Propagation of transients in a random mediumAlan R. Wenzel Citation: Journal of Mathematical Physics 16, 44 (1975); doi: 10.1063/1.522412 View online: http://dx.doi.org/10.1063/1.522412 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/16/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Propagation of transient waves in a random medium J. Acoust. Soc. Am. 94, 1874 (1993); 10.1121/1.407583 Propagation through an anisotropic random medium J. Math. Phys. 15, 1901 (1974); 10.1063/1.1666555 Fluctuations of waves propagating in a random medium J. Acoust. Soc. Am. 55, S57 (1974); 10.1121/1.1919811 Propagation of Transients in a Random Medium J. Acoust. Soc. Am. 55, 392 (1974); 10.1121/1.3437178 On Wave Propagation in a Random Inhomogeneous Medium J. Acoust. Soc. Am. 29, 197 (1957); 10.1121/1.1908828

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Propagation of transients in a random medium Alan R. Wenzel

NASA-Ames Research Center, Moffett Field, California 94035 (Received 25 June 1974)

The propagation of transient scalar waves in a three-dimensional random medium is considered. The analysis is based on the smoothing method. An integro-differential equation for the coherent (or average) wave is derived and solved for the case of a statistically homogeneous and isotropic medium and a delta-function source. This yields the coherent Green's function of the medium. It is found that the waveform of the coherent wave depends generally on the distance from the source measured in terms of a certain dimensionless parameter. Based on the magnitude of this parameter, three propagation zones, called the near zone, the far zone, and the intermediate zone, are defined. In the near zone the evolution of the waveform is determined primarily by attenuation of the high-frequency components of the wave, whereas in the far zone it is determined mainly by dispersion of the low-frequency components. The intermediate zone is a region of transition between the near and far zones. The results show that, in general, the randomness of the medium causes a gradual smoothing and broadening of the waveform, as well as a decrease in amplitude of the wave, with propagation distance. In addition, the propagation speed of the wave is reduced. It is also found that an oscillating tail appears on the waveform as the propagation distance increases.

INTRODUCTION

The subject of wave propagation in random media has been studied extensively over the past two decades (see, e. g., the review article by Frisch!). Most of the work in this area has been concerned with time-harmonic waves rather than with transient phenomena. Recently, however, interest in transient waves has been stimulated by a desire to understand how sonic booms propagate through atmospheric turbulence, and a number of theo­retical papers dealing with this phenomenon have ap­peared. 2-7 A review of current research on this topiC has been given by Pierce and Maglieri. 8

Our main objective here is to study the propagation of transient waves in random media from a general view­point. Consequently, we have adopted a more general (but also more idealized) analytical model than those which have been used previously to study sonic-boom propagation. Our approach is based on the smoothing method, which has been discussed by Frisch. j This leads to what is essentially a linear treatment; i. e. , effects such as nonlinear steepening of the wave, consi­dered in Refs. 4, 5, and 7, are neglected. Our analysis does include multiple- scattering effects, however.

In Sec. I we formulate the problem in terms of an in­tegro-differential equation for the coherent wave. In Sec. n we solve this equation by transform methods for the case in which the medium is statistically homogeneous and isotropic and the source term is a space-time delta function. This yields the coherent Green's function of the medium. The main result of Sec. II is given by Eq. 35, which is an integral expression for the Green's func­tion. In Secs. IIA and lIB we evaluate this expression approximately for two propagation regions which we call the near and far zones. In addition, in Sec. IIC we eval­uate it numerically for the region which we call the in­termediate zone. The results of the latter calculation are presented in Figs. 1-5.

As noted in Sec. IIA. our near-zone results are sim­ilar to those of Cole and Friedman. 6 However, our re­sults for the intermediate and far zones do not appear to have been obtained previously.

I. FORMULATION OF THE PROBLEM

We wish to consider the propagation of transient scal­ar waves in an unbounded, three-dimensional, random medium. As our mathematical model of this phenomenon we choose the scalar wave equation

(1)

where X= (Xj,X2,X3), and the "acoustic speed" c(x, l) is assumed to be a random function of space and time. The problem we are concerned with can be formulated, gen­erally, as follows: Given the (nonrandom) source function f and some appropriate statistical properties of c, find some specified statistical properties of u. Here we shall be concerned with the ensemble average of u, denoted by (u), which is called the coherent wave.

We now proceed to get an equation for the coherent wave. We begin by assuming that the random inhomo­geneities of the medium are small; i. e., we write

C (x, t) = C 0[1 +EiJ. (x, t)], (2)

where Co is the average acoustic speed (assumed con­stant), and iJ. (x, t) is a random function defined so that (iJ.) = 0 and (iJ. z:> = 1. The parameter E is a measure of the deviation of C from its average, and is assumed to be smalL

By substituting for C from Eq. 2 we can write Eq. 1 in the form

where the operators La, L!, and 1.2 are given by

Lo=Cii2a~-V'2,

L! =- 2C(j2iJ.()~,

L2 = 3Cii2J.l2a~.

(3)

(4)

(5)

(6)

Keller9 has shown that the ensemble-averaged solution of Eq. 3, i. e., the coherent wave, satisfies the equation

(7)

(In deriving Eq. 7 it has been assumed that (Lj; = 0, which is the case here. ) Thus, for our case the equation

44 Journal of Mathematical Physics, Vol. 16, No.1, January 1975 Copyright © 1975 American Institute of PhYSics 44

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for the coherent wave is obtained by substituting Eqs. 4-6 into Eq. 7 and noting that the inverse operator Lr/ can be written in the form

(8)

(Here, and henceforth, an integral sign without limits de­notes an integral taken over all of three-dimensional space. ) The result, after terms of order E3 are dropped, is

(c(j2oi - v 2)w(x, t) +E2{3c(j2Wtt (X, t)

- 1T-1C 04 J r-1[R TT(r, co1r)w tt (x + r, t - c o1r)

- 2R T(r, colr)wttt(x + r, t - co1r) +R(r, co1r)

XWtttt(x + r, t - co1r)] dr} = f(x, t). (9)

The letter subscripts denote differentiation.

In deriving Eq. 9 we have set w=(u). Also, we have assumed that the medium is statistically homogeneous, and we have introduced the correlation function

(10)

We now assume that the random fluctuations of the medium are independent of time. (This is equivalent to assuming that the characteristic time associated with the wave is much less than that associated with the fluctua­tions of the medium. In the case of waves propagating in real media, this condition is generally satisfied. In par­ticular, it is satisfied in the case of sonic-boom propa­gation through atmospheric turbulence. ) Then J.l = J.l (x), and Eq. 9 Simplifies to

(Ci"?2~ - V2)w(x, t) + E2{3c0211'tt(X, t)

- 1T-1cii4 J r-1R(r)w tttt (x + r, t - cii1r) dr} = f(x, f), (11)

where now

(12)

The procedure by which Eq. 9 is obtained from Eq. 1 is due essentially to Keller, 9 and is referred to as the smoothing method by Frisch. 1 It has been used previous­ly to study the propagation of time-harmonic waves in various types of random media. 9-13

II. THE COHERENT GREEN'S FUNCTION

We now proceed to solve Eq. 11 whenf(x, t) = o(x)o(t). This will yield the free-space coherent Green's function of the medium [i. e., if lU(X, t) is the solution of Eq. 11 withf(x, t) = 6(x)6(t), then the Green's function G is given by G(x, t;x', t') = lO(X - x', t - t')]. This function describes the propagation of a spherical delta-function pulse.

We begin by introducing the Fourier time and space transforms, defined by

Tfl (w) =jl (w) = r: exp(iwt)fl (t) dt,

Sgl (k) =gl (k) = J exp(- ik· x)g1 (x) dx,

(13)

(14)

respectively, where fl (t) and gl (x) are any square-inte­grable functions. The inverse transforms are given by

(15)

45 J. Math. Phys., Vol. 16, No.1, January 1975

s-lil(X) =gl (x) = (21Tt3 J exp(ik· x)i! (k) dk.

Applying the operator T to Eq. 11 yields

- (V2 +k~)zii(x, w) - E2[3k~w(x, w)

+1T-lk~ J r-1 exp(ikor)R(r)w(x +r, w) dr] = 6(x),

where k 0 = w / co. Operating on Eq. 17 with S gives

D(k, w)~(k, w) = 1,

where

x exp(ik· r) dr].

(16)

(17)

(18)

(19)

We now assume that the medium is isotropic, so that R(r) = R(r). Then we can carry out the angular integra­tion in Eq. 19, after which we find that

D(k, w) =D(k, w)

=k2 - k~- E2k~[3 +4k~-1 Jo~ exp(ikor)R(r) sinl?rdr]. (20)

Hence, from Eq. 18,

~(k, w) = ~(k, w) = [D(k, w)]-I. (21)

The function w(x, w) is obtained by operating on Eq. 21 with S-1 and carrying out the resulting angular inte­gration in k space. The result, after some algebra, is

w(x, w) = iiJ(x, w) = (21T2xt1 fo~ [D(k, w )]-lk sinkxdk. (22)

We can evaluate the integral in Eq. 22 by means of con­tour integration, after which the expression for w becomes

(23)

Here Dk is the derivative of D with respect to k, and kl is the root of the dispersion equation D(k, w ) = 0 which lies in the upper half of the complex k plane, and which has the property that k 1 - k 0 as E - O. To lowest order in E, it is given by

kl = ko + (1/2)E2[3ko + 4k~ fo~ exp(ikor)R(r) sinkor dr]. (24)

That kl' as given by Eq. 24, indeed has a positive imag­inary part is easily proved using an analysis similar to that of Keller (Ref. 9, p. 152). The same analysis shows that I Rek 11> I k 0 I, and that the root near - k 0 has a neg­ative imaginary part.

In deriving Eq. 23 we have neglected all other zeros of D(k, w) lying in the upper half-plane. The justification for neglecting these zeros (if they exist) is that, as shown in Appendix A, they have large imaginary parts and hence correspond to rapidly-attenuating waves. Such waves will be important only in a region near the source. Since we are not interested in the behavior of the solu­tion in this region, we disregard these waves.

We now rewrite Eq. 23 by inserting the formula for kl given by Eq. 24 into the expression for Dk(k, w) ob­tained from Eq. 20. Upon neglecting higher-order terms in E, we obtain

(25)

Alan R. Wenzel 45

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where we have defined

>11 (k o) = k~ 10<0 exp(ikor)rR(r) coskor dr

- ko 10<0 exp(ikor)R(r) sink or dr. (26)

Note that the quantity w(x, w) exp(- iwt), with w given by Eq. 25, is just the coherent wavefield radiated by a time-harmonic point source; i. e., it is the solution of Eq. 11 whenj(x, t) = {i(x) exp(- iwt). Since, as noted above, Imk t > 0 and I Rekll > Ikol, we see that this wave decays exponentially with distance from the source, and also that its phase speed is less than co. These proper­ties of the coherent wave are similar to those found in previous studies of plane, time-harmonic waves. 9,10

The solution of Eq. 11 can now be obtained by operat­ing on Eq. 25 with ']"'1. This yields

w(x, t) = w(x, t) = (81T2X)"1 1: [1 + 2e2>11(k o)]

x exp[ikoX(1 + ~e2) + 2ie2koXiJJ(ko) - iw t] dw. (27)

Here we have defined

iJJ (k o) = ko 10<0 exp(ikor)R(r) sink or dr, (28)

and we have substituted for kl from Eq. 24.

We can write Eq. 27 in a more convenient form as follows. We first note that

(29)

Next, after substituting Eq. 29 into Eq. 27, we introduce K, a new integration variable, which is defined so that K=k o(1 +~e2). Upon dropping terms of order e4 in the re­sulting expression for w, we obtain

w(x, t) =C*(81T2X)"11: exp{iK(x - c*t)

+ 2e2[>I1(K) + iKxiJJ(K)]}dK,

where

c * = (1 + ~e2)"lc o.

As a check on our results, we note that by setting

(30)

(31)

e = 0 in Eq. 30 we obtain the well-known Green's function solution for the case of a uniform medium.

In order to simplify Eq. 30 further, we now assume that x is so large that we can neglect the term >I1(K) com­pared to the term iKXiJJ(K) in the square brackets in that equation. In dimensionless terms this means that we must have x/Z» 1, where 1 is the correlation length of R. Then Eq. 30 becomes

w(x, t) = C*(81T2X)"1 1: exp[iKY + 2ie2KXiJJ(K)] dK, (32)

where we have defined Y =x - c*t.

We now write the integral of Eq. 32 in dimensionless form. We begin by introducing the normalized correla­tion function S(s), defined by

R(r) = S(r/l).

Next we define the length scale 1> by

1> = E(lx)1 /2.

46 J. Math. Phys., Vol. 16, No.1, January 1975

(33)

(34)

Then in terms of the integration variable p = K(), Eq. 32 becomes

w(x, t) = c * (81T21>X)-t 1: exp[i1)p - p2¢ (a- lp)] dp, (35)

where we have defined 1) = y/1>,

a = e(x/l)t!2, (36)

and

¢(q) = 10<0(1- exp(2iqs»S(s) ds. (37)

Equation 35 shows that the waveform associated with the Green's function, expressed in terms of dimension­less coordinates, is determined by the parameter a. De­pending on the magnitude of this parameter, we can iden­tify three propagation zones which we call the near zone, the far zone, and the intermediate zone. They are dis­cussed in detail below.

A. The near zone

The near zone is defined by the condition that a« 1. Referring to Eq. 35 we see that, in this case, I a-lp I »1 over the entire range of integration, except for a small interval near the origin which we shall neglect. Hence, we can evaluate the integral of Eq. 35 approximately by substituting for the function ¢ its asymptotic expansion for large val ues of the argument. This is obtained by in­tegrating by parts in Eq. 37, and can be written

¢(q) = mo + (2iqr l + O(q-3), (38)

where

mn = 10<0 sns(s) ds, n=0,1,2, .... (39)

Upon inserting Eq. 38 into Eq. 35 (after dropping the term of order q-3 in Eq. 38) and noting that the resulting integral is tabulated, we obtain, after some manipulation,

w(x, t) = cl (81Tx1>ltl (mo1T)-1/2 exp[ - (x - Clt)2/4mo1>iJ, (40)

where

1>1 = (1 + tE2)"11>, (41)

and

(42)

Equation 40 describes the waveform associated with the Green's function in the near zone. It shows that, near the wavefront, i. e., near X=Clt, the waveform is given approximately by a Gaussian curve. We see from Eqs. 34 and 41 that the waveform broadens in proportion to the square root of the propagation distance, while the amplitude of the wave decreases (in addition to the de­crease due to spherical spreading) as the inverse square root of the propagation distance. Equations 40 and 42 show that the wave propagates with a speed equal to cl, and that cl is less than co. Thus, the propagation speed of the wave is reduced by the randomness of the medium.

Replacing the function ¢ in Eq. 35 by the approximate form given by Eq. 38 is equivalent to considering the ef­fect of the randomness of the medium on only the high­frequency components of the wave. This effect consists of an attenuation which is proportional to the square of the frequency, as well as a frequency-independent reduc-

Alan R. Wenzel 46

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1.0

.8

.6

.4

W(,I

.2

o~----------------~~--~------

-.2

FIG. 1. The waveform associated with the Green's function, computed using Eq. 49, for the case a ~ 0.1. The function W(~) is related to the Green's function by Eq. 48. The stretched coordinate ~ is defined so that ~ = 0:;1 (x - c*t), where 03 and c* are given by Eqs. 50 and 31. The wave is propagat­ing from left to right.

tion in propagation speed. Hence, in the near zone, the effect of the randomness of the medium on transient waves, insofar as the waveform is concerned, is essen­tially that of a pseudoviscosity, with the waveform being determined mainly by attenuation of the high-frequency components of the wave.

Our near-zone solution (Eq. 40) is similar to one which was obtained by Cole and Friedman (Ref. 6, p. 71, Eq. 11) for the case of a plane wave propagating in a tur­bulent medium. (Their solution is expressed in terms of an error function since they considered a step-function, rather than a delta-function, pulse). These authors ob­tained their result by, in effect, solving the linearized Burgers' equation. Since it is well known that this equa­tion governs (approximately) the propagation of small­amplitude sound waves in a viscous fluid, 14 their formu­lation better illustrates the pseudoviscous character of the medium noted above.

B. The fat zone

The far zone is defined by the condition that ct »1. In this case I ct-1p I «1 over that range of p which yields the major contribution to the integral of Eq. 35. Hence we can evaluate that integral approximately by substituting for ¢ the first few terms of its power series expansion. This is obtained by expanding the function exp(2iqs) in Eq. 37 in a power series and integrating term by term. Upon dropping all but the first term of the expansion we find that the integral of Eq. 35 can be evaluated in terms of the Airy function. After some manipulation, the re­sulting expression for w can be written

w(x, t)=C*(47TXI53)-lAi(~)

where Ai denotes the Airy function,

03 = (6 t 2m l l2x)1/3,

47 J. Math. Phys., Vol. 16, No.1, January 1975

(43)

(44)

and ~ = o;ly. Equation 43 describes the waveform associated with

the Green's function in the far zone. We see that, near the wavefront, i. e., near x = c*t, the waveform is given approximately by an Airy function. Equations 43 and 44 show that the waveform broadens in proportion to the cube root of the propagation distance, while the ampli­tude of the wave decreases (in addition to the decrease due to spherical spreading) as the inverse cube root of the propagation distance. Note that the propagation speed of the wave is equal to c*. This is slightly greater than the propagation speed in the near zone. However, from Eq. 31, c* is less than co; hence, as in the near zone, the propagation speed is reduced by the randomness of the medium.

More detailed information regarding the waveform de­fined by Eq. 43 can be obtained from any standard re­ference work on the Airy function (see, e. g., Ref. 15).

Replacing the function ¢ in Eq. 35 by the first few terms of its power-series expansion is equivalent to ne­glecting all but the low-frequency components of the wave. Moreover, in keeping only the first term of this expansion we are, in effect, ignoring attenuation of the lOW-frequency components, and considering only disper­sion. Hence, in the far zone the high-frequency compo­nents of the wave have effectively attenuated to zero, with the waveform being determined primarily by dis­perSion of the low-frequency components.

C. The intermediate zone

The intermediate zone is defined by the condition that ct '" 1. In this case it is necessary to evaluate the integral of Eq. 35 numerically. In order to do this, we must first assume a particular form for the correlation function of the medium. Here we shall assume a Gaussian correla­tion function; i. e., we write

(45)

By substituting Eq. 45 into Eq. 37 we find that the func­tion ¢(q) for this case can be written

1.0

.8

.6

-.2

-.4 L...l-.....L---1_-L---"---1._L-..-L......L---1_-'-, ~ --'

-10 - 8 - 6 - 4 - 2 0 2 4 €

FIG. 2. Same as Fig. 1, except that a ~ 0.5.

Alan R. Wenzel 47

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Wr .8

.6

.4

W(O

.2

0

-.2

-.4 -10 -8 -6 -4 -2 0

~ FIG. 3. Same as Fig. 1, except that O! = 1. O.

¢(q):=; (1T1 / 2 /2)[1 _ exp( _ q2)] - iD(q),

where D(q) is Dawson's integral:

D( q) :=; exp( - q2) Ia q exp(z2) dz.

2 4

(46)

( 47)

With the aid of Eq. 46, Eq. 35 can be put into the form

41Td0 3xw(x, t) = W(~),

where

W(O=1T-1 (cos[~p+ typ2D(P/Y)]

expl-;t 1T l /2.yp2(1_ exp( _ p2/y2»] dp,

y=(3Q'2)1/3, and, in this case,

°3

= (3E2l2X)1 /3.

( 48)

( 49)

(50)

The function W(~) describes the waveform associated with the Green's function in the intermediate zone. Using Eqs. 47 and 49, we have made numerical calcula­tions of W(~) for a range of ~ near the wavefront (i. e., near ~ =0), and for several values of Q'. These results are plotted in Figs. 1-5. The figures describe the tran­sition of the wave profile from the near-zone to the far­zone form as the wave propagates through the interme­diate zone (i. e., as Q' increases). The most prominent feature of this transition is the development of an oscil­lating "tail" on the profile which becomes more pro­nounced with propagation distance. This is the result of dispersion of the low-frequency components of the wave which, as we have seen, becomes increasingly impor­tant in determining the wave profile as the wave propa­gates into the far zone.

ACKNOWLEDGMENTS

Research supported by the Atmospheric Sciences Sec­tion, National Science Foundation, NSF Grant GA-31561. and by the National Research Council. Part of this work was done while the author held an NRC Resident Re­search Associateship at NASA-Ames Research Center. The author is grateful to Henry Poor of the Division of Atmospheric Science, University of Miami, and to

48 J. Math. Phys .• Vol. 16. No.1. January 1975

1.0

.8

.6

.4

W(~)

.2

0

-.2 i r

-.4 I -10 -8 -6 -4 -2 0 2 4

~ FIG. 4. Same as Fig. 1, except that o! = 2. O.

Margaret Johnson of the Computer Sciences Corporation, Mountain View, California, for programming the numeri­cal calculations.

APPENDIX A

We wish to show here that all the roots of the disper­sion equation

D(k,w)=O (AI)

which lie in the upper half of the complex k plane, and which are bounded away from ± ko as E - ° (i. e., all the roots except k1) have the property that Imk- + 00 as E- 0.

To see this, we assume that k = k(E) is such a root. By using the definition of D(k, w) given by Eq. 20, we can write Eq. Al in the form

k2 - k~ =E2k~l3 + 4k~k-l Io'" exp(ikor)R(r) sinkrdr]. (A2)

Since the left-hand side of Eq. A2 is bounded away from

1.0

.8

.6

-.4 L--'----"------'_"----'-------"-_'---'------L-----'-_L-.~ -10 -8 -6 -4 -2 0 2 4

FIG. 5. Same as Fig. 1, except that O! = 10. O.

Alan R. Wenzel 48

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zero as E- 0, the right-hand side must be similarly bounded. This requires that the second term in the brackets in Eq. A2 be unbounded as E - O. This term, however, is an entire function of k; hence we must have Ik 1- 00 as E- O. As a consequence, the integral of Eq.

A2 must be unbounded as E - O. But this is possible only if 1m k - co as E - 0, as can be seen by writing the term sin kr in exponential form. Thus, the result is established.

lU. Frisch, "Wave Propagation in Random Media," in Prob­abilistic Methods in Applied Mathematics, edited by A. T. Bharucha-Reid (Academic, New York, 1968).

2S.C. Crow, J. Fluid Mech. 37, 529 (1969). 3A. D. Pierce, J. Acoust. Soc. Am. 49, 906 (1971).

49 J. Math. Phys .• Vol. 16. No.1. January 1975

4A. R. George and K. Plotkin, Phys. Fluids 14, 548 (1971). oK. J. Plotkin and A. George, J. Fluid Mech. 54, 449 (1972). 6W. J. Cole and M. B. Friedman, "Analysis of Multiple Scat-tering of Shock Waves by a Turbulent Atmosphere," in Third Conference on Sonic Boom Research, edited by 1. R. Schwartz, NASA SP-255 U97l).

1M. S. Howe, J. Sound Vib. 24, 269 (1972). B A. D. Pierce and D. Maglieri, J. Acoust. Soc. Am. 51, 702 (1972).

9J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964). loA.R. Wenzel andJ.B. Keller, J. Acoust. Soc. Am. 50,

911 (1971). llR.E. Collin, Radio Sci. 4,279 (1969). 12F.C. KaralandJ.B. Keller, J. Math. Phys. 5,537(1964). 13J. B. Keller and F. Karal, J. Math. Phys. 7, 661 (1966). 14D. T. Blackstock, J. Acoust. Soc. Am. 41, 1312 (1967). 15H. A. Antosiewicz, "Bessel Functions of Fractional Order,"

in Handbook of Mathematical FUnctions, edited by M. Abramowitz and 1. Stegun (National Bureau of Standards, WaShington, D. C., 1964), p. 446.

Alan R. Wenzel 49

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