonically 
 on the time, with frequency gd, the displacement E, is taken proportion- 
 al to e""- . From equation (4.83)^ a solution satisfying the boundary 
 condition ?= for x = is readily found: 
 
 ■ ?= ?(. sin Kx K = V ^ - i'^S (4.88) 
 
 c'^ 
 
 For small damping (5 «gd) the following may be assumed: 
 
 K=k-ix k = ^ ^=1^ (4.89) 
 
 c 2 
 
 By the substitution of equation (4.88) in the second boundary condition, 
 equation (4.86), the amplitude Cq is determined: 
 
 p 
 
 ^ = - (4.90) 
 
 ° iX)pc(X + iY) sin Kl + pc^ K cos Kl 
 
 When the real and imaginary parts are separated and the fact that 
 
 xZ « 1, X, Y « 1 is taken into account, the following is obtained, 
 
 for the amplitude Cg- 
 
 P (4.91) 
 
 cope 
 where 
 
 y [cos k2 - Y sin kl]^ + sin kl ^ 
 
 2 
 h = |x + — (4.92) 
 
 is the damping coefficient of the radiator. 
 
 On the basis of equation (4.84) the amplitude of the pressure at the 
 bottom of the resonator at x = is equal to KnPi^- The required am- 
 plification factor of the resonator is therefore equal to: 
 
 q(cD) = ^ — (*-93) 
 
 ■y/ [cos kZ - Y sin kZ]^ + sin kZ — 
 
1^^ NACA TM 1399 
 
 The points cos kZ - Y sin k2 = determine the position of the reso- 
 nance frequencies. With the fact that Y is small taken into account, 
 this condition may be represented in the form: 
 
 (4.94) 
 
 cos i== = 
 c 
 
 oOg = 1^ (2s + 1), s = 0, 1, 2, . • . 
 
 where L is the effective length of the resonator : 
 
 L = J + 0.7 /^ (4.95) 
 
 At the points of resonance, the value of the amplification factor is 
 equal to 
 
 q(cD) = ^ (4.96) 
 
 (since sin(k 2) = l). Hence the amplitude of the vibrations of the 
 pressure in the resonator under the action of an external stream is: 
 
 p = Pp ^-^ (4.97) 
 
 2 hgc 
 
 This equation may also serve for estimating the value of p. Considera- 
 tion will now be given to the computation of the intensity of the sound 
 radiated by a resonator excited by an air stream. Evidently, it is suf- 
 ficient to compute the energy radiated through the mouth of the resonator 
 and, in what follows, to make use of the law of the inverse square of 
 the distance. The mean flow of energy through the mouth of the resonator 
 according to the general equation (3.3), is equal to: 
 
 N = p|S = i v^yi • S (4.98) 
 
 '0 
 
 where pJL is the amplitude of the velocity of the air vibrations near 
 the mouth and ^ is the amplitude of that part of the air pressure near 
 the mouth which is produced by the radiation. This part is equal to 
 pcX$„. Hence, the mean flow of energy through the entire mouth is 
 
 N = i pc • X^qS (4.99) 
 
 and the energy flow through 1 square centimeter at the distance r from 
 the resonator will be 
 
 N^ = -^ PcXSS^ (4.100) 
 
 In order to obtain the final result, the value of L. at the point 
 of resonance must be taken. According to equation (4.91), for the tube 
 
NACA TM 1399 "'"'^^ 
 
 Sq = — T— -7 aJ^<3. for the Helmholtz resonator^ according to eqiiation (,4.79j; 
 
 P c 
 ^ = — "^TT- Hence, for the tube the following expression applies: 
 
 ■(. #)■ 
 
 ^6 = -^-#2 P'^'Ip^-' (^-101) 
 
 s 
 
 and for the Helmholtz resonator: „ 2 
 
 No = -^-4-2 P'^fp v) ('^•1°!') 
 
 Sjrr^DC Sh'^L'^ V 2 / 
 
 ijtr pc 2h L 
 
 where v is the velocity corresponding to the excitation of the s ^ 
 vibration of the tube and Vq is the velocity at which the Helmholtz 
 resonator is excited. 
 
146 NACA TM 1399 
 
 CHAPTER V 
 
 ACTION OF A SOUND RECEIVER IN A STREAM 
 
 23. Physical Phenomena in the Flow about a Sound Receiver 
 
 A sound receiver placed in a stream of air or water will register 
 periodic changes of pressure brought about not only by the arriving 
 sound signal but also by the flow around the body. 
 
 Such periodic pulsations are termed "pseudosound. " It is clear 
 that the pseudosound will act as an obstacle for the successful re- 
 ception of the useful signal^ an obstacle which may possibly be very 
 significant. It is well known in practice how strongly the audibility 
 of the sound of a distant airplane may be lowered in a wind. Such 
 lowering of the audibility occurs also in the work of hydrophones of a 
 ship (in this case the noises of the ship are intermixed). 
 
 For this reason the case of the action of a sound receiver in a 
 stream is of a practical interest. The phenomena due to the unsteady 
 flow must be distinguished from the phenomena that take place in a 
 steady flow. The phenomena that take place in a steady flow are con- 
 sidered first. A steady flow does not contain pressure pulsations 
 periodic in time, but such pulsations arise on the receiver body because 
 of the vortex formation. 
 
 The vortex formation is, in the case considered, the only cause of 
 the pseudosound. The predominant frequency of this sound is determined 
 by the formula of Strouhal which was used previously: 
 
 f = X I (5.1) 
 
 Lffuse near the f: 
 
 If the Reynolds number is large I Re = — > 10 j, the spectrum of 
 the vortex pseudosound may be very diffuse near the frequency equation 
 (5.1). The pressure of the pseudosound will be proportional to the 
 dynamic pressure: 
 
 v2 
 P = PP -g- (5.2) 
 
NACA TM 1399 1'^'^ 
 
 where (3 is a numerical coefficient that depends on the shape of the 
 body. 
 
 If the flow is unsteady, further pressure pulsations characteristic 
 of the flow are superposed on the pressure pulsations determined hy the 
 vortex formation. This pseudosound of the flow was partly considered 
 previously (section 24) . In this case it is necessary to distinguish 
 "between the pressure pulsations "brought about by the local change in 
 the velocity of the flow and the pressure pulsations associated with 
 the momentum transfer of the flow. This question was previously dis- 
 cussed in part (section 24), but now it will be considered in greater 
 detail. A simple example may serve to illustrate the pseudosound of 
 the flow. The receiver is assumed to have the shape of a sphere and 
 to be placed in a stream in the direction of the OZ-axis (fig. 48) . 
 The flow velocity V is assumed to pulsate periodically with the 
 frequency oj = 2jt/T: then 
 
 V = Vq + SV . cos cot (5.3) 
 
 The vortex formation is disregarded, and the flow is assumed to be 
 potential . The equation for the potential # is : 
 
 S^.^.^=0 (5.4) 
 
 Sx^ By2 hz^ 
 
 The radial component of the velocity v^ = - S<l/9r on the surface of 
 the sphere (r = a) must be equal to zero, and the velocity at a large 
 distance from the body must become V = - S'I'/Sz (eq. (5.3)). A 
 solution of equation (5.4) satisfying these boundary conditions, as 
 easily verified by substitution, will be 
 
 V ^ 2r2J 
 
 § = V cos Ir + ——] r cos = z (5.5) 
 
 V 2r2y 
 
 By the formula of Bernoulli, the pressure at such a point will be 
 
 P=|i-l(V^)^ (5.6) 
 
 p St 2 
 
 and on the surface of the body (r = a), on the basis of equation (5.5), 
 
 I p = constant + | p cos . a ||. - i sin20 . p ^ (5.6) 
 
 From this formula, it is seen that the pressure is made up of two com- 
 ponent parts, namely, the term p' which is also present in the steady 
 flow: 
 
 Q ? 
 
 p' = constant - - sln^e . p Z_ (5.7) 
 
 I 
 
148 NACA TM 1399 
 
 and the terra p"; determined "by the acceleration of the flow: 
 
 p" = 2 p ^Q3 . a . ^ (5.8) 
 
 2 St 
 
 From a comparison of equations (5.7) and (5.8)^ it follows from equation 
 (5.3) that the variahle part of the pressure p' determined hy the 
 dynamic pressure considerably exceeds the part p" determined by the 
 local acceleration if 
 
 a ^ « V 5V 
 St 
 
 or ^ (5.9) 
 
 2na 
 — — < V 
 
 ] 
 
 that is, p'« p" if the dimensions of the receiver a are sufficiently 
 small. The velocity pulsation considered is uniform over the entire 
 space. If the pulsations have the dimension A, then Ts A/v, and 
 equation (5.9) reduces to 
 
 a « A (5.10) j 
 
 This condition was obtained in section 24 by a different method; it is 
 apparent that;, for small dimensions of the receiver in comparison with 
 the dimensions of the pulsations, the changes in dynamic pressure have 
 a much greater significance than the acceleration of the flow. 
 
 The spectral distribution of the pseudosound of an unsteady flow 
 is entirely determined by the nature of the flow. If the nonsteady 
 condition of the flow is produced by the flow about certain bodies 
 placed near the receiver so that the receiver is in the vortex street 
 of these bodies, the spectral composition of the pulsations is deter- 
 mined by the Strouhal frequencies and their overtones, as has already 
 been shown in the example of the ideal Karman street (section 24). 
 
 At a large distance from the bodies, the Karman street undergoes 
 a breaJcdown and the flow will be turbulent. A natural wind likewise 
 represents a turbulent flow. The fundamental features of this turbu- 
 lence were described in section 10. 
 
 As previously stated in section 24, the computation of the magnitudes 
 and spectral distribution of the pressure pulsations on the surface of 
 a receiver in any unsteady flow is at the present time an insurmountable 
 problem. It was pointed out that a partial analysis of this problem 
 IS possible on the basis of dimensional considerations in th<= applica- 
 tion to the fundamental equation of hydrodynamics. In the general case, 
 the pressure on the surface of the receiver is assumed to be determined 
 
NACA TM 1399 149 
 
 P = apa ^^ + 3 ~ (5.11) 
 
 This equation is a generalization of equation (5.6)^ which holds for a 
 particular case. The fact that the pressure at any point and at any 
 instant of time depends on the dynamic pressure (pv^/2) and on the 
 local change in velocity 3v/c5t is expressed in equation (5.11). Since 
 the velocity varies not only in magnitude but also in direction^ the 
 angle of attack will vary with the velocity fluctuations of the flow. 
 Because of this variance^ the numerical coefficients a and p,, which 
 depend only on the shape of the hody and the angle of attack, will also 
 he functions of the time. If the magnitude of the pulsations 5v is 
 much less than the mean velocity of the flow v, the changes in a 
 and 3 will he slight. Further, the derivative Bv/ot is of order of 
 magnitude equal to &v/T = V5v/A; and therefore, with equation (5.10) 
 satisfied, Bv/ot may be rejected. For the variable part of the pres- 
 sure p, 
 
 p. = 3„pv5..(|)^5tp£ (5.12) 
 
 where f is the angle of attack and the subscript denotes the value 
 of (3 and Sp/S^j/ for the angle of attack of the main flow (ijf = ^q) . 
 
 (The variation of the angle of attack 6\|/ is equal to &v, /v, 
 where Sv-(- is the fluctuation of the velocity in the direction per- 
 pendicular to v) . 
 
 For isotropic pulsations, Sv^ = 5v and therefore 
 
 P' = 1^0 + J fir) ) P^^ = eP^Sv (5.13) 
 
 The spectrum of the pressure p' therefore coincides with the spectrum 
 of the velocity pulsations 5v, and its magnitude may be computed from 
 the stationary flow about the body under consideration. This evidently. 
 is the only rational conclusion which can be drawn from equation (5.11). 
 For the mean square of the pressure pulsations from equation (5.13) 
 
 p'2 = e2p2-v-25^2 (5.14) 
 
 and the spectral distribution is obtained from 
 
 &v2 = ^[Sv(a3)]2dm (5.15) 
 
;L50 NACA TM 1399 
 
 where &v((jd) is the amplitude of the pulsation "belonging to the frequency 
 CD. Hence, for the mean-square pressure of all pulsations, the frequencies 
 of which lie between cn^ and cDg^ 
 
 ^p2^2 / 
 
 p'2((j:);i_,aj2) = e2p2v2^ (&v(aL>y] qcD (5.16) 
 
 The amplification factor of the receiver is assumed equal to q(a)) in 
 order that the signal received "by the receiver is measured "by the magni- 
 tude P : 
 
 ^ =J q(cD)p'(a3)e^'^^da3 (5.17) 
 
 Squaring and averaging equation (5.17) with respect to time yield 
 
 p2 = / q2(a)) . p'2(aj)daj (5.18) 
 
 where, on the basis of equation (5.14), 
 
 p'2(a3)dm = e2p2v2^5v(aj)3 2dai (5.19) 
 
 If the receiver has sharp resonances to that, for example, there is a 
 natural vibration with the frequency u) = ooq and damping coefficient 
 of h, then for ao = cdq the amplification factor becomes particularly 
 large and, as is known from section 24, is equal to q(cuQ) = q'ojQ/h, 
 where the coefficient q' is of order of magnitude equal to 1 (section 
 27). Integrating equation (5.18) with respect to a:>Q between the 
 limits of the resonance line (cdq - h/2, cdq + h/2), yields 
 
 o^O+h/Z 2 
 
 >0-h/2 h 
 
 X 
 
 2^,.„^ _ / n2^,.„^^,2r,„„^^,„ _ „,2 '^o ^,2 
 
 P ("^) = / / q'^(a3o)p"^(aoo)da3 = q'^ -0 . p'^(cDo) (5.20) 
 
 For small values of h;, this part of the magnitude p2 may predominate 
 over the remaining parts to such an extent that practically the entire 
 effect of the pseudosound on the receiver may be reduced to the emitting 
 of a sound from the receiver at the resonance frequency ojq . Hence, 
 receivers with sharp resonances will be particularly subject to acoustic 
 disturbances. 
 
 This case is characterized by the possibility of reducing the action 
 of the nonsteady flow to that of a steady flow. The fundamental result 
 is the fact that the spectrum of the pressure pulsations reduces to th,e 
 spectrum of the velocity pulsations . If the approaching flow is a well 
 developed turbulent flow, the approaching flow may be applied to the 
 turbulence theory described in section 10. 
 
NACA TV 1399 IFl 
 
 This theory was developed for homogeneous isotropic turbulence. 
 The 2/3 law, which determines the spectral distribution of the veloc- 
 ity of the turbulent motion over the pulsations of different scales, 
 was obtained. Wow, however, the distribution over the frequencies is 
 of interest. The problem of associating the distribution of the fre- 
 quencies with the distribution over the spatial scales has been solved 
 only for linear vibrations of the medium (for example, for sonic noise) . 
 
 In the case of the turbulent motion of a gas or liquid, this 
 relation has not as yet been established. 
 
 For the determination of the velocity spectrum over the frequencies, 
 the same considerations which were applied to the diffusion of sound in 
 a turbulent flow (section 12) may be employed. As was explained, the 
 important fact is that the frequency of the turbulent pulsations is in 
 itself very small. The high frequencies, which are of significance in 
 acoustics, are obtained in virtue of the fact that the large-scale 
 velocity pulsations transfer the small-scale pulsations. If the large- 
 scale velocity pulsations which change slowly are included in the mean 
 velocity v (in this way v will have the sense of a mean velocity 
 over a time during which this velocity does not undergo considerable 
 changes and which is much greater than the period of those frequencies 
 which are received by the receiver) , then v will be precisely the 
 velocity with which the small velocity pulsations are displaced. For 
 these small pulsations the 2/3 law holds, in accordance with which the 
 mean value of the square of the velocity u for the pulsations which 
 have a scale less than A = 2jt/q will be (see section 10, eqs . (2.63) 
 
 and (2.64)) 
 
 1 .2 .„. _ 1 .„-2/3 _2 3/^(^y/^ 
 
 E(q) = |u^ (q) = irq""/^ r = ^V^[—) (5.2l) 
 
 The magnitude u(q) is precisely 6v(q) . Thus, 
 
 [&-r{l)]^ = -r. q"^/^ (5.22) 
 
 3 
 
 Since these spatial pulsations of velocity are transferred with velocity 
 V, the frequencies of the corresponding pulsations will be f = v/a or 
 (w = qv. Thus the intensity of the velocity pulsations with frequencies 
 between co and *» will be 
 
 [8v(cu,»)]2 =10'^' =^ [6v(a.)] ^d,^ (5.23) 
 
152 NACA TM 1399 
 
 Differentiating this magnitude with respect to o) yields the required 
 /alue: 
 
 [^.i^)f = ±rr-f^^ (5.24) 
 
 Substituting this result in equation (5.19) gives the equation for the 
 spectral distribution of the pressure acting on a receiver placed in a 
 turbulent stream. 
 
 p'2(a3)dco = I reSpSvZM^'^^ ^ (5.25) 
 
 The constant r, as has already been pointed out, is evidently a function 
 of the velocity v. Its value has been considered in connection with the 
 diffusion of sound in a turbulent flow (section 10^ eqs . (2.58) and (2.59)) 
 
 Since r and e may be assumed as known, equation (5.25) permits 
 computing the spectrum of the turbulent noise. As is seen from this 
 eqi.'ation, the noise of a turbulent flow is concentrated around low fre- 
 quencies; the intensity of sound near the frequency cd is proportional 
 
 to en . For cu = 0, equation (5.25) is not valid since slow pulsations 
 in the mean velocity v have been included. With regard to the de- 
 pendence on the wind velocity, if y is assumed constant, the dependence 
 
 on the velocity is obtained as v^'^ . It was pointed out, however, that 
 r increases with the velocity in a manner which is as yet difficult to 
 determine precisely but which, from all the data available, may be taken 
 approximately as v. If this dependence is taken into account, the noise 
 
 should increase as v^^/^. Finally, it must be borne in mind that equation 
 (5.25) IS not suitable for high frequencies since in its derivation it 
 was assumed that the dimensions of the receiver were a « A- Hence, 
 it is applicable only to cd < 2nv/a. (Otherwise, terms are added which 
 are due to the local acceleration bv/bt.) Undoubtedly it gives a lower 
 limit^of the sound. The fact that the intensity of the sound increases 
 as ca' is evidently one of the greatest obstacles to acoustic direction 
 
 finding, since predominantly low frequencies (80 to 100 hertz) which will 
 be masked by the turbulent sound arrive from a distant airplane. 
 
 29. Shielding a Sound Receiver from Vortical Sound Production '■ 
 
 No universal method of shielding a sound receiver from vortical 
 sound production is possible. The question depends essentially on the 
 dimensions of the receiver and on the working frequency range, the choic 
 of which is determined by the character of the signal that is to be re- 
 ceived. It is nevertheless possible to indicate certain methods that 
 may be found useful. 
 
 e 
 
WACA TM 1399 -^^^ 
 
 In the first place; it is possible to vary the dimensions of the 
 receiver so that the Strouhal frequency may he displaced either toward 
 the lower frequencies (hy increasing the dimensions of the receiver) or 
 toward the higher frequencies ("by reducing the dimensions), depending on 
 the purpose. This method is hased on the fact that for the same velocity 
 of the approaching stream and shape of receiver the characteristic Strouhal 
 frequencies are inversely proportional to the linear dimensions of the 
 receiver: 
 
 f" d' 
 
 (5.26) 
 
 In those cases where the change in dimensions of the receiver is not a 
 rational procedure , a sound -transparent screen F of netting or fabric 
 (fig. 49) may be applied. The principal air flow in this case tends 
 to pass around the screen and the velocity of the flow within is con- 
 siderably lowered. The Strouhal frequency formed on the vortex deflector 
 is then lowered and will be equal to 
 
 f ' = f . - (5.27) 
 
 D 
 
 where f is the Strouhal frequency on the body of the receiver M, d 
 the dimension of the receiver, and D the dimension of the deflector. 
 It is necessary to avoid angles, projections, and so forth, on the body 
 of the deflector, since they may become the cause of vortex formation in 
 an undesirable range of frequencies . 
 
 In addition to the effect of lowering the frequency, which was due 
 to the screen, the region of the vortex formation is farther removed 
 from the body of the receiver, another useful -result of this method. 
 A part of the flow will nevertheless pass through the screen, but its 
 velocity v' will be less than the velocity of the approaching flow 
 V. Because of this lower velocity, the frequency of the vortex formation 
 immediately at the receiver is likewise lowered in the ratio 
 
 f " = f . — (5.28) 
 
 V 
 
 while the amplitude of the pressure will drop by (v'/v) . The value 
 of V can not be computed exhaustively, but reasonable estimates may 
 be made. For this purpose, the resistance of the nettings to the flow 
 of air (or of water) must be considered. 
 
 In view of the fact that nettings or screens are widely applied for 
 various wind-shielding apparatuses, it is necessary to discuss them in 
 more detail. If the difference in pressure on both sides of the screen 
 is set equal to Ap, and the volume rate of flow of the air through it is 
 set equal to Q cubic centimeters per second, then 
 
3_54 NACA TM 1399 
 
 Ap = WQ (5.29) 
 
 The magnitude W defined ty this equation is the resistance of the 
 screen. This magnitude may be represented in the form 
 
 W = w 1 (5.30) 
 
 S 
 
 where S is the cross-sectional area of the flow, I the effective 
 thickness of the screen (or other porous partition, for example, fabric), 
 and w the resistivity. The characteristic magnitude for a screen is, 
 of course, the product wZ . The resistance W is considered as the 
 resistance of a system of parallel ducts (or small tubes), the length of 
 which is equal to I and the cross-sectional area of which is equal to 
 0} the area of the screen on which, on the average, there is one opening 
 is denoted by Z . If the ducts are not identical, then I, o, and Z 
 must be considered as the characteristics of the mean representative of 
 the ducts. In order to determine the pressure drop Ap over the length 
 of the duct Z, the equations of Navier-Stokes reduced to nondimensional 
 form were used. For measuring the coordinates along the duct, the length 
 scale was taken as the length of the duct I, and for the transverse 
 scale, the magnitude V^. The scale of velocity would be the velocity 
 of the flow u, and the scale of the acceleration the magnitude cdu, j 
 where oj is the frequency of the flow pulsations. These equations are: "i 
 
 — + (v,7)v = - ^ + o . vv (5.31) J 
 
 dt p I 
 
 where u = M./p is the kinematic viscosity. The derivatives S/Sx, • 
 o/3y, and o/Sz may be reduced to the derivatives along the duct (S/Ss) f 
 and transverse to the duct (S/Sn) . Then, setting S/St = cjoS/St ' , ■! 
 S/Ss = (l/z)B/Bs', d/Sn = (l/-|/a')o/on' , v = u-v', and p = Ap-p', equa- 
 tion (5.3l) reduces to the form: 
 
 _, ^ /v' Sv' V' -,\ ; 
 
 Sv ' 2 / s s n Bv ' \ 
 
 cuu + u + 1 
 
 St' V 7 cts' Vo Sn-y J 
 
 ^ Ss' ^ ^n'J a \^^,2 i2 ^^,Z) 
 
 Ip 
 
 where all the stroked magnitudes are nondimensional and the magnitudes 
 and their derivatives are of the same order; s' and n' are unit 
 vectors along and transverse to the duct. The case where the viscosity 
 is the predominant factor is considered first. In this case the last 
 term predominates over the others . Dividing the entire equation by 
 ou/a, the desired pressure drop Ap will be measured in the units 
 Zpou/a {Qn\xlu/a could be used as this corresponds to the Poiseuille law) 
 
NACA TM 1399 
 
 in equation (5.31') dimensionless parameters on which Ap may depend, 
 namely, uxr/o; \xo/ol, and a/l , will enter. Thus, 
 
 
 where F-, is a certain dimensionless coefficient depending on the in- 
 dicated parameters . The term ua/) I represents the Reynolds number and 
 determines the ratio of the inertia to the viscosity forces. The value 
 of this term is small for small u, hut for these values of u the 
 equation becomes linear; hence for small velocities of flow Y-^ practical- 
 ly does not depend on ua/ol. Further, for long ducts {a/l^ « l) the 
 pressure drop should be proportional to the length of the duct Z; hence, 
 F-, should likewise not depend on a/l^. Thus, 
 
 Ap 
 
 87t\xl 
 a 
 
 
 UCT 
 I 
 
 «1, 
 
 ^ < 1 
 
 72 
 
 for }E. «1, a_< l (5.32') 
 
 I ^2 
 
 Finally, for small frequencies (ooa/o « i) the Poiseuille law must be 
 
 obtained so that \|f(0) = 1. For large axr/o, the coefficient ^ = -^ojcr/o 
 (see, e.g., Crandall, ref. 53). For ua/oZ,>l, the forces of inertia 
 will predominate over the viscous forces, and therefore it is convenient 
 to use the dynamic pressure pu^/S as the measure of the pressure. In 
 an analogous manner^ the following equation is obtained in place of 
 equation (5.32) : 
 
 2 
 
 Ap = Hii- 
 
 Va 
 
 Fp f"^, ^, '^^\ (5.33) 
 
 \\ia i2 u / 
 
 For small values of the parameters entering the function Fp this 
 coefficient will only slightly depend on them. The square law of the 
 resistance is therefore obtained. The effect of the acceleration is 
 now determined by the parameter ca-^a/u. The volume rate of flow Q 
 is computed as 
 
 Q = ua - (5.34) 
 
 2 
 
 When u from equation (5.32) is substituted and when equation (5.34) is 
 compared with equation (5.29), 
 
 
 (-) 
 
 for 
 
 ^ » 1, — < 1 (5.35) 
 
■j_56 NACA TM 1399 
 
 In the same manner, substituting u from equation (5,34) in equation 
 (5.33) yields 
 
 2 Va a2 ^ \ ZQ / S S 
 
 V 
 
 for 
 
 H.» 1 ^< 1 (5.36) 
 
 where (p is the value of Fg for small values of u l/ua and a/l . 
 If the frequencies of the pulsations are not large, W will increase lin- 
 early with an increase in Q. In the case W = 0, by the author's meas- 
 urements (ref. 54), the numerical value of the coefficients \|/ and (p 
 is such that 
 
 -3 Z Z g 
 
 for 
 
 and 
 
 for 
 
 W^ = 2.5 . 10" -^ - • - 
 
 ° a2 S (3ec)(cm)^ 
 
 Q < 1.5 . — (5.37) 
 
 Z 
 
 -2 2 Q 
 
 ■y/o S (sec)(cm)' 
 
 Q » 1.5 — (5.38) 
 
 For the correct application of equations (5.37) and (5.38), it is 
 necessary to take 
 
 CT = ab "\ 
 
 \ = (a + d)(b + d) S (5.39) 
 
 I = 2d J 
 
 where a and b are the lengths of the sides of the openings and d 
 is the thickness of the fibers of the netting or fabric. It is necessary 
 to bear in mind that the last equation is valid only for the condition 
 W» Wq . In the intermediate region, the resistance must be considered 
 as the sura W = (W + W„)/2. This intermediate region corresponds, as 
 
 experience shows, to the values of the Reynolds number \ia/\}l = 10, 
 which corresponds to the values of the volume rate of flow Q indicated 
 
NACA TM 1399 ^^'^ 
 
 in equations (5.37) and (5.38). Figure 50 shows the form of the resist- 
 ance W as a function of Q for fahric with a = b = lxiO-2 centimeter 
 and d = S'^IO"^. The transition from the resistance Wq^ independent 
 of the velocity, to the resistance W proportional to the velocity, as 
 predicted by theory, is apparent in figure 50. The computation of the 
 resistance of a fabric or netting permits the evaluation of the velocity 
 u of the flow of air through a meshed screen. ^9 The value of the pres- 
 sure drop of the air flowing through the meshed screen will be 
 
 Ap = 3 P^ (5.40) 
 
 where 3 < 1; this pressure drop is equal to 
 
 3p 1_ = WQ = WS H = wZ Si (5.41) 
 
 2 S S 
 
 The magnitude Q/S is the mean velocity of the flow v'. For 
 
 ua/oZ < 10, the unit resistance wZ (it has the dimensions mechanical 
 
 ohm/ cm , and 1 mechanical ohm = 1 g/sec) is constant and equal to wqZ. 
 
 Hence from equation (5.41), 
 
 V ' _ 3P"V" 
 
 V 2wqZ 
 
 Z^for H^ < 10\ 
 
 u = ^ (5.42) 
 
 a 
 
 From these equations for a fabric with WqZ = 10 mechanical ohm/cm^ and 
 V = 5 m/sec, v'/v = 33x10"^ << 1. The term ua/ol = v'Z/\3l = 7. 5*3 « 10 
 
 (assuming Z = 10" and 2 = 2'10" ) . For larger values of ua/oZ, 
 equations (5.38) and (5.4l) yield 
 
 "' ' {^f\'°" ^»^o) (5.43) 
 
 V \2r \ UZ 
 
 where x is the coefficient of proportionality between the unit resist- 
 ance wZ and the velocity v': 
 
 wZ = rV (5.44) 
 
 39 
 
 All these considerations refer also to the flow of water, but the 
 
 numerical coefficients in equations (5.37) and (5.38) will be different. 
 
 Further, the fabric will swell up in water so that its dimensions will 
 
 change considerably. 
 
158 NACA TM 1399 
 
 On the "basis of equation (5.38) r = 2'10~'^QlZ/-^a . For the same data 
 
 and a = 10 
 
 -4 
 
 f 
 
 r = 2-io"2 
 
 v'/v = 0.17P ' 
 
 and 
 
 v'l/vl = 8-10" P 
 
 2,1/2. 
 
 V 
 
 1/2 
 
 so that for v = 10 meters per second, v'Z/^Z = 803 • In this case 
 
 the velocity v' of the flow through the screen is linearly connected 
 with the velocity of the approaching flow v. From equations (5.42) 
 and (5.43) it is apparent that a considerahle lowering of the velocity 
 within the screen for moderate resistances (ahout 10 mechanical ohm/cm ) 
 of its fahric can "be obtained. The extent such a screen will lower the 
 intensity of the sound of the arriving useful signal has to be con- 
 sidered. Actually, the resistance in this case must be computed by 
 equation (5.35) for cd / 0. The magnitude \(f(axr/o), although it in- 
 creases also in this case, nevertheless still remains a magnitude on the 
 order of 1 (for medium frequencies and small values of a) . It is 
 therefore possible to take the value of W for en = 0. If the resis- 
 tivity is w, the pressure drop of the sound wave will be 
 dp = - w dx'Q/s = - w dx 5", where "f" is the velocity of the fluc- 
 tuations. This magnitude is equal to p/pc where c is the velocity 
 of sound. Thus 
 
 dp = - JL p dx (5.45) 
 
 pc 
 
 therefore for the total thickness I of the screen, 
 
 _wZ_ 
 P = Poe P"" (5.46) 
 
 That is, the drop in intensity of the sound in decibels will be 
 
 I(db) = - 20 log e . ^= - 0.20wl (5.47) 
 
 pc 
 
 that is, for example, for wZ = 10 mechanical ohms per centimeter^ only 
 about 2 decibels.. Thus, without conflicting with the sound transparency, 
 it is possible to lower the velocity of the flow within the screen and 
 thereby lower the frequencies and intensities of the vortices . 
 
NACA TM 1399 1^'^ 
 
 In connection wifh screens, it is of interest to point out another 
 method of their application for eliminating the vortex formation. This 
 elimination depends on the fact that a flow passing through a sufficiently 
 transparent screen becomes turhulent, the frequencies of the vortices 
 heing then determined by the dimensions of the meshes of the screen 5 
 and the velocity of the flow (f" -= kv/5) . These frequencies may be so 
 high that they appear beyond the limits of the frequency range of the 
 receiver. The placing of such a screen near (or around) the receiver 
 will not, of course, shield the receiver from the pressure pulsations 
 in the flow if it is nonstationary, but the vortex formation on the 
 body of the receiver will be artificially displaced toward the region 
 of high frequencies f". This effect is shown, for example, by the 
 protective screens that cover the mouths of loudspeakers (fig. 5l) . If 
 there were no screen, the frequencies of the vortices would be deter- 
 mined by the dimensions of the opening of the loudspeaker. The screens 
 displace the spectrum of the vortices toward the higher frequencies . 
 This breaking down of the vortices is very well shown in the excitation 
 of resonators by an air stream, discussed in section 27 . If at the 
 mouth of such an excited resonator a screen S'S" is placed inter- 
 secting the stream (fig. 52), the excitation of the resonator is im- 
 mediatedly cut off, even for the case of a very rough screen (meshes of 
 the order of 1 cm ) . This change in the scale of the vortices is also 
 employed for the absorption of vibrations in open wind tunnels by 
 placing near the opening of the tunnel from which the vortices are 
 shed projecting lugs which break down these vortices into smaller ones 
 (ref . 50) . 
 
 30. Shielding of Sound Receiver from Velocity 
 
 Pulsations of Approaching Flow 
 
 If the approaching flow is not steady, the problem arises of 
 shielding the sound receiver from the pressure pulsations brought about 
 by the nonsteady flow. In section 28 it was explained that, for the 
 condition where the mean velocity of the flow v is much greater than 
 the velocity pulsations 5v and for the condition where the dimensions 
 of the receiver d are much less than the dimensions of the pulsations 
 A, the nonsteady flow about the body of the receiver may be considered 
 on the basis of a knowledge of the flow picture for the steady flow. 
 This permits making use of the important results from the theorem of 
 Bernoulli applied to the flow about a body. In the flow about bodies, 
 due to the compression of the stream, the velocity of the stream on the 
 lateral sides of the body increases, while ahead of and behind the body 
 it is slowed down. As a result, by the law of Bernoulli, 
 
 v2 
 p = constant - p __ (5.48) 
 
^,-,0 NACA TM 1399 
 
 the pressure on the lateral surfaces of the "body drops ^ while it in- 
 creases ahead of and hehind the hody. On figure 53 is shown the pres- 
 sure distribution over the surface of a sphere and of a streamlined body. 
 A particularly interesting picture is evidenced in the case of the stream- 
 lined body. 
 
 The pressure in the middle part of the surface is not only negative 
 but is very small in absolute value. Hence, if the sound receiver is 
 placed in this part of the body, the change in pressure produced by the 
 velocity pulsations of the flow may be very small. For a suitable 
 choice of the shape of the body, the local coefficient (3 could be made 
 to attain a value between the pressure p and the dynamic pressure 
 pv^/2 up to 0.02. We may note that for a mean velocity of the flow 
 directed along the axis of the body (dp/di|;)Q = so that equation (5.13) 
 reduces to 
 
 p' = pQPV 6v (5.49) 
 
 The receiver diaphragm may be made flush with the surface of the de- 
 flector at the place where 3o ^^ minimum, or the receiver may be . 
 placed within the deflector, making a part of its screen transparent to 
 sound. The sound-transparent surface must be very smooth and not too 
 transparent to the flow; otherwise the flow about the body may change 
 considerably. 
 
 The pressure distribution over the body with maximums ahead of and 
 behind it and a minimum at the lateral sides suggests still another 
 method for dealing with the pressure pulsations, namely, the principle 
 of compensating the pressures . The essential character of this principle 
 will be described in a simplified idealized form by imagining a body of 
 the type illustrated in figure 53 placed in a stream. Inside the body 
 there will be a chamber with a pressure receiver in it. A part of the 
 surface of the deflector will be made transparent to the flow, for 
 example, at the forward part where the pressure is positive and at the 
 sides where it is negative (fig. 54) . Under these conditions there 
 will be a stream of air through the chamber. The velocity of this 
 stream normal to the surface of the deflector will be denoted by Vn, 
 and the difference in the pressures outside and inside the transparent 
 partition by Ap = p^ - p. . The flow of air passing through the area 
 ds will then be equal to: 
 
 JT J Ap Pa ~ Pi , , 
 
 dL = Vj^ . ds ""n = -^ = ^ (5.50) 
 
NACA TM 1399 161 
 
 where W is the resistance of the partition. The total flow through 
 the entire transparent parts of the surface of the deflector will be: 
 
 v^ds = r^^^ • ^s (5.51) 
 
 On account of the incompressihility of the fluid, this flow must he 
 equal to zero; hence j 
 
 p . ds / p. . ds 
 
 ■ a 
 
 W 
 
 ;5.52) 
 
 If W is constant, it follows that the mean outside pressure is 
 equal to the mean inside pressure: 
 
 Pg^ = Pj_ (5.53) 
 
 The mean is taken over the transparent parts of the deflector. If 
 W is large, the interior velocities will he small and the pressure 
 p^ may he assumed as practically constant over the entire volume so 
 that p. = p. . By choosing the position of the transparent surfaces 
 and their resistance W and making use of the fact that at some places 
 p > and at others Pn ^ 0, p„ may he made equal to zero: from 
 equation (5.33) it then follows that p-j_ = . Thus, a chamher of 
 constant pressure is derived. An example of this type of chamher is 
 illustrated in figure 54 . The parts of the deflector transparent to 
 the air flow in the given case are located forward (screen s-^) and at 
 the sides (screen sg) . At s-i_, p -* and at sg^ p < 0. The third 
 
 screen S3 hreaks up the additional stream entering the chamber through 
 the opening S]_. In this case, it was found possible to attain the 
 value (3q = 0.001 for the position of the microphone M shown in the 
 figure so that the pressure near M was only a thousandth of the dynamic 
 pressure. The screens were still entirely transparent to the sound. 
 The screens are Important also from the viewpoint that, using acoustical 
 terminology, they represent only active resistances since the enclosed 
 chamber possesses no resonances. If for example, the deflector is made 
 rigid with a small number of openings, a resonator of the Helmholtz 
 type is obtained which strongly distorts the frequency characteristic 
 of the receiver device. 
 
 The preceding discussed principle of pressure compensation leading 
 
 to the formation of a constant -pressure chamber in many cases partially 
 
 acts, so to speak, by itself. In fact, if the receiver is placed inside 
 
 a deflector provided with walls transparent to the flow, it is sufficient 
 
 that a part of the flow enter the chamber at p„ > P- and issue at 
 
 a 1 
 
 Pg^ < V± foi" a-'t least partial compensation to take place. Such partial 
 compensation will be obtained, for example, in a screen deflector having 
 
152 NACA TM 1399i 
 
 the shape of a sphere (see fig. 49) on the surface of which, for a 
 sufficiently thick screen, there will occur "both positive and negative 
 pressures (see diagram of pressures in fig. 53) . 
 
 As an example there may be cited the shield of a loudspeaker 
 (fig. 55) made in the form of a sound-transparent sphere which encloses 
 the opening of the speaker. If the transparency of such a sphere for 
 the air flow is small, the pressures will be distributed as shown in 
 the figure by the + and - signs, and compensation of the pressure 
 pulsations due to the velocity pulsations will to a certain extent 
 be obtained. 
 
 It must, however, be borne in mind that all the conclusions refer 
 to that part of the pressure pulsations which is produced by a change 
 in the dynamic pressure. The local changes of the velocity, as has 
 been explained, likewise lead to pressure pulsations of the form 
 
 p" = pg^ rZ. This part of the pulsations remains even on the lowering 
 
 of the magnitude p' = p'pv /2 so that it will form a certain back- 
 ground serving as the limit of the lowering of the acoustic inter- 
 ferences brought about by an unsteady flow. It is possible, of course, 
 to suppose that this part of the pressure may likewise be subject to 
 compensation, but for this there is no rational data because very 
 little is known of the flow about bodies in a nonsteady stream. More- 
 over, the possibility of eliminating the interferences due to this 
 part of the pressure may be doubted since they are essentially the 
 same type as the pressure changes produced by a sound wave in a wave- 
 less region. The elimination of such interferences will therefore 
 probably be in contradiction to the requirements of the sound receiver. 
 
 1 
 
 31. Sound Receiver Moving with Velocity 
 
 Considerably Less Than Velocity of Sound 
 
 The fundamental problem which is encountered in the mathematical 
 theory of a moving receiver is that of computing the variable pres- 
 sure produced by an approaching sound wave on the surface of a re- 
 ceiver, in particular its working part. This problem includes the com- 
 putation of the flow about the body of the receiver, a computation 
 associated with well-known difficulties. 
 
 A particularly difficult problem is that of the vortex formation 
 arising behind the body. If the receiver body is of a well-streamlined 
 shape, however, then, at least in its forward part, the flow may be 
 considered as potential. If the working diaphragm is located in this 
 part, the application of the potential-flow theory may be entirely 
 practicable. In this section the idealized case of potential flow for 
 V « c will be considered. 
 
^ACA TM 1399 1^3 
 
 In this case the equation for the potential of the sound wave (p , 
 if terms of the order of v^/c^ are neglected, is (see eq. (1.85)): 
 
 1 S^^ A^ + 2(-^, ^1 (5.54) 
 
 ;2 St2 
 
 Vc2 ' ht) 
 
 where ^q is the potential of the undisturbed flow about the body of 
 the receiver (v = - V^q) • This potential satisfies the equation 
 
 A*n = (5.55) 
 
 '^O 
 
 and the boundary condition 
 
 Sn 
 
 - = (on the surface of the body) ' (5.56) 
 
 where S/on is the derivative along the normal to the surface of the 
 body. The required sound potential Q must satisfy equation (5.54) 
 and the boundary condition 
 
 — =0 (on the surface of the body) (5.57) 
 
 Sn 
 
 (the yielding of the diaphragm is ignored) . For a harmonic sound of 
 frequency oo (the frequency go is considered in a system of coordinates 
 in which the receiver is stationary and the medium is in motion) , set 
 
 o = cp^e 
 
 jicDt_ 'Yhen from equation (5.54) 
 
 AcPq + k^Q + 2ik - (V<?Q, VcPq) = ^ = '^ (5.54') 
 
 The potential of the sound wave ^ in the absence of the flow is assumed 
 j known. This potential must then satisfy the equation 
 
 Ai|;q + k^to = (5.54") 
 
 and the condition Si|/Q/Sn = on the surface of the receiver. Setting 
 
 k*o 
 <P0 = *0 • ^ (5.58) 
 
 where §q is the potential of the flow and then substituting this 
 solution in equation (5.54') satisfy the equation (5.54") when terms of 
 the order of v /c^ are neglected. Equation (5.58) may therefore be 
 considered as the solution of equation (5.54) with the assumed accuracy. 
 
164 NAC^ TV 1399 
 
 This solution satisfies also boundary condition (5.57). In fact^ 
 
 Sep- ^'^o io^t H^^-T~y I °^o 
 
 e = e 
 
 on on 
 
 
 (5.57') 
 
 on the surface of the body, since B\|;Q/Sn and S^^^/Bn are equal to 
 zero. Thus, if the wave field ^q is known near the stationary re- 
 ceiver, then with an accuracy up to v/c the field near the moving 
 receiver tp is obtained with the aid of the simple equation (5.58). 
 
 In particular, if there is considered a plane wave propagated, for 
 example, in the oz -direction, the solution for the stationary receiver 
 will be 
 
 ^Q = Ae'^^^ + S(r,e,(p) (5.59) 
 
 where Ae"""" is the incident wave and S is the dissipated wave. For 
 
 large r, S must have the form S = B(e,(p)e""^ /r, where B(0,cp) is 
 the amplitude of the dissipated wave at a large distance from the body. 
 It depends only on the angles and C), which determines the direction 
 of the dissipated beam. The solution for the moving receiver will be: 
 
 r *oi r *oi 
 
 i aot-kz-k — i aot-k-;^ 
 
 L ^J + S(r.0,cp)e L ^J 
 
 cp = A . e •- ^-J + S(r,0,cp)e •- -^ (5.60) 
 
 ik^ /c 
 At a large distance from the body, the factor e 0' essentially 
 
 gives the Doppler effect. In fact, for illustration take the case of 
 
 a plane wave moving on the body with velocity v along the z-axis. 
 
 Then at a large distance from the body 'J'q = - vz, and therefore the 
 
 phase of the approaching wave is [aat-k(l-v/c) z ]. Consider next a 
 system of coordinates in which the flow is at rest, ^ = z - vt . In 
 this system the phase of the wave will be 
 
 [a:)t-k(l-v/c)z ] = [co(l-v/c)t-k( 1-v/c)^], and therefore the frequency is 
 equal to cjDq = (u(l-v/c) . Hence the frequency in the system in which 
 the body is at rest will be 
 
 CD = ^— = oj^ h + I j + ... (5.61) 
 
 0^ 
 
 /c 
 
 v/c 
 
 as must be the case by the equation for the Doppler effect. The 
 variable pressure on the surface of the receiver will now be computed. 
 For this purpose use is made of the equation of Bernoulli, according 
 to which 
 
 =; 
 
 dp Sep [v(<^o^)3 
 
 ^ = constant + — (5.62) 
 
 P St 2 
 
NACA TiV 1399 ^ 6.S 
 
 Op+n Pp 
 
 Since the small change Sw = J dp/p - J dp/p = Jt/p, where it 
 is the sonic pressure^ there is obtained for the variable part of the 
 pressure 
 
 -i^ 
 — = [iGD\(fQ-(V*Q,V\(f )] e c + terms of higher order (5.63) 
 
 Far from the body where the flow becomes uniform^ this equation 
 does not give any results of interest. It confirms only the fact that 
 the pressure n in a system of coordinates in which the body is at 
 rest is the same as that in a system in which the body moves (for 
 checking this statement, it is necessary to take into account the 
 Doppler effect, by virtue of which, in a system of coordinates con- 
 nected with the flow, S(p/St = cMq'-'? and not cd<P) . 
 
 Near the body the situation is otherwise. The magnitude ^q near 
 the surface of the body is of the order of magnitude equal to the ampli- 
 tude of the incident wave A (see eq. (5.59)) and v^q is of the order 
 of magnitude equal to A/a, where a is the dimension of the body 
 (here the tangential component of \t|/q is considered; the normal com- 
 ponent is equal to zero). Hence the first term in equation (5.63) is 
 approximately equal to (x>A and the second = vA/a. For v/a > oo 
 the pressure on the surface of the receiver will be determined not by 
 the first but by the second term. The amplitude of the potential A 
 is connected with the amplitude of the pressure of the incident wave by 
 the equation A = Uq/Ip'S). Hence, according to equation (5.63), the 
 pressure on the surface of the receiver due to the first term on the 
 right side of equation (5.63) will be rt' = n^ and that due to the 
 second term will be rt" = v«„/aa:) and for v/a > gd may be greater than 
 «'. That is, the characteristic amplification effect occurring in a 
 moving receiver is obtained provided its dimensions are sufficiently 
 small and the frequency of the sound is not too high. The condition 
 of the presence of such amplification may, on the basis of what has 
 been said, be written in the form 
 
 V ^ 2jta /,- ^ . \ 
 
 - > — - (5.64) 
 
 C A. 
 
 where v/c « 1. The dimensions of the receiver must thus be very much 
 smaller than the length of the sound wave X. 
 
166 l^^ACA TM 1399 
 
 32. Sound Receiver Moving with Velocity Exceeding Velocity of Sound 
 
 This case of the motion of a receiver presents special interest 
 and at the same time special difficulties for theoretical computation. 
 These difficulties are connected with the fact that to all the com- 
 plexities of the problem of the flow about a body there is added the 
 further feature of supersonic motion, the existence of density jumps 
 (or shock waves) the occurrence of which was discussed in section 19. 
 Instead of a solution of the problem posed, this section will be re- 
 stricted, in addition to a few general remarks, to the discussion of 
 the idealized simplest case which may serve as an orientation for a 
 more detailed analysis of the problem of a receiver moving with super- 
 sonic velocity. 
 
 This problem has been the subject of frequent discussions (see, 
 e.g., ref . 34) and various questions have been raised: Will the re- 
 ceiver in general receive the sound signal; will there exist a re- 
 flected wave; and so forth. There is, in fact, no basis for assuming 
 that a receiver moving with supersonic velocity will not receive the 
 variable pressure of a sound wave as soon as it is incident in its 
 field. It is evident that it will always fall in its field provided 
 the sound does not issue from a source located behind the receiver 
 so that the sound is forced to overtake the receiver which for v > c 
 it cannot do . 
 
 The wave dissipated by the receiver will possess the characteristic 
 that its entire field will lie behind the receiver in the Mach cone and 
 moreover will be double (see section 20) ; that is, there will be two 
 fields of different frequencies. For supersonic velocity of the re- 
 ceiver, however, the transmitted wave before reaching the receiver body 
 must pass through the shock wave separating the part of the medium 
 undisturbed by the motion of the body from the undisturbed part. Figure 
 56 illustrates what has been said for the case of a sound wave radiated 
 by the source Q and received by the receiver P moving with velocity 
 V > c. The curve M'MM" represents a section of the surface of the 
 shock wave. 
 
 In connection with this, the question arises of the passing of a 
 sound wave through a shock wave which is, in a way, a second screen of 
 the receiver. 
 
 Under the usual conditions the presence of a sharp change of state 
 of the medium would necessarily lead to the occurrence of "two new waves, 
 the reflected wave and the transmitted one. In this case, however, a 
 reflected wave, as it is known, cannot be formed since the shock wave 
 moves with supersonic velocity and without doubt would overtake the wave 
 reflected from it. A certain light is thrown on this paradoxical 
 
KACA TM 1399 ^^'^ 
 
 situation 'hy the consideration of the simpler prohlem, namely, the 
 transmission waves through a plane shock wave. It will he shown that 
 in this case two transmitted waves arise of which one is a particular 
 type. 
 
 Assume a straight density jump (shock wave) lying in a plane parallel 
 to the plane x = and moving in the direction of the positive x-axis 
 (fig. 57) with velocity V. As was explained in section 19, the velocity 
 V is greater than the velocity of sound in the medium at rest (V > cg), 
 in which the shock wave is displaced. The case will he considered in 
 which a plane wave (from x = + ") is propagated so as to meet this shock 
 wave. Since in the shock wave a jump in entropy occurs, recourse must 
 he made to the general equations of the acoustics of a nonhomogeneous 
 moving medium (eqs. (l.70), (l.7l), (1.72) and (1.73)) if the propagation 
 of sound is considered under these conditions. These equations for the 
 one -dimensional prohlem which is being considered are 
 
 (5.65) 
 (5.66) 
 
 St + - Sx = 
 
 p Sx Sx VSp^g Sx ' \.hsj p Sx 
 
 
 St Sx Sx 
 
 
 Sa ^ ^ da _ Q 
 
 Bt B 
 
 (5.67) 
 
 x 
 
 In these equation ^ is the velocity component of the sound vi- 
 brations along the x-axis (£„ = ^^ = O); v is the velocity of the medium 
 
 along the x-axis (vy = v^ = O); 5, n, and a are the changes in density 
 
 of the gas, its pressure, and entropy, respectively, produced by the 
 sound wave. The terms vp, yp, and ys are neglected because p, p, 
 and s are assumed constant on each side of the shock wave. If the 
 entropy of the medium were everywhere constant, then, as was shown 
 earlier (see section 4), a = 0. In a shock wave, however, the entropy 
 itself changes discontinuous ly so that it must not be assumed that 
 s = constant and it is not legitimate to assume a = for the entire 
 medium. In the incident wave, of course, propagated in a medium at rest 
 (eq. (5.66)), a = 0, since this wave may be considered as a usual adiabatic 
 sound wave. With regard to the secondary waves arising as a result of 
 the interaction of the incident sound wave with the shock wave, the 
 question whether these waves are accompanied by changes in enthropy or 
 not can be decided only on the basis of the consideration of the 
 boundary conditions . 
 
-icc NACA TM 1399 
 
 166 
 
 For the solution of the previously mentioned prohlem, it is con- 
 venient to use a system of coordinates in which the shock wave is at 
 rest (x' = X - Vt) . In this system the velocity of the medium is 
 
 u = V - V (5.68) 
 
 and equations (5.65)^ (5.66)^ and (5.67) become 
 
 tc2 SS_ h bo \ 
 p Sx' p Sx'i 
 
 |.u|,= -(--.^-l (S.SS.) 
 
 56 S5 a£ , ^ 
 
 Br-^^ar^ + p §7^ = • (^-se-) 
 
 ^ + u ^ =0 (5.67') 
 
 at a 
 
 X' 
 
 where in place of (Sp/^p) and (Sp/^s) are used the values 
 
 s p 
 
 c2 = r £ 
 p 
 
 (^] =h = -^ (5.69) 
 
 \ds/p c^ 
 
 With the assumption that the incident wave is a harmonic vave, C^ 5; 
 
 (oj't+k'x') 
 and a may he set proportional to e-j^ , where oj' is the 
 
 frequency (in the system x') and k' is the wave numher. For such a 
 
 wave; carrying out the differentiation in equations (5.65')^ (5.66'), 
 
 and (5.67'); yields 
 
 (cD' + uk')£ = - ^^ 6 - - k'a (5.70) 
 
 (cD' + uk')S = - pk'£ (5.71) 
 
 (o)' + uk')a = (5.72) 
 
 From the preceding two solutions are obtained: either a = or 
 (cu' + uk") = 0. In the first case, from equations (5.70) and (5.7l) 
 there is obtained 
 
mCA TM 1399 
 
 16^ 
 
 k' = ^' 
 
 +c - u 
 
 p 
 
 a = (5.73) 
 
 This solution represents the usual adiahatic sound wave^ the phase 
 velocity of which is equal to V^. = ± c - u^ as should be the case for 
 a moving medium (if a system of coordinates is used in which the medium 
 is at rest, i.e.^ u = 0, the Vj = ± c), where c is the adiahatic 
 velocity of sound in the medium under consideration. 
 
 The second solution of equations (5.70), (5.7l), and (5.72) reads 
 
 k' = -^ 
 
 u 
 
 C = 
 
 c2 
 
 a = - — 
 
 h 
 
 6 (5.74) 
 
 In this wave the velocity of the sound vibrations is equal to zero; 
 while changes occur in the entropy a and in the density & of the 
 medium, this wave does not, however, give rise to changes in pressure 
 in the medium. In fact, it = c^S + ha. From equation (5.74) it follows 
 
 that 
 
 (5.74') 
 
 This conclusion is evident also from the fact that for the wave under 
 consideration the velocity ^ = so that the moving force must like- 
 wise he equal to zero. It is convenient to call this wave an entropy 
 wave. As is seen from equation (5.74), this wave is propagated with a 
 velocity equal to the velocity of motion of the medium u; that is, it 
 is essentially simply carried along hy the medium. Thus there are two 
 types of wave. At first glance it appears that the discussion could he 
 restricted to the usual isentropic waves (eq. (5.73)) and, with the 
 possession of two independent solutions, the boundary conditions on the 
 shock wave could be satisfied. 
 
 It could, in fact, easily be shown that from these solutions it is 
 not possible to construct a solution satisfying the initial data which 
 represent, for example, a restricted train of waves encountering the 
 shock wave. On the contrary, in the problem presented herein, there 
 must be unavoidably recourse to the solution (5.74); that is, in the 
 shock wave there occur, as was already mentioned, irreversible 
 
170 NACA TM 1399 
 
 processes, and the disturbances of this shock wave will give rise to 
 entropy fluctuations which will be propagated as a wave of the form of 
 eqiiation (5.74). 
 
 All these results are obtained automatically if the conditions on 
 the surface of the discontinuity (see section 19) are used with the sub- 
 stitution of the differential equations of the hydrodynamics of a com- 
 pressible fluid. According to equations (3.93), (3.94), and (3.95), 
 these conditions are 
 
 MlPl = U2P2 
 
 2 2 
 
 p^ui + p^ = PgUg + Pg 
 
 u? 4 
 
 ^1 + ^= ^2 + 2 (5.75) 
 
 where the subscripts 1 and 2 refer to the medium behind the shock wave 
 (1) and to the medium at rest ahead of the shock wave (2). The first 
 of these conditions expresses the law of the conservation of matter; 
 the second, of momentum; and the third, of energy (w is the heat 
 function) . In the transmission of a sound wave, these conditions change j 
 since all the magnitudes receive small increments (?, 6, n, and a) . It i 
 must also be taJcen into account that u = v - V and that the velocity 
 of motion of the shock wave V must likewise be varied. The change in 
 this velocity will be denoted by A. With the use of only linear approxi- 
 mation, the varied conditions which will be the boundary conditions for 
 the sound wave on the shock wave will be obtained from equation (5 75) 
 Thus, replacing in equation (5.75) u by .? - A, p by p + 5, and 
 p by p + 7t yields 
 
 Vl + (^1 - ^)Pl = ^2&2 + (?2 - ^)P2 (5.76) 
 
 u^S^ + 2(C^ - A)u^p^ + c^S^ + h^a^ = ^2^^ ^ 2(^2 " ^)^2P2 + 4&2 "^ ^2 
 
 (5.76') 
 
 (since n = c 6 + ha) and finally from the third equation of equations 
 (5.74), it is borne in mind that &w = c%/p + c^a/r (r is the gas 
 constant, p = rpT) there is obtained 
 
 c? c? 2 2 
 
 Pl ^1 + T ^1 + ^1 (Cl - A) = ^ 62 + — ag + U2 (C2 - A) 
 
 (5.76") 
 
KACA TM 1399 
 
 171 
 
 I 
 
 In the undisturbed medium (x > 0^ subscript 2) there is only the 
 incident wave. For this wave Og = and Cg = " <=2&2/p2^ where cg 
 is the adiabatic velocity of sound in the undisturbed medium. In the 
 medium behind the shock wave (x < 0^ subscript l) the change in density 
 
 belongs to the transmitted isentropic sound 
 
 IT 
 
 8]_ belongs to the entropi wave for 
 
 - t/{t - l) X rS-^/p-, . With the use 
 
 of these relations^ C and a are eliminated from equations (5.76)^ 
 (5.76')^ and (5.76")^ and after simple algebraic transformations there 
 are obtained 
 
 is 
 
 ^1 = Si 
 
 + e 
 
 2} where 
 
 &-]_ belong 
 
 wai 
 
 re for which 
 
 Cl 
 
 = - C 
 
 -|6-|/p2 and 
 
 wh: 
 
 LCh Ci = 
 
 
 
 and 
 
 ^1 = 
 
 - c^&i/h = 
 
 !1 + !i 
 
 P2 Pi 
 
 (U]_ - C]_) 
 
 2 
 
 - ^1 
 
 &]_ + U]^ -t- U]^0]^ = (U2 - C2J 6, 
 
 (5.77; 
 
 ^Pl P2J 
 
 Si + 
 
 _1 
 P2 
 
 r - 1 p- 
 
 2 
 
 ^2 
 
 Pi 
 
 2 
 = 2 
 
 ^"^ (^ " 4) 
 
 (5.77') 
 
 From conditions (5.75) it is possible to express the magnitudes 
 characterizing the state of the gas behind the shock wave in terms of 
 the ratio of pressures P-i/po^ iri the shock wave, and ahead of it (see 
 section 19); there are then obtained 
 
 (r - 1) + (r + 1) Hi 
 
 P2 
 
 Pi 
 
 (r + 1) + (r 
 
 Pi 
 1) ^ 
 
 = Cr 
 
 Pi 
 
 P2 
 
 (r - 1) p| + (r + 1) 
 
 (r + 1; 
 
 -i- 
 
 (5.78) 
 
 li. = 
 
 cg [(r - 1) ^ + (r + 1 
 2r ~ 
 
 Pi 
 
 P2 
 
 ^ 
 
 (r + 1) ^ + (r 
 P2 
 
 1) 
 
 u„ = 
 
 '2 [jr + 1) ^ + (r - 1] 
 
 2r 
 
 (5.79) 
 
172 
 
 NACA TM 1399 
 
 vfhere u. 
 
 Ug < 
 
 0, and Cg = TPci/pg = '^ ^^ '^^^ square of the adiabatic 
 
 velocity' of sound in the undisturbed medium. The medium ahead of the 
 
 shock vave will be assumed at rest so that 
 
 = Vr 
 
 V 
 
 V(v2 = 0) 
 
 The magnitude &2 is given by the amplitude of the incident wave. 
 
 Hence, from equations (5,78) and (5.79) it is possible to find 6| and 
 
 5n' for the transmitted waves. With these values it is possible, according, 
 
 to equations (5.73) and (5.74), to obtain the remaining characteristics | 
 
 of the transmitted waves. The pressure in the incident wave will be 
 
 denoted by Jt„ = c So and the pressure in the transmitted acoustic wave, 
 
 2 I , , " 
 
 by n' = c-,6x (since a-i = 0). With the elimination of S-i from equa- 
 
 pip 
 tions (5.77) and (5.77'), the ratio ^'/^ri - "^i^l/^ Sg ^^ obtained: 
 
 ^1_ 
 
 1 
 
 2 r 2 
 
 ui 
 
 2 
 
 ^1 
 
 U2 !_ 
 2 p, 
 
 
 
 2 r 2 
 
 n 1 
 
 ^1 1 
 
 )■ 
 
 c2 P2 r - 1 Pi 
 
 ^c2 P2 Pi '^l \Pl P2 
 
 (5.80) 
 
 This ratio for small shock waves approaches 1, as should be the case, 
 and is equal to 
 
 ILL = 1 + 5 . r + 1 . ^1 ^ + 
 
 jtQ 8 r 
 
 P2 
 
 (5.81) 
 
 (for (p., - Po)/P2 « l) • For large shock waves there is obtained 
 
 
 ^ 1 
 
 r 
 
 1 + 2 
 
 (^) 
 
 1/2 
 
 Pi - P2 
 P2 
 
 + 
 
 (5.82) 
 
 (here (p - Vn) /v-p » l) • In both cases, the sound pressure in the 
 
 transmitted wave increases in comparison with the pressure in the in- 
 cident wave. The pressure of the entropy wave, as has already been 
 pointed out, is equal to zero. Hence it is of less interest to find 
 
 and 
 
 u then it is to find the changes in temperature which occur 
 
 because of the passage of the entropy wave. From the identity 
 
 P(P,T) = p(p,s; 
 
 (5.83) 
 
NACA TM 1399 
 
 there is obtained 
 
 173 
 
 o 
 
 With the knowledge that (Sp/Sp)rji = p/p = ^ > where a is the isothermal 
 velocity of sound and (Sp/ST)p = p/T^ and with the fact that for the 
 entropy wave (Sp/Sp)g6^ + (Sp/Ss) a-]_ = 0, the temperature fluctuations 
 produced by the sound wave are ohtained in the form 
 
 e.' = T,(r - 1) . — (5.85) 
 
 J- -■- Q, 
 
 and "by the entropy wave 
 
 ^1 / s 
 
 e^ = - T^ . -^ (5.86) 
 
 P 
 
 From the preceding it is seen that the role of "both waves will he com- 
 parable only in the case where 6-, and S" are of the same order. If 
 from equations (5.77) and (5.77') B^/Sg is eliminated, ^l/^? ^^ obtained. 
 For small shock waves (p-, - P2)/P2 « -L) there is obtained 
 
 P^=r_^.^J-l^^ ... (5.87) 
 
 &2 2r P2 
 
 and since in this case S-|/52 is near 1_, the entropy wave does not 
 play a marked role. Farther on its value increases, and with increase 
 in the shock wave S-i/Bg approaches 
 
 For the sound wave SJ/Sg = ^^/'^q • c /c-^. As a result of equations 
 (5.78) and (5.82), for large shock waves this ratio is equal to 
 
 ^1 1 1 
 
 &2 r , ,^1/2 
 
 (5.87") 
 
]_74 NACA TM 1399 
 
 Ttius for (p, - Vo)/Vo » 1 tl^e value of 'both waves in relation to the 
 fluctuations- of the temperature of the medium behind the shock wave 
 becomes of the same order: 
 
 
 Pi 
 
 e ' = - T. i V_£L_Z 2 g^gg,) 
 
 ^ ^ , X 1/2 Pi 
 
 From equation (5.82) it follows that for large shock waves the 
 sound pressure behind the shock wave is intensified. From this, of 
 course^ there is not to be drawn any final conclusion as to the pres- 
 sure on the sound receiver itself. It may be assumed that in the case 
 of supersonic velocity of motion of the receiver there will also occur 
 the intensification which was considered in section 29, based on the 
 theorem of Bernoulli . This side of the question is not possible to 
 analyze in greater detail because the supersonic flow about a body 
 presents, as yet, a far from solved problem. 
 
 The very simple case considered herein leads to an explanation of 
 the absence of a reflected wave when a sound wave passes through a 
 shock wave, and there is no basis for thinking that this aspect of the 
 matter would be subject to essential modification for shock waves of 
 more complicated form (of the type illustrated in fig. 57). 
 
 A similar remark may be made on the existence of two waves behind 
 the shock wave: the sound and entropy waves. With regard to the quanti- 
 tative relations, the fact that for small shock waves the transmitted 
 wave is almost undisturbed by the shock wave should likewise not de- 
 pend on the shape of the shock wave and probably has a more general 
 significance than follows directly from the special case considered. 
 
 Translated by S. Reiss 
 National Advisory Committee 
 for Aeronautics 
 
WACA TM 1399 175 
 
 REFERENCES 
 Chapter I 
 
 1. Leontovich, M. A.: ZhETF, vol. VI, 1936, p. 561. 
 
 2. Ackeret, I.: Ed . d . Exp. Phys . , Bd . 1, 1927, p. 289. 
 
 3. Umov, N. : On the Movement of Energy in Bodies. Dissertation Odessa, 
 
 1873. 
 
 4. Blokhintsev, D.: ZhTF, vol. XV, 1945, p. 84. 
 
 5. Knudsen, V.: Jour. Acous. Soc . Am., vol. 6, 1935, p. 199. 
 
 6. Kneser, G. : Jour. Acous. Soc. Am., vol. 5, 1933, p. 122. 
 
 7. Konstantinow, V. P., and Bronstein, I. M. : Sow. Phys . vol. 9, 1936, 
 
 p. 630. 
 
 8. Andreev, W. N.: Jour. Phys., USSR, vol. II, 1940, p. 305. 
 
 9. Blokhintsev, D.: DAN, vol. XLV, 1944, p. 343. 
 
 10. Andreev, N. N., and Rusakov, I. G. : Acoustics of a Moving Medium. 
 
 GTTI, 1934. 
 
 11. Ohukhov, A. M. : DAW, vol XXXIX, 1943, p. 40. 
 
 12. Blokhintsev, D. : ZhTF, vol. XV, 1945, p. 71. 
 
 Chapter II 
 
 13. Blokhintsev, D.: DAN, vol. XLV, 1944, p. 343. 
 
 14. Emden, R.: Meteorolog. Zs . , vol. 35, 1918, p. 13. 
 
 15. Hadamard, J.: Lecons sur la Propagation des Ondes , etc., Paris, 1903. 
 
 16. Chibisov, S. V.: Isv . Ak. Nauk SSSR, Seriya geofiziki, no. 1, 1940, 
 
 p. 33 . 
 
I'' 6 rjACA TM 1399 
 
 17. Richardson, H. F.: Proc . Roy. Soc . (London), A, no. 1, ser. vol. 33 
 
 1926, p. 110. ' 
 
 18. Kolmogorov, A. N.: DAW, vol. XXVI, 1940, p. 6. 
 
 19. Obukhov, A. M. : DAW, vol. XXXII, 1941, p. 29. Izv.Ak.Wauk SSSR, 
 
 Ser. geog. No. 4-5, 1941, p. 453. 
 
 20. Kolmogorov, A. W.: DAN, vol. XXX, 1941, p. 299. 
 
 21. Brent, D.: Physical and Dyanmical Meteorology. Gidrometizdat, 1938. 
 
 22. Krasilnikov, V. A.: Dissertation, Inst. teor. geof. AN SSSR, 1941. 
 
 23. Gedicke: Ann. d. Hydrographie, vol. X, 1935, p. 400. 
 
 24. Findesen: Meterorolog. Zs . , vol. 53, no. 1, 1936. 
 
 25. Dahl, H., and Devick, 0.: Nature, vol. 139, 1937, p. 550. 
 
 26. Stewart: Phys . Rev., vol. 14, 1919, p. 376. 
 
 27. Sieg, H.: Elektr. Nachr. Techn . , vol. 17, 1940, p. 193. 
 
 28. Obukhov, A. M. : DAN, vol. XXX, 1941, p. 611. 
 
 29. Blokhintsev, D.: DAN, vol. XLVI, 1945, p. 136. 
 
 30. Wood, A., Brown, H., and Cochrane, C: Proc. Roy. Soc. (London), 
 
 ser. A, vol. Ill, 1923, p. 284. 
 
 31. Shuleikin, V. V,: Physics of the Sea. 
 
 Chapter III 
 
 32. Blokhintsev, D. : Trudy Fizichesk. In-ta. 
 
 33. Nepomnyashchii, E. A.: Investigation and Computation of the Sound 
 
 of an Air Propeller. Trudy TsIAM, Oborongiz, 1941. 
 
 34. Karman, T.: Gas Dyanmics. OOTI, 1939. 
 
 35. Hugoniot, H. : Jour, de L'Ecole Polytechnique (Paris), Cahier LVII, 
 
 1886, pp. 3-97; Deuxie'me Partie, Cahier LVIII, 1889, pp. 1-125. 
 
 36. Rankin, I.: Phil. Trans. Roy. Soc. (London), vol. 160, 1870. p. 277. 
 
INACA TM 1399 177 
 
 37. Durand: Aerodynamics. Vol. Ill, 1939. 
 
 |i38. Prandtl, L.: Phys . Zs . , vol. 8, 1907, p. 23. 
 
 i39. Prandtl, L.: Schrift. d. Deutsch. Akad. d. Luftfahrtforsch. , Heft 7, 
 \ 1937. 
 
 i 
 
 |40. Eksklagon, E.: Acoustics of Guns and Shells. Izd . Voennotekhn. 
 akad. RKKA, L. 1929. 
 
 I 
 
 i Chapter IV 
 
 I 
 
 ijll. Prandtl, L., and Tietjens, 0. S.: Hydro- und Aeromechanik. Vol. 2, 
 1935. 
 
 '42. Loitsianskii, L. G. : Boundary Layer . 
 
 43. Boltze, E.: Dissert., Gottingen, 1908. 
 
 1 44. Strouhal, V.: Wied. Ann. vol. 5, 1878, p. 216. 
 
 !45. Lehnert, R.: Phys. Zs . , vol. 38, 1937, p. 476. 
 
 j46. Holle, W.: Akust. Zs . , vol. 3, 1938, p. 321. 
 
 j47. Undin, E. Y.: Investigation of Vortex Sound. CAHI Report, 1942. 
 
 I 48. Karman, T., and Rubach, U. : Phys. Zs . vol. 13, 1912, p. 49. 
 
 149. Kochin, N. E., Kibel, I. A., and Roze, N. V.: Theoretical Hydro- 
 
 I mechanics. Part 1. ONTI, 1937. 
 
 I 
 
 ! 50. Strelkov, S. P.: CAHI Report. 
 
 51. Blokhlntsev, D. : ZhTF, vol. XV, 1945, p. 63. 
 
 52. Gutin, Y. L,: ZhTF, vol. VII, 1937, p. 1096. 
 
 ■ ■ Chapter V 
 
 53. Crandall: Acoustics. 
 
 54. Blokhintsev, D. : ZhTF, vol. XII, 1942, p. 484. 
 
 55. Blokhintsev, D.: DAN, vol. XLVII, 1945, p. 22. 
 
178 
 
 NACA TM 1399 
 
 0) 
 
 60 
 
 (D 
 
 s 
 
 •H 
 
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 179 
 
 m 
 
 
 H 
 
 
 . 
 
 
 ^^ 
 
 ^^^ 
 
 'Z:, 
 
 \ 
 
 -/v 
 
 \ 
 
 •^1 31 
 
 i'T:il'ti 
 
 ^^.N ^. 
 
 u 
 * 
 
 Cm 
 
 ^^^\^ 
 
 » 1 B 
 
 ^\\ \ 
 
 \ V, \ 
 
 \\ * 
 
 \\ ' 
 
 v^ 
 
 v\ 
 
 \, \ 
 
 \ \ 
 
 V\ 
 
 \ 
 
 QJ 
 
 6- 
 
 o © 
 
 0:1 u 
 
180 
 
 MCA TM 1399 
 
 I i 1 1 11 Sound shadow ib 
 
 v}}h>yM .:. : >,^.rT>7r 
 
 //J, km' 
 
 90^ 
 
 Figure 9. 
 
 Weakening of wind 
 
 Strengthening 
 of wind 
 
 Q 
 S 
 
 Wind 
 *^ 
 
 6 
 
 M 
 
 Figure 11. 
 
MCA TM 1399 
 
 181 
 
 !"/ 
 
 Figure 12. 
 
 f 
 
 ^«=yP* 
 
 Figure 13. 
 
182 
 
 MCA TM 1399 
 
 ff ■=■ constant 
 lip 
 
 tr ^Q , g : 
 
 Figure 14 , 
 
 Figure 16. 
 
 -^/i 
 
 Z 
 
 
 Figure 17. 
 
MCA TM 1399 
 
 183 
 
 I 
 
 Figure 18. 
 
 n^i%i^- 
 
 e = 
 
 Figure 20. 
 
184 
 
 MCA TM 1399 
 
 II 
 
 CM 
 00 
 
 CD 
 
 s 
 
 •H 
 
 m 
 
 s 
 
 •H 
 
M.CA TM 1399 
 
 185 
 
 « 
 
 Figure 25. 
 
 Figure 26. 
 
186 
 
 MCA TM 1399 
 
 Figure 27. 
 
 B 
 
 ^(J sec 
 
 Figure 28. 
 
 N 
 
MCA TM 1399 
 
 187 
 
 • _r • 
 
 r-2: 
 
 i 
 I 
 
 i 
 
 f 
 
 t 
 
 i-3 : 
 
 I 
 j 
 
 Figure 29. 
 
 Figure 30. 
 
188 
 
 MCA TM 1399 
 
 a 
 
 ■H 
 
 &4 
 
 •H 
 
MCA TM 1399 
 
 189 
 
 Figure 35. 
 
 Figure 36. 
 
190 
 
 MCA TM 1399 
 
 /jy^hz 
 
 Circular profile 5 mm diam. 
 
 3m hz 
 
 (-SO 
 
 -40 
 
 -3D 
 
 -20 
 
 ~W 
 
 
 I I I I I I I ) ri I I I I I I I I I I I I I I I I I 1 I 1 I I I -I 1 ^ • « 1 I I « I I I f 1 
 
 I \ \ 
 
 sn§ 
 
 mo 
 
 mo 
 
 zm 
 
 hz 
 Figure 37. 
 
 lOuU 
 
 SO 100 150 200 
 
 V, m/sec 
 Figure 38. 
 
 Figure 39. 
 
NACA TM 1399 
 
 191 
 
 
 SV' 
 
 .-^ 
 
 6 
 
 cS 
 
 u 
 
 ^ 
 
 c 
 
 ^ 
 
 
 JO 
 
 to 
 
 0) 
 
 CQ 
 
192 
 
 NACA TM 1399 
 
 7" 'i 
 
NACA TM 1399 
 
 193 
 
 Figure 49. 
 
 Figure 51. 
 
 .• Screeil 
 
 b 
 
 a = b = 1-10" cm 
 p = 2-10'^cm 
 
 20 40 60 80 m , 120 q 
 
 Q, cm /sec 
 Figure 50. 
 
 
 Figure 52. 
 
 Figure 53. 
 
194 
 
 MCA TM 1399 
 
 in 
 
 LO 
 
 (D 
 U 
 3 
 bD 
 •H 
 
 > 
 
 tlli 
 
 
 m 
 
 -p 
 
 
 l+H 
 
 
 
 
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 o 
 
 
 
 S 
 
 
 CVl 
 
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 NACA - Langley Field. Va. 
 
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