Multiple scattering of sound by turbulence and other inhomogeneities

Journal of Sound and Vibration (1973) 27(4), 455-476

MULTIPLE SCATTERING OF SOUND BY TURBULENCE AND OTHER INHOMOGENEITIESt M. S. HOWE Engineering Department, University Cambridge, England

0/ Cambridge,

(Received 24 January 1973)

This paper describes a multiple scattering theory of the propagation of sound through an inhomogeneous atmosphere. Two cases are considered separately. The first is that of propagation in an extensive region of low Mach number turbulence, and the second is that of propagation in an atmosphere in which the speed of sound varies randomly owing to fluctuations in temperature. Kinetic equations for the diffusion of energy in wavenumber space are derived. When account is taken of the relatively slow temporal evolution of the turbulence or temperature fields, these equations are shown to be unsymmetric in the energy exchange integrals, and indicate that there is a net transfer of energy between the sound fieldand the medium . Thekineticequations are analysed for the case ofwaves longcorupared to the correlation length of the inhomogeneities and for the important case of the propagation of high frequency waves. Here the three principal conclusions relate to (i) absorption of sound by the turbulence, (ii) spectral broadening of the acoustic spectrum due to interaction with a temporally evolving field, and (iii) scattering by spatial inhomogeneities. The analysis constitutes what is essentially a scauering theory ; nevertheless, at very high frequencies results such as (ii) and (iii) are shown to agree precisely with predictions based on the theory of geometr ical acoustics.

1. INTRODUCTION When sound propagates through a medium whose physical properties vary randomly in space an initially coherent signal is gradually distorted and scattered by the inhomogeneities. If the dimensions of the region occupied by these inhomogeneities are small compared with a certain length scale determined by the wavelength of the incident sound, the correlation scale of the inhomogeneities, and the relative magnitude of the inhomogeneities, it is generally possible to analyze the interaction in terms of a single scattering approximation (i.e., the Born approximation). In the opposite limit, however, the scattering region can be so large that the original coherence of the wave form is completely destroyed by passage through the inhomogeneous region. In th is case multiple scattering of waves is certainly an important issue, and in general an approach based on the Born approximation cannot be expected to give even a rough estimate of the properties of the scattered field . Several rather more sophisticated approximate schemes have been devised for treating such problems. They are based on the supposition that only mean properties of the field are relevant for analytic investigation. For example, the coherent component of the wave field, defined as an average of the wave field taken over an ensemble of statistically equivalent random media, can be studied by means of a renormalized wave equation [I, 2]. Here the effects on the coherent wave of the random properties of the medium are taken into account

t Presented at the Euromech 34 Conference,

G~ttingen.

455

West Germany 4-6 October 1972.

456

M. S. HOWE

by means of a modified non-random form of the original wave equation. This equation contains terms accounting for dissipation due to scattering, and wave speed reduction due to random "buffeting" by the inhomogeneities. In many situations, however, especially those in which the wavelength is short compared to the correlation scale of the inhomogeneities, deductions based merely on a knowledge of the evolution of the mean field can give a completely false impression of the state of the system. This is because the decay in amplitude of the coherent wave can be due almost entirely to phase interference between the different statistical realizations of the medium. For this reason it is often preferable to study the mean square properties of the field amplitude which is phase independent. There exists a general analytical scheme for examining the evolution of the mean square amplitude, and involves the use of the Bethe-Salpeter equation [3]. That equation describes the propagation of correlations within a random medium, and is discussed at some length by Frisch [4]. This approach has been criticized [5,6] on the grounds that the Bethe-Salpeter equation appears to violate the principle of conservation of energy, at least in its lowest order approximation, which is normally the only case which can be treated analytically. Moreover that equation only reduces to an energy transport equation in the limit of wavelength large compared to the correlation scale of the inhomogeneities, which specifically excludes the limit of geometrical acoustics. Actually Rybak [7] has applied the BetheSalpeter formalism to the propagation of long waves along a randomly inhomogeneous flexible beam but used an incorrect form of the Bethe-Salpeter equation, so that the apparent agreement with energy conservation was purely fortuitous. An amusing pro babilistic approach has also been developed by Kuttruff [8], who applies his results to calculate, for example, the acoustic reverberation time of a forest! But the analysis here lacks the simplicity and generality of the Bethe-Salpeter theory. Practically all recent work on wave propagation in random media has been confined to problems involving inhomogeneities which vary in space alone, or to situations in which it is permissible to assume that temporal variations are negligible during the time of passage of an incident wave. Thus in the case of the propagation ofsound through turbulence it has generally been assumed that the turbulence isfrozen for the duration of the interaction. This case has been analysed by several authors from the point of view of single scattering theory [9-13]. In the limit of high frequency such theories predict strongforward scatter and are incapable of distinguishing between the relatively weak scattered sound (wavenumber rotation effects) and the effect of phase shift. Crow [14] has proposed a procedure to effect this separation, but again his theory is still of limited validity because of the single scattering approximation. Modified ray theories are discussed by Chernov [15] and by Tatarski [16]. More recently Plotkin and George [17] and the author [18] have attempted to analyse the effects of extensive regions of frozen turbulence on finite amplitude sound waves, with particular application to the propagation of shock waves. Physically the assumption of frozen turbulence simplifies the theory because it restricts all the scattered waves to be of the same frequency as the incident wave; the presence of temporal fluctuations actually tends to broaden the spectrum of frequencies, a phen omen on essentially distinct from scattering. It is often convenient to think of the effect on a given wavenumber component of the incident field as follows. There will be a tendency for the inhomogeneities of the medium to rotate the vector from its original direction; this scattering is practically independent of the temporal fluctuations of the medium. The latter introduce a length dilatation of the wavenumber vector, however, an effect which could occur even when the spatial properties of the medium are uniform. Of course the situation in a turbulent medium is more complicated than this because it is well known [19] that the turbulent fluctuations themselves give rise to an acoustic field. This

MULTIPLE SCATIERING OF SOUND

457

generation of sound by turbulence naturally leads one to enquire into the possibility of the inverse process occurring, and this aspect of the problem has been examined heuristically by Crow [20], who proposes a viscoelastic theory applicable to fields whose wavelengths greatly exceed the correlation scale of the turbulent motions. Actually Crow's theory does not appear to take complete account of multiple scattering so that it probably needs to be modified in order that it can be considered in the context of aerodynamic noise theory. The heuristic step in Crow's analysis enters because the interaction terms responsible for the transfer of energy between the turbulent fluctuations and the acoustic field are third order, and it is difficult to see how they can be incorporated into a mathematically tractable description of the process. Crow introduces "memory" functions at this stage. This emphasizes the difficulties encountered in trying to deal with the scattering of sound whose wavelength is comparable with that of the "self-sound" of the turbulence. There are many instances, however, in which the incident sound is both more intense and of much shorter wavelength than that characteristic of the self-sound of the turbulence. The energy spectrum of the acoustic field then consists of two widely separated and weakly coupled spectral bands, one occupied by the incident field and its scattered and diffused products, and the other by the aerodynamic sound. This wide separation both in frequency and intensity generally permits one to analyse separately their interactions with the essentially incompressible turbulent motions. The present paper sets out to examine in detail the multiple scattering of such an incident sound field with turbulent fluctuations of the medium. The theory will also be considered for cases where the fluctuations of the medium are caused by variations in temperature in space and time, producing random fluctuations in the speed of sound. The analysis involves the derivation of a kinetic equation governing the distribution of energy in wavenumber space brought about by multiple scattering of the incident sound. The method is not subject to the limitations associated with the use of the Bethe-Salpeter equation discussed above, but on the contrary apparently leads to a unified theory of scattering valid over the whole spectrum of incident wavelengths. It has already been noted that classical scattering theory (see, for example, references [9-13]) ceases to be valid at high frequencies (when the incident wave actually propagates according to geometrical acoustics): that theory predicts an increasingly large intensity of forward scatter as the wavelength diminishes. This forward scatter is the "phase shift" mentioned above and is the manifestation of the variable velocity ofpropagation of the sound wave caused by local convective effects and variations in the speed of sound. Such singular behaviour will be seen to be absent from the present approach and moreover predictions regarding scattering and spectral broadening are found to agree precisely with corresponding results derived on the basis of geometrical acoustics. Similarly at the opposite extreme of wavelength large compared with the correlation scale of the turbulence, the present method recovers the well known results of single scattering theory. It is hoped that this asymptotic agreement of the theory with well established results will give confidence in the basic idea underlying the present discussion and stimulate applications to problems involving scattering at intermediate wavelengths. In section 2 certain well known results in the theory of sound propagation through random media are recalled, in particular the equations describing scattering by turbulent velocity fluctuations and by fluctuations in the speed of sound. This is followed in section 3 by a detailed derivation of the kinetic equations. The equations for time dependent fluctuations of the medium are unsymmetric, and indicate that the interaction ofsound with the medium involves a net transfer of energy (section 4). It is shown in section 5 that at high frequency the effect of a turbulent medium is essentially threefold: (i) there is a net transfer ofenergy from the acoustic field to the turbulence; (ii) temporal fluctuations of the turbulence produce spectral broadening of the acoustic field; (iii) spatial fluctuations are responsible for scattering. In the limit of very

458

M. S. HOWE

high frequency conclusions (ii) and (iii) are shown to agree exactly with those of geometrical acoustics (sections 6 and 7). 2. THE EQUATIONS OF MOTION

We begin by recalling the basic form of the wave equation appropriate to sound propagation through an inhomogeneous atmosphere. The most convenient starting point is to write the equation for the perturbation density p in Lighthill's form [19]. If V, is the fluid velocity and a is the speed of sound in the uniform quiescent medium, then Lighthill shows that the exact inviscid equation of motion can be expressed in the form

a ot

2

2

I 1 0 ) -2 -2- \72 P = -2 - - ( p Vr Vj)

(a

a aXr OXj

I +\72(p 2

a

a2 p).

(I)

Here x, (i = 1,2,3) is the space coordinate referred to rectangular Cartesian axes, t is the time; repeated suffixes imply summation, and \72 = oZlox, ax,. It will be assumed that variations in pressure p and density are related by all adiabatic law of the form pip1 = constant. This means that the second term on the right of equation (I) is only significant when the speed of sound a in the quiescent atmosphere differs from that in the turbulence: i.e., when the temperature of the atmosphere is not uniform. Such temperature variations may be produced, for example, by uneven convective heating from the ground. Actually it is conven ient to cons ider the two cases separately. We shall examine first the effect of turbulent velocity fluctuations alone, the temperature being uniform and the second term on the right absent. Then effects of thermal scattering are considered, in which the second term 011 the right is more impor tant. Thus let us set (2)

where til is the turbulent velocity in the absence of an incident acoustic field, and V, is the velocity perturbation associated with the presence of the sound. If the undisturbed fluid density is Po, and is assumed to be independent of position and time, then with pi = P - Po, equation (I) becomes correct to first order in pi and VI : 1

a ot 2

)

_ _ - "ij2

( a2

2

2p 02 pi = _ 0 _ _ (Ul VJ) a Z OX,OXj

U a pi p a + U_I_j - _ +~ - - (UI Uj) Z 2 2

a

aX l aXJ

2

a

aX I OXj

(3) .

In this equation the third-order term u, Uj pi has been approximated by tI, tlj' pi, where an overbar denotes an ensemble average. This retains an effect on the acoustic field which is correct to second order in the fluctuations of the medium. The first term on the right of equation (3) describes the interaction between the turbulent velocity fluctuations and those induced by the acoustic field. The remaining term on the right of equation (3) is essent ially the Lighthill [19] quadrupole SOurce of aerodynamic noise . Let U denote the root mean square turbulent velocity, and I a typical correlation scale in the turbulence; then it is known that this aerodynamic self-sound has a frequency ~UII. It will be assumed that this frequency and the corresponding acoustic intensity are both small in relation, respectively, to those associated with the incident sound . The wavelength of the aerodynamic noise is actually of order 11M, where M = U]a, which is very much greater than I for the small values of M encountered in the atmosphere. For example, in the atmospheric boundary layer M rarely exceeds 0·5 x lO- z. Thus the incident acoustic field and the self sound of the turbulent motions occupy widely separated spectral regions in the theory to be discussed here. The interaction between the two sound fields is

459

MULTIPLE SCATTERING OF SOUND

contained in the first term on the right of equation (3), but here it is the incompressible, rotational component of UI which is significant, and which dominates the energy exchange processes. For this reason the presence of the aerodynamic sound will be ignored, and the following equation taken as the starting point of the discussion: 2 I 2;-ZV' (

a

a ot

2 2) P =-2 2po a2 UtUJ a p -;--;-(UtVJ) +-2 - a a '

a

UXI

ax,

a

(4)

XI XJ

where the prime on the density perturbation has been dropped. Next suppose that the inhomogeneities in the properties of the medium are due to random variations in the temperature. Letpo and Po denote, respectively, the pressure and density in the absence of the incident sound. These quantities now vary randomly with position and time, and if c denotes the local speed of sound then

" ' ='YPo c -,

(5)

Po where j is the ratio of the specific heats of the atmosphere. Hence, assuming the pressure and density perturbations to be related by the adiabatic law plpo = (pi Po)~ during the passage of a sound wave, we have, to first order in the acoustic field,

(6) Substituting into equation (I) and subtracting out the steady-state terms which are independent of the incident sound yields, in the absence of turbulence scattering, 2

-1 -a, - V'2 ) P = V'2(ep), ( a2

at-

(7)

e

where = c21a2 - I is a random function of position and time describing the fluctuations in the speed of sound. Again the prime has been dropped from the density perturbation. It will be assumed that the fluctuations represented by ~(x, t) have zero mean, and constitute a stationary random process. A consequence of this is the existence of a correlation function, R(x, t), which is an even function of each of its arguments, and satisfies R(x - X, t - T)

=

e(x, r) e(X, T),

(8)

the overbar denoting an ensemble average. The space-time Fourier transform of the correlation function, i.e., the spectrum of the fluctuations, is then defined by

JJ

OJ

4>(k w) ,

=

_1_ (2n)4-

R(x t) e-/(Ic,x-wr) dx dr. '

(9)

-OJ

This is an even function of the wavenumber vector k and the frequency w, and can also be shown to be non-negative [21]. Similar assumptions will be made regarding the turbulent velocity fluctuations, and so, in an obvious notation, (10)

and 4> (k I),

'" W)=_I_JJR (271f -00

(x t)e-H,,·x-Wrldxdt. l)'

(11)

460

M, S. HOWE

Actually, it will be found convenient to assume further that the turbulent fluctuations are homogeneous and isotropic, so that, since these motions are essentially incompressible for sufficiently small turbulent Mach number M, (PIJ(k, w)

E(k, w) = --

4nk

4

2

[k OIJ - k, k)],

(12)

where E(k, OJ) is the energy spectrum function of the turbulence [23]. When the time scale associated with temporal variations in the turbulent motions is large on a scale determined by the incident sound, the energy spectrum function E(k,w) is strongly peaked at OJ = 0 with a band width -U/l. Similar remarks apply to the relatively slow temporal evolution of the thermal fluctuation spectrum given by equation (9). 3. THE KINETIC EQUATIONS Before embarking on a detailed analysis of the scattering equations (4) and (7), it is necessary to decide precisely what kind of information is required for a useful description of the wave field in the random atmosphere. It has been argued above that a knowledge of the mean square acoustic field is likely to be of particular importance, since it gives a direct measure of the acoustic intensity. Let us suppose that the sound field can be expressed as a sum of plane wave modes of the form P = L h(k., WN) e/("n'>:-WHt), (13) n.N

where k_" = -k", OJ_N = -OJ". h*(k",OJN) = h(-k/,,-OJN), and, initially at least, no restriction is placed on the values of'k, and WN' Introduce a dimensionless parameter e, say, to represent the order of magnitude of the random inhomogeneities; for example, in the case of thermal scattering described by equation (7), s will be proportional to ~. When 6 vanishes the medium is no longer random, and the wave modes (13) propagate according to the unperturbed wave equation, The introduction of randomness, however, leads to a redistribution of energy amongst these modes because of the continuous interactions with the inhomogeneities. Experience in the theory of the propagation of the coherent component of the wave field (see, e.g., reference [5]) suggests that the distances and times over which significant changes in the Fourier amplitudes h(k",w N ) occur are of order 1/62 • This implies that the effects of the inhomogeneities may be taken into account by introducing suitable "slow" variables X = e2x, T= e2 t, say. Now the representation (1) of the field in the random medium actually contains more Fourier components than would be necessary to describe propagation infree space. Mathematically this is because the eigenfunctions in free space are not complete with respect to the random medium. Physically it is because when a wave interacts with an inhomogeneity local, non-propagating disturbances are generated which have no representation in terms of the free space eigenfunctions. These local variations in the wave field are really contained in that part of the Fourier decomposition which is correlated with the scatterer. It is natural therefore to attempt to re-write the decomposition (13) in the form P=

~

L.

/I,N

h(k

"'

W

N

) el("n'>:-WHI)

+

~

L n.N

h'(k OJ ) el)"n'>:-WNl) m

N

,

(14)

where the first summation involves Fourier components which are locally statistically independent of the scattering inhomogeneities. The second summation contains those components which are correlated with the inhomogeneities, and thus takes account of the

MULTIPLE SCATTERING OF SOUND

461

local interaction of the acoustic field with these inhomogeneities. This decomposition will be made more precise in due course. But it is easy to see that, since the interaction between the sound and the random fluctuations of the medium must vanish as 6 -)- 0, then h'(k",w,...) -)-

°

as

8-+0,

while liCk", WN) remains finite . In other words,for sufficiently small 6, h'(k., W,v) ~ h(k.. WN) '

(I 5)

Of course the smallness of the local components h'(k", wo\') does not imply that their effect is negligible . On the contrary, in an extensive random medium cumulative interactions can result in a dramatic change in the large scale properties of the sound field. These preliminary remarks can be clarified as follows . We partition the medium into a set of rectangular parallelopiped "cells". The dimensions of each cell depend on the value of 8 and will be specified more precisely below, but they are assumed to be small on a scale -O(l/e 2 ) . We also associate a time interval fI with each cell which is also small on a scale -0(1/e 2 ) but which tends to infinity as e ~... 0: e.g., fI - O(1/e). In each cell and over the time interval fI it is then possible to express the various field variables exactly as Fourier series such as equation C13). The wavenumber vectors k, are now determined by the dimensions of the cell (i.e., the Fourier components must "fit into" the cell) and the W N by the time interval fl. Now of the Fourier components of the field within the cell some will be correlated with the inhomogeneities in the cell and some will be statistically independent We can therefore partition the Fourier coefficient hCk n,WN) by averaging it over an ensemble of realizations of the cell as distinct from an ensem ble of realizations of the whole random medium. Naturally each realization of the cell will also affect conditions outside of that cell, but only over distances of the order of the correlation scale of the inhomogeneities; beyond that conditions in the medium are statistically independent of the ensemble of realizations. Thus the representation (14) can be made precise by defining n(k", WN) as the average value of h(k,u WN) taken over an ensemble of realizations of the cell in question. Then h'(k",WN) is given by (16)

and has zero mean. The field described by the locally averaged Fourier components is expected to vary slowly over distances and times of order 1/£2, and to describe the physically important characteristics of the sound. The random components describe the details of the weak interaction of the locally averaged field with the cell, and vary with each realization of the cell. It will be necessary to determine these random Fourier coefficients h'Ck,1l WN) in terms of the hCk",wN) and of the inhomogeneities of the cell. A method involving the so-called local Born approximation will be described below. The validity of this approximation can generally be shown to requ ire that £2 V ~ I, where V is a suitable non -dimensional measure of the volume of the cell 14]. Th is imposes a limitation on the size of the cell. However, one of our objectives is to arrive at a continuum representation of the acoustic field and this requires that the dimensions of each cell tend to infinity asymptotically as e -+ 0, so that the discrete Fourier series representations pass over to Fourier integrals. Both requirements can clearly be satisfied by taking the diameter of each cell to be of order 1/e I / 3 , say, in which case the magnitude of a locally random Fourier coefficient is guaranteed to be of order e times that of the locally averaged coefficients, and therefore negligible as 8 -+ O. Let us now derive expressions for the random Fourier components of the sound field . We shall confine attention to thermal scattering, governed by equation (7); the details for the

462

M. S. HOWE

more complicated case of turbulent scattering are similar but more cumbersome. The corresponding results for turbulence are presented at the end of this section . Consider a typical cell C. say, and denote the ensemble average ofthe density perturbation p over C-the "cell average"-by p and the corresponding random component by p' . Then taking the cell average of equation (7) for points inside C gives 1 a2. ) - - - V2. p- = V2(ep') ( a2 Bt2 '

(17)

since ~ =' O. Subtracting this from equation (7) gives (18)

This equation describes the generation of the locally random field p' by means of the interaction between the temperature inhomogeneities specified by and the cell averaged field p. The second term on the right of equation (18) represents effects of multiple scattering within C and also interactions between component waves of p' correlated with C but initially scattered at points outside of C . The latter arise from regions lying within a correlation scale of the inhomogeneities from the boundary of C, and give a contribution which is negligible compared with that from C itself as the dimensions of the cell increase (i.e., as s -+ 0). Similarly the dimensions of the cell have been chosen in such a manner that the effects of multiple scattering within C are negligibly small. This means that the following local Born approximation should be adequate for determining the locally random field inside C :

e

(19) Now, within the cell C,

p = 2: n.N

h(k no CON) el(k.·"'-"'Nf),

(20) and also the random inhomogeneities can be Fourier analysed in C, the result being expressed in the form e(x. 1) = L e(kl/' cuo) el(k. "x-OJ,,!) Llk Llco. (21) q.O

In this expression the interval zlk between the wavenumber components k q is determined by the dimensions of the cell . Similarly, the interval Llco is fixed by the time interval ?7 of the Fourier time analysis. As e -+ 0 equation (21) passes over to a Fourier integral representation of~. and in particular ~(k.co) becomes the (generalized) Fourier transform of ~(x. t). In this limit it is known that the ensemble average (;(k p • w p ) e(k q , coo) £lk £lco -+ iP(kq , coo) Ok p • -k; o"'P' - WQ'

(22)

where
h'(k no CON

""" - k~ eCk q • coQ) Mk n - kill CON - COo) Llk£lco LII,Q [k 2 (CON +- :iO)2] • -_ "

a2

(23)

463

MULTIPLE SCATTERING OF SOUND

Here the notation CON + iO implies that CON has been given a small positive imaginary component to avoid the zeros of the denominators. This ensures that the causality condition is satisfied in the appropriate manner [22]. These preliminary results will now be used to derive a kinetic equation governing the energy exchange processes among the Fourier coefficients h(k,,,w,v). The method is based on the observation that in order to derive an energy equation for a wave system satisfying equation (7), say, one could proceed by first multiplying that equation by ap/at. The result could then be rearranged to form a time derivative of the energy density plus a divergence term giving the energy flux, together with terms representing the interaction of the wave system with an external source or sink of energy. We are actually interested in the energy content of individual Fourier components of the sound field, and this suggests that the procedure be modified slightly by multiplying the wave equation simply by the time derivative of a single Fourier component as distinct from the whole field. To do this first substitute the expansions (13), with h(k", CON) expressed in terms of equation (16), and (21) into the wave equation (7), viz., (7)

Note that when applying the differential operators to the Fourier expansion of p, the Fourier coefficientsdepend on the "slow" variables X = e2 x, and T = e2 t, describing the evolution of these coefficients over distances and times which are large compared with those characteristic of the cell, and this can be taken account of by formally substituting

a

a

ax

ax ax at

a a

a

a

-~-+-;-~-+-.

at aT

Next multiply equation (7) by (24)

where the notation c.c. denotes the complex conjugate of the preceding quantity. This is real and proportional to the local time derivative of the Fourier component. Take the statistical cell average, and then average over the volume of the cell and the time interval!T by integrating with respect to x and t and dividing by the cell volume and!T. Finally average with respect to an ensemble of realizations of the whole random medium. The complete averaging procedure will be denoted by the angle brackets O. To carry out this programme consider first the left-hand side of equation (7). Multiplying by expression (24) we have to determine I, say, where 1= ({-ih(kn , roN)el(k.'''-WN') + c.c.} X

io)2 [(k.. - oX

I (.

a2

COM

io + aT

2:

e/(k m '"-w,,,t)

m.M

)2] h(k

>

n" COM)

•

(25)

The space and time average ensures that the only contributions from the summation come from the terms kill = ±k,,, ())M = ±WN, since the Fourier components are mutually orthogonal. Also since h, h' vary only over scales ~O(l/e2), the term in square brackets in equation (25) may be expanded in powers of the derivatives and only the first non-trivial (0(8 2 ) ) terms retained. This gives (26)

464

M. S. HOWE

Now is a measure of the energy density of the corresponding Fourier component, As e --+ 0 this must pass over to a continuum representation, and we therefore write

(27)

Now turn attention to the interaction term on the right of equation (7). Substituting the formal Fourier expansions, multiplying by (24) and averaging gives JI:

II = ({-ih(k., CON) el(kn'z-wNl)

+ c.c.} 2: s,»

(k, + k.,}2 ~(kp, COp) h(k," COM)

m,M

(29)

Note that in this case the derivatives of the Fourier coefficients have been neglected since in the final result these would introduce contributions which are uniformly of order 6 2 smaller than the terms retained. Denoting the cell average by an overbar we have: II =

(2: ik~ h(k.. CON) ~(kp, wp) h(-kn p,p

k p, -CON - COp) LlkLlw

+ c.c.),

(30)

Now

=

h'(k,,, CON) ~(kp, (,(,lp)·h(-k. - kp,-WN - Wp)

+ h(k,,, CON)' e(kp,COp) h'(-k" -

k p , -CON - COp) (3 I)

and the first two terms on the right are of order e2 , whereas by construction the third is uniformly of 0(6 3 ) and may be neglected. Thus we have

II =


+L

p, p

WN) e(kp, cop) h(-k. - k p, -CON - COp) zlk Llw

ik~ h(kn , CON) e(k p,Wp) h'(-k. -

+

kp,-co N - COp) zlk Llw

+ c.c.),

(32)

and these interaction terms can be determined by substituting for the locally random Fourier coefficients from equation (23). To illustrate this consider the details for the first term on the right of equation (32), JIll say. Using the result (23) we have 111 =

465

MULTIPLE SCATTERING OF SOUND

°

But as e -+ the dimensions of the cell and the time interval fT ultimately exceed, respectively, the correlation length and correlation time of the inhomogeneities, and therefore in this asymptotic limit the cell average ';(k p , wp)';(kq, w Q ) may be replaced by the ensemble average with respect to the whole random medium. This enables us to use the result (22) to simplify the above expression to give _ "" -ik~ cP(k q , wQ)
q. Q

_

(w N

+1 a2

•

Now we have already seen that h'(k, w) :: O(e)h(k, w), so that if <[h(k,w)12 ) is replaced by

<[h(k, wW>

=

([h(k, wW>

+

the resulting error is less than that already introduced by the neglect of the third term on the right of equation (31). Hence asymptotically as s -+ 0, and using equation (27), we have

2:

III =

-ik~ cP(k", wQ) ,B(k" - k q , WN - wQ)(Llk)2(Llw)2 X

q, Q

(34)

It is an easy matter to show that the term in curly brackets in this expression is just equal to 21tisgn (WN) <5

(k;,- :~)

so that equation (34) assumes the concise form III =

2:

2n sgn (WN)

q,Q

k~ cP(k

q,

wQ) ,B(k" - k q , WN - wQ) () (k7. _

In anticipation of the continuum limit obtaining as

II Ii

111 = 21tAkLlw

sgn

(wN)k~cP(k. -

B -+

W:) (Llk)2(Llw)2, a

0, we may formally write

K, WN - Q)P(K, Q)15

(k~ - :~)dK dQ.

(35)

Similarly the second term on the right of equation (32) can be evaluated to give liz

=

-2n Ak LlW

sgn (Q)

k~ K

2

cP(k. - K, WN - Q) ,B(k.,WN) s ( K 2

-

~:) dK dQ.

(36)

Hence since I = III + II2 , and with the suffixes 11 and N dropped, the results (28), (35) and (36) imply the following continuum kinetic equation for the mean square Fourier amplitude P(k, w):

ap

a2

op

.aT(k,w) +-;;-k' ax(k,w) =

n;z

If

k 2 cP(k - K, tu

-Q){sgn (w) k ,B(K, Q) (k 2

<5

2

-

::) -

-ex>

- sgn (Q)

K2 f3(k, w) 15 (K ~:)) dK dQ. 2

-

(37)

466

M. S. HOWE

This is the desired integra-differential equation describing the evolution ofthe mean square Fourier components of the acoustic field asymptotically as I: -'>- O. Before discussing the implications of equation (37) we present the corresponding results for scattering by turbulent velocity fluctuations described by equation (4). Actually the situation is not quite as straightforward since equation (4) contains the acoustically induced velocity VI as well as the density perturbation p. These quantities are related by the linearized momentum equation

ov, QV, a 2 op -+Uj-=---' at OXj Po ax,

(38)

and this can be used to relate the Fourier components of v, and p in an iterative scheme, and so enable the kinetic equation for p to be derived in the manner described above. Carrying through the analysis then leads to the following kinetic equation for (J(k, w):

op

- ( k w)

aT'

a2

op

(J)

oX

+ -k· -(k

'

w)

2)}dKdQ.

__ 1 P(k W)O(K 2 1.01'

-

Q a

(39)

2

This describes the energy exchange processes between the Fourier components of the acoustic field due to interactions with the turbulent velocity fluctuations.

4. SOME IMPLICATIONS OF THE KINETIC EQUATIONS

The kinetic equations (37) and (38) are integra-differential equations for the spectrum, P(k,w), of the Fourier coefficients of the acoustic field. It is clear from the form of the equations that on an average the energy containing wave modes correspond precisely to those values ofk and (J) which satisfy w2 = a'k". This is perhaps an obvious result, but it is noteworthy that it has not been assumed in the analysis leading to the equations. Thus it is permissible to set (40)

When this is substituted into the kinetic equations they can then be integrated with respect to (J) over a small interval about w = -i-ak, say. No loss in generality results from this since P(k,w) is clearly an even function of w. This leads to the following reduced forms for the kinetic equations: (i) thermal scattering

Ot!

atff

er

ax

-(k) +ak'-(k)

=

nak

II l/J(k - K, ak: eo

-e

co

2

.0 ) dK ciQ; Q)(kl e(K) - sgn (.0) K2 et(k)] () ( K 2 - 2" a

(41 )

467

MULTIPLE SCATTERING OF SOUND

(ii) turbulent scattering

aff

er =

08 + ak·-(k) A

-(k)

4n

ak 2

ax

JOOJ "J(2k,k (k- K)2 ( 2 _ j
Q2) a dKdQ, 2

(42)

-00

where J{ = k/k. Now when the random properties of the acoustic medium are time dependent there is no reason to suppose that there should not be an interchange of energy between the sound field and the agency responsible for the temporal fluctuations, e.g., the incompressible turbulent motions. This is because it is not possible to derive an acoustic energy conservation equation from wave equations such as (4) and (7); in other words, it cannot be assumed that the sound field evolves as a closed, conservative system. These statements are, of course, reflected in the forms of the corresponding kinetic equations (41) and (42). To see this first suppose that the inhomogeneities of the medium do not depend on time. F or exam ple, in the case of thermal scatteri ng the spectrum functi on 4J(k, OJ) of the tern perature inhomogeneities then has a delta function dependence on frequency: i.e.,
(43)

say, corresponding to an infinitely long correlation time. Substituting this expression into equation (41) and performing the trivial integration over Q gives

a8'

a8

A

-(k) + ak'-(k) aT ax

J x

na

=

k 3 lfI(k - K)[6"(K) - r1'(k)] D(F - K 2 ) dK.

(44)

-:
As expected this now shows that frequency is conserved on scattering, since in a time independent medium interacting waves must satisfy k = K. But the symmetry of the integrand also implies that acoustic energy is conserved. The total acoustic energy equation may be obtained by integrating (44) over all values of k; the symmetry then results in the vanishing of the double integral on the right, and this leads to the following energy conservation equation: X

fr

J

G'(k) dk + div

-C\:".

J 3,•.

ak8(k) dk = O.

(45)

-:(

Such symmetry is certainly absent in the energy exchange integrals of the time dependent problems described by equations (41) and (42), and in general there is no reason to suppose the existence of a corresponding conservation equation (cf. section 5). It is also of interest to see how the kinetic equation reproduces the results of classical scattering theory. For example, when the turbulence is assumed to be frozen the spectrum tensor

af!

A

acff

-(k) + ak·-(k)

aT

=

ax

a;3 J(k' K)2k, kj

co

-co

K)[6"(K) - 6"(k)] <5(K2 - k 2) dK.

(46)

468

M. S. HOWE

Consider the case of an incident wave packet ofwavenumber ko. Let dldt denote the derivative following the motion of this wave; then equation (46) implies that

dtB' dt (k o) =

47r

ak~

f'" (K· k

o)2 k OI k OJ cP lJ(ko - K)[0"(K) - ~(ko)] O(K2 - k6) dK.

(47)

-.., If initially all of the acoustic energy is contained in the incident wave then it might be anticipated that for sufficiently small distances of propagation

(48) for K :1= ko. When this is the case it seems appropriate, therefore, to neglect cS'(K) in the integrand of (47) in comparison with tS'(k o), and this gives deS'

dt (k o) = -ar8(ko),

(49)

where (50)

r is the fraction of the energy of the incident wave that is lost by scattering per unit length traversed by the wave packet. But this result is precisely that obtained from single scattering theory (cf. equation (48) of reference [13]). It is known to be an adequate approximation for the decay of the incident wave provided that the wavelength is long compared with the correlation scale of the turbulent fluctuations. To work out the "scattered" field in the same approximation consider again equation (46) with k:l= ko. Then for small propagation distances of the incident wave the dominant contribution to the square bracket ofthe integrand is from that part of tB'(K) due to the incident wave. Now the spectrum function II(K) of the incident wave, which must, of course, be regarded as a distribution in wavenumber space, will possess a delta function singularity at k o: i.e., (51 ) say. Then for k i' k o it can be deduced from equation (46) that the approximate formula describing the generation of the scattered field is aft

-(k)

er

atS'

+ ak'-(k) =

ax

27rlo

- ( k ' k o)2 k j k J cPfj(k - ko)fJ(k - ko).

akt

(52)

The delta function on the right of this result implies that all the scattered waves have the same frequency as the incident wave. It is clear also that the formula (52) is secular in that it predicts an unlimited growth of the scattered field. It corresponds precisely to the single scattering result of classical scattering theory (cf. the critical review given in reference [13]), and is subject to the same limitations: that is, it is only expected to be an adequate approximation to the scattered field provided that the region occupied by the turbulence is sufficiently small. When this is not the case it is necessary to solve the full kinetic equation (46). Classical scattering theory thus involves the neglect of one of the terms representing multiple scattering in the square brackets of the integrands of eq uations (41) and/or (42), and it is precisely because of this that it ceases to be valid at high frequencies. In the next section the high frequency limit wil1 be examined with the ultimate objective of establishing the identity between results derived on the basis of the kinetic equations and those of the theory of geometrical acoustics, which is known to be valid in that limit. This should give confidence in the

469

MULTIPLE SCATIERING OF SOUND

ability of the kinetic equations to describe scattering phenomena at intermediate frequencies for which there is currently no well established method of analysis. 5. PROPAGATION OF SHORT SOUND WAVES THROUGH A TURBULENT ATMOSPHERE

Here attention will be confined to the more interesting case of scattering by turbulent velocity fluctuations. A "short" sound wave is one whose wavelength is short compared to the correlation scale I, say, of the turbulent motions. If k is a typical acoustic wavenumber this will imply that kl ~ I. Now in terms of the root mean square turbulent velocity U, the time scale of the temporal fluctuations of the turbulence is of order 1/ U, which is large compared with the period of a typical acoustic wave provided that (l/U)/(I/ak) ~ 1; i.e., if Ik

-~

M

I

(53)

'

where M = Uja is the Mach number of the turbulent fluctuations. It has already been pointed out that in atmospheric applications M _10- 2 • The integral with respect to Q on the right of equation (42) can be carried out immediately by using the properties of the <5-function, which shows that the only non-trivial contributions are from Q = ±aK. The result involves E(k,ak - Q) (see equation (12» evaluated at these two points: i.e., E[k,a(k ± K)]. But in the present high frequency case the relevant values of k and Kwould be expected to satisfy the inequality (53) and, since the frequency dependence of E(k, w) must in fact be in terms of a non-dimensional variable such as cal] U (so that E(k, w) is small when w ~ U/l), this means that as a function of k ± K, E [k, aik ± K)] must be sharply peaked at k ± K = O. Thus the only significant contribution can be from the term involving E[k, a(k - K)]. This can be used to effect an approximate evaluation of the right-hand side of equation (42), which now assumes the form 2n J= P

JIX>

(k'K)2

-;(3 k l k J C/>u(k - K, a(k - K»[ktR'(K) - KtR'(k)] dK.

(54)

-al

The method of approximation is straightforward, but rather tedious, and the details will be omitted. Let edenote the angle between the vectors k and K, and adopt spherical polar coordinates with the pole along K = k. Then, by also using the definition (12), equation (54) can be expressed as

J Jde J 7C

lXl

J= P 2

271

3

2

d¢ K3 sin 9 cos eE(lk -

dK

o

0

XI, a(k -

K»)[kl(K) - Ktf(k)] .

(55)

[k-K/4

0

From equation (55) it is clear that the main contribution to the K-integral must be from the region about K=k, where E[[k-Kf,a(k-K)] is not small. Also E[lk-KI,a(k-K)]/ [k - KI4 remains finite and non-zero as k - K ~ 0 (see reference [23], p. 38), but is significant at high frequencies for small values of e only. This means that the integral can be approximated further by _k 2

J- 2

J Jde J eo

00

2n

dK

o

d¢

0

0

K"sin3 ()cos:z eE [k 9, a(k - K )][k tf(K ) - K 8 (k )]. [2k sin(O/2)]4

(56)

470

M. S. HOWE

This is now evaluated by expanding the term

KJ sin J () cos 2 ()[kt9'(K) - KtS'(k)) [2k sin (eJ2>]4

about K = k, and going to second order in the approximation. The calculation is straightforward and similar to that of LighthiII [II]. The result of this analysis is to transform the integro-differential kinetic equation (42) into the following diffusion equation for t9'(k): 0t9' 0t9' -(k)+a-a- (k) XI

aT

12u2 !

= -

- - S(k)

aD 2

2u z lk' az tS' a(j ok ~

12u2lk at9' (k) a1Jz ok II

u2 k 2

+ - z - - - (k) + - ' Vi tf(k).

+---

2aLl

(57)

In this equation the following notation has been adopted: XI is the space coordinate parallel to the wavenumber vector k; %k M denotes differentiation in the direction parallel to k, and Vi is the two-dimensional Laplacian operator in wavenumber space in the plane normal to k, Set IX) E(,,)

f

=

(58)

E(K, co) dill;

-IX)

then E(K) is the energy spectrum function of the turbulence in wavenumber space [23]. The lengths I, Ll are then defined by the relations 1= .!!2uz

0:>

f E(K) d x K

'

o

I

7l'

L1

2u

-=-2

(59)

fIX)

KE(IC)dK ,

a

where I is just the longitudinal correlation length familiar in the theory of isotropic turbulence ([23], p. 51), and u Z = tUI u.. To obtain the first three terms on the right of equation (57) evaluation of the integral S~ Z E(K,aOdC is also required . Since E(K,aO decays rapidly to zero when exceeds a small quantity of order -M, the behaviour of E(K, co) in the neighbourhood of ill = 0 can be approximated by

t

Ie

T

] ] 4

E(K, ill) = - - E(K) e- w T

2V7r

/

•

(60)

Here r is a correlation time and generally depends on the wavenumber magnitude K . However, correct to the present degree of approximation (i.e., to second order in the expansion about K = k), only the value of 1" for 1C -+ 0 is required. The length fJ is related to this correlation time by 0 = at. The first term on the right of the diffusion equation is negative and represents a net decay in acoustic energy; it does not describe decay by scattering since its integral over all wavenumber vectors k is negative definite (as distinct from zero; cf. the discussion of the previous section). Since it also vanishes when the temporal fluctuations of the medium vanish (i.e., when 0 -+
MULTIPLE SCATTERING OF SOUND

471

The remaining terms on the right of equation (57) represent the diffusion of energy in wavenumber space. The coefficient of the first depends on the temporal fluctuations, and vanishes when the turbulence is assumed to be "frozen" (~ -+ 00). This term is responsible for length dilatation of wavenumber vector: i.e., for spectral broadening. The coefficient of the second diffusion term involves only the spatial properties of the turbulence and describes scattering: i.e., rotation of vectors in wavenumber space.

6. SCATTERING OF A HIGH FREQUENCY ACOUSTIC BEAM; SPECTRAL BROADENING

The compatibility of equation (57) with results derived on the basis of geometrical acoustics can be investigated in the context of the problem of the propagation of a high frequency acoustic beam through turbulence. Consider a steady situation in which a beam of sound propagating in the positive direction of the x-axis is generated at x = 0 with initial wavenumber vector k = ko = (k o, 0, 0). Equation (57) is difficult to solve in full generality because the coefficients of the terms on the right depend on the current wavenumber magnitude k. However, for propagation over sufficiently short distances, k may be approximated by k o in these terms, and also it may be assumed that the dominant contribution to the space derivative on the left-hand side is in the direction parallel to k o. Then, in the steady state, equation (57) reduces to

12m 2 1k o Or! 2m2 /k582 t! m 2 k5 2 8x=-~r!+ 82 8k ll +~ 8k~ +2'TY'.!.tS', Or!

12m2 I

(61 )

where m = uja. N ow the effects of the spatial inhomogeneities are contained in the final term on the right of this equation, and are essentially independent of temporal effects represented by the terms involving ~. Without loss of generality, therefore, the two effects can be considered separately. Thus steady-state scattering will be associated with the diffusion equation

(62) The spectral distribution of the monochromatic acoustic beam at x = 0 must have the form tS' = lo(k.!.), where I depends only on the component k II' The appropriate solution of equation (62) is readily derived by Fourier analysis, and is r!

=

f iJ exp ( -iJkf } 2rr:m2k~X 2m 2 k 5 X '

(63)

This result illustrates the lateral diffusion of the beam about the original direction of propagation caused by multiple scattering. It can be used to determine the root mean square angular divergence of the beam at distance X: that is,

_~ 2v v =

<(p>

ko

,

(64)

approximately. To this end equation (63) is interpreted as a probability distribution which is normalized by dividing by the integral of r!(k) over all k: i.e., by f. Then (65)

472

M. S. HOWE

i.e.,

f2X

V (£]2> = 2m ,.JT ·

(66)

Thus the angular divergence ultimately becomes so large that the approximation (62) of the diffusion equation (57) is no longer adequate. Next consider the effects of the temporal fluctuations, scattering being neglected. The appropriate equation is now

12m 2 1
oB

-=--- +

ax

12m 2 Iko ss 2m21k~ a2 1! -+---, lJ2 ok I lJ2 ok;

(67)

subject to the initial condition B = glJ(k" - k o) at X = 0, where g is a function of k.L alone . Again the solution is straightforward, and the result is

= d

2 2 ex {_ 12m2 /X _ o2(k n - k o + 12m 1k o X/lJ 2mkov2rrlX p lJ2 8m2 kVX

)2 }.

glJ

(68)

This solution exhibits the decay of wave energy with distance due to absorption by the turbulence (first term in the exponential). It is independent of the wavenumber at these high frequencies, and, since lJ = at - 1/M, the dissipation length is proportional to ljm" which is clearly very large in the case of propagation through the atmospheric boundary layer. This suggests that molecular relaxat ion effects will be of greater importance in determining the attentuation of the acoustic field [24]. The wavenumber drift contained in the second term of the argument ofthe exponential is also an O(m 4 ) effect, and will be seen to be negligible in the region of validity of the present approximation. The mean square dilatation in wavenumber, «k i - k O)2) , can be calculated by using a formula analogous to (65) above, and turns out to be

V«k"-

k

J!I.

O)2>'" 2mk ov lX/o2~ 2mko J';/:1 2= 2kom 2

(69)

This determines the effective spectral width of the beam as it propagates through the turbulence. The initial beam was concentrated at k i = k o, but interactions with the slow temporal fluctuations of the medium have generated sound waves of slightly different frequency. This is the phenomenon of spectral broadening, and has recently been considered in the acoustic context by Ffowcs Williams and Fitremann [25] by an alternative method. Finally it should be noted that the validity of the present approximation to the diffusion 2 equation (57) requires that n - kO}2>/k~ and >are small. This means that the effects of absorption and of the spectral drift are an order of magnitude smaller than these quantities.

«k

<0

7. COMPARISON WITH GEOMETRICAL ACOUSTICS

The results derived in the previous section regarding scattering and spectral broadening are really rather unsophisticated applications of the general kinetic equation (42). Nevertheless it has been repeatedly stressed here that the kinetic equation approach constitutes what is essentially a scattering theory. In the high frequency limit classical scattering theory is known to be of doubtful validity because of the prominence of "forward scatter", whereby most of the "scattered" field actually corresponds to a phaseshift in the incident wave. This is caused by local convective variations in the wave propagation velocity. The theory presented here, however, is phase independent so that there is no reason to suppose that it ceases to be valid

MULTIPLE SCATTERING OF SOUND

473

at these high frequencies. Hence it should be possible to obtain results such as equations (66) and (69) by means of the theory of geometrical acoustics, which is known to be valid in that limit. That this is indeed the case will be demonstrated below. The equivalence is not purely of academic interest, however. Rather it shows that at high frequencies the kinetic theory has a region of overlap with the well founded theory of geometrical acoustics, and may therefore be cited in support of the somewhat involved analysis leading to the kinetic equations. The equivalence with classical scattering theory in the limit of long wavelength has already been established (section 4). The real interest in the kinetic equations therefore lies in their ability to make predictions in regions of wavenumber space not covered by the classical theories. Now in a turbulent atmosphere, where Uj denotes, as before, the turbulent velocity, the frequency w of a very short sound wave is given in terms of the wavenumber vector k, by

w =±ak +k·u.

(70)

Consider the problem posed and solved in section 6 of an acoustic beam of initial wavenumber k o = (k o, 0, 0) emitted in the positive direction of the x-axis at X = O. It is convenient to regard the beam as consisting of a succession of elementary wave packets. Then, according to the theory of geometrical acoustics [26], such a packet propagates essentially as a particle of "energy" wand "momentum" k. These, together with the position vector X of the packet, are determined by the following system of Hamilton characteristic equations: dX

Ow

d"t= ok'

dk

ow

dw

ax

"dt=ai"'

-=--, dt

ow

(7Ia, b,c)

where d/dt is the derivative following the motion of the packet, and w satisfies equation (70). There is no loss in generality in taking the positive sign in equation (70), and then dX

dt = ak + u.

(72)

In order to effect the lowest order approximation to the solution of the system (71), it is sufficient to neglect the convective term, u, on the right of equation (72), and to set k = (1,0,0), so that, in the first approximation,

x = (at, 0, 0)

(73)

at time t after emission of the packet. To determine k II' the component of k parallel to the X-axis, note first that from equation (71b) it satisfies OU oU I dk ll (74) -=-k·-:=::-ko -

ex

dt

ex'

where Ul is the X-component of u. This velocity is a function of the position X = (at,O,O) of the packet and of the time t, so that integrating expression (74) along the path gives, in the first approximation,

' Jax °

OUt

k , -ko=-k o -(as,O,O;s)ds.

(75)

Multiply this equation by equation (74) and take the ensemble average:

f I

1d

2dt «k u -kO)2)=-k~

°

2

0 R lI OX2 (a[s-t],O,O;s-t)ds,

(76)

474

M. S. HOWE

where R I1(x , t) is the velocity correlation function defined in equation (10). When the packet has travelled more than a correlation scale / of the turbulence this becomes essentially

JaaX 00

d 2 2 d/(k " -. kon = -k o

2

Ru 2

(as, 0,0; s) ds.

(77)

-'" To evaluate this integral the spectral representation of RtJ given by the Fourier inverse of equation (11), and the special form (12) for the spectrum function
-«(k -k dt II

o)>=-nk~ a JdKIz lO

2

00

2

3

o

:::: -nk5 3

a

I'" dx o

Z2

)

E(K,Z)

( 1a-2 K-2 - dz K

-00

I

oo

Z2 -E(lC, z) dz

1C

-00

(78) where, as before, E(K,W) has been approximated by the expression in equation (60) near

w=o. Hence, integration of equation (78) finally yields (79)

in complete accord with the expression (69) obtained from the lowest order approximation to the diffusion equation (57). Similarly, the equivalence of the respective expressions for the angular spread <82 can be established by using the Hamilton equations to determine . Note that it was necessary to assume that the path length satisfied the condition x~ / in the above analysis. This restriction is implicit in the theory of the kinetic equation, since the limiting process 8 -';- 0 ultimately means that the elementary "cells" have diameter exceeding 1. This indicates that the theory is suitable only for determining changes in the mean square amplitude over distances which are large compared with the correlation scale. For shorter distances the "fluctuations" in the exact solution about the mean square will be important.

>

8. CONCLUSION

The kinetic theory of multiple scattering as presented in this paper appears to constitute a unified scheme for tackling acoustic scattering problems for arbitrary incident wavelengths. The theory has been shown to be superior to that based on the Bethe-Salpeter equation inasmuch as there is no limitation on the wavelength of the sound that can be treated . Further, i n the case of a "frozen" random medium, the kinetic equation conserves total acoustic energy. For long waves the kinetic theory reproduces the well known results of single scattering analysis, and at very high frequencies the equivalence with geometrical acoustics has been established. This should give confidence both in the basic ideas underlying the theory, and also in the results of possible future applications in frequency ranges not amenable to analysis by classical methods. A case of topical interest to which it is hoped to apply the theory con-

MULTIPLE SCATTERING OF SOUND

475

cerns the controversial question of sonic boom propagation through a turbulent atmosphere (see, e.g., references [17] and [18]). More general applications of the integro-differential kinetic equations will almost certainly involve the use of electronic computers, but even then the analysis is expected to be more straightforward than that of an approach based on a direct numerical integration of the random wave equations (4) or (7).

ACKNOWLEDGMENT The work reported in this paper was supported by the Bristol Engine Division of Rolls Royce (1971) Ltd., and was conducted while the author was a Research Assistant in the Department of Mathematics, Imperial College of Science and Technology.

REFERENCES 1. J. B. KELLER 1964 Proceedings 0/ Symposia on Applied Mathematics 16,145-170. Wave propagation in random media. Providence, R.I: American Mathematical Society. 2. M. S. HOWE 1971 Journal 0/ Fluid Mechanics 45,769-804. Wave propagation in random media. 3. E. E. SALPETER and H. A. BETHE 1951 Physical Review 82, 1232-1242. A relativistic equation for bound state problems. 4. U. FRISCH 1967 in Probabilistic Methods in Applied Mathematics, Vol. 1. (ed. A. T. BharuchaReid). Wave propagation in random media. New York: Academic Press. 5. M. S. HOWE 1973 Proceedings 0/ the Royal Society A331, 479-496. Conservation of energy in random media, with application to the theory of sound absorption by an inhomogeneous flexible plate. 6. M. S. HOWE 1972 Journal 0/ Sound and Vibration 23, 279-290. Multiple scattering of bending waves by random inhomogeneities. 7. S. A. RYBAK 1972 Soviet Physics-Acoustics 17,345-349. Waves in a plate containing random inhomogenei ties. 8. H. KUTfRUFF 1967 Acustica 18, 131-143. Uber Nachhall in Medien mit unregelmassig verteilten Streuzentren, insbesondere in Hallraumen mit aufgehangten Streue1ementen. 9. D. I. BLOKHINSTEV 1946 NACA Technical Memorandum No. 1399. Acoustics ofa non-homogeneous moving medium. 10. T. H. ELLISON 1952 Journal 0/ Atmospheric and Terrestrial Physics 2, 14--21. The propagation of sound waves through a medium with very small variations in refractive index. 11. M. J. LIGHTHILL 1953 Proceedings of the Cambridge Philosophical Society 49,531-551. On the energy scattered from the interaction of turbulence with sound or shock waves. 12. R. H. KRAICHNAN 1953 Journal 0/ the Acoustical Society 0/ America 25, 1096-1104. Scattering of sound in a turbulent medium. 13. G. K. BATCHELOR 1957 in Naval Hydrodynamics (Chapter 16). Washington, D.C.: National Academy of Science (Publication 515 of the National Research Council). 14. S. C. CROW 1969 Journal 0/ Fluid Mechanics 37,529-563. Distortion of sonic bangs by atmospheric turbulence, 15. L. A. CHERNOV 1960 Wave Propagation in a Random Medium. New York: McGraw-HilI Book Company, Inc. 16. V. I. TATARSKI 1961 Wave Propagation in a Turbulent Medium. New York: McGraw-Hill Book Company, Inc. 17. K. J. PLOTKIN and A. R. GEORGE 1972 Journal 0/ Fluid Mechanics 54, 449-467. Propagation of weak shock waves through turbulence. 18. M. S. HOWE 1972 Journal 0/Sound and Vibration 24, 269-272. Non-linear theory of sound propagation through a random medium. 19. M, J. LIGHTHILL 1952 Proceedings 0/ the Royal Society A211, 564-587. On sound generated aerodynamically: (I) General theory. 20. S. C. CROW 1967 Physics 0/ Fluids 10, 1587-1589. Visco-elastic character of fine-grained isotropic turbulence. 21. R. L. STRATONOVICH 1963 Topics in the theory ofrandom noise, Vol. I. New York: Gordon and Breach.

476

M. S.

HOWE

22. M. J. LIGHTHILL 1960 Philosophical Transactions 0/ the Royal Society A2S2, 397-430. 23. G. K. BATCHELOR 1953 The theory 0/ homogeneous turbulence. Cambridge Univers ity Press . 24. M . J. LIGHTHILL 1956 in Surveys ill Mechanics (eds, G. K. Batchelor and R. O. Davies). Viscosity effects in sound waves of finite amplitude. Cambridge University Press. 25. J. E. FFOWCS WILLIAMS and J. M. FITREMANN 1972 Private communication. 26. M. J. LIGHTHILL 1965 Journal ofthe Institute 0/ Mathematics and its Applications 1,1-28. Group velocity.