Using JMP format

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 40, NUMBER 10

OCTOBER 1999

Diffusive energy scattering from weakly random surfaces Guillaume Bala) and George Papanicolaoub) Department of Mathematics, Stanford University, Stanford, California 94305

Leonid Ryzhikc) Department of Mathematics, University of Chicago, Chicago, Illinois 60637

共Received 17 March 1999; accepted for publication 19 June 1999兲 We derive transport theoretic boundary conditions for acoustic wave reflection at a weakly rough boundary in an inhomogeneous half space. We use the Wigner distribution to go from waves to energy transport in the high frequency limit. We generalize known results on the reflection of acoustic plane waves in a homogeneous medium. We analyze higher order corrections, which include a enhanced backscattering effect in the back direction. © 1999 American Institute of Physics. 关S0022-2488共99兲02009-5兴

I. INTRODUCTION

Wave propagation in weakly fluctuating random media over distances large compared to the wavelength can be described by incoherent energy transport. This is the radiative transport regime.1,2 Near boundaries and interfaces, waves undergo coherent or partially coherent reflection. Angularly resolved energy reflection and transmission in homogeneous media in average, has been studied extensively in the past.3–6 Recently, the problem has been revisited in the transport theoretic context using a plane wave decomposition; see Ref. 7. There, the boundary conditions for the transport equations in a domain with rough boundaries of small amplitude and homogeneous background are derived. In this paper, we derive boundary conditions in the case of inhomogeneous domains. We systematically use the Wigner transform to study the reflection of angularly resolved acoustic energy density from a Dirichlet surface. The scale of the volume inhomogeneities is large compared to the wave length. The boundary conditions are a direct generalization of those obtained in the case of a homogeneous medium, with a reflection operator depending upon the position at the boundary. Our main ingredient is a perturbation analysis around the flat boundary case studied in Ref. 8. A. Transport equations for the energy density

As we recall in Sec. II, the phase space acoustic energy density ␮共x,k兲 satisfies the transport equation ⵜ k␻ ⫹ •ⵜ x␮ ⫺ⵜ x␻ ⫹ •ⵜ k␮ ⫽0.

共1兲

Here ␻ ⫹ (x,k) is the eigenfrequency of the acoustic waves given by

␻ ⫹ ⫽c 兩 k兩 ,

c⫽

1

冑␬ ␳

,

共2兲

where ␳共x兲 is the density and ␬共x兲 is the compressibility of the background medium. These equations hold in the high frequency regime for monochromatic waves, when the wavelength is a兲

[email protected] [email protected] c兲 [email protected] b兲

0022-2488/99/40(10)/4813/15/$15.00

4813

© 1999 American Institute of Physics

Downloaded 18 Apr 2006 to 171.64.38.75. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

4814

J. Math. Phys., Vol. 40, No. 10, October 1999

Bal, Papanicolaou, and Ryzhik

much smaller than the variations of the density and compressibility. They were derived in domains without boundaries or interfaces by several authors in the context of geometrical optics 共see for instance Ref. 9兲. B. Scattering from a rough boundary in a homogeneous medium

In the preceding section, the free transport equation 共1兲 is posed in the whole space, with no boundary. In realistic applications, it is important to consider problems in bounded domains with correct boundary conditions. We assume that the domain is given by H ⑀ ⫽ 兵 x苸R n :x n ⬎ ⑀ ␩ h(x⬘ / ⑀ ) 其 . The function h(x⬘ ) is a mean zero stationary random process with covariance function R(y⬘ ) defined by

具 h 共 x⬘ ⫹y⬘ 兲 h 共 x⬘ 兲典 ⫽R 共 y⬘ 兲 .

共3兲

The power spectrum Rˆ (k⬘ ) is the Fourier transform of R(y⬘ ): R 共 y⬘ 兲 ⫽

冕

dp e ip⬘ •y⬘ Rˆ 共 p⬘ 兲 . 共 2 ␲ 兲 d⫺1

共4兲

The boundary of our domain is varying on the scale of the wavelength ⑀, which gives rise to scattering of incoherent, or diffuse, energy from the boundary. The small parameter ␩ Ⰶ1 is measuring the height of the surface relative to the wavelength. One finds by a direct computation using a plane wave decomposition that the average energy of reflected waves going in the direction k⫹ (k⬘ ) is given, for Dirichlet boundary conditions, by

冉冕

spec diff ␮ out共 x⬘ ,k⬘ 兲 ⫽ ␮ out ⫹ ␮ out ⫽ 1⫺ ␩ 2

⫹␩2

冕

冊

dp⬘ ⫹ ˆ 共 k⬘ ⫺p⬘ 兲 p ⫹ R n k n ␮ in共 x⬘ ,k⬘ 兲共 2 ␲ 兲 n⫺1

dp⬘ 2 ˆ 共 k⬘ ⫺p⬘ 兲共 p ⫹ R n 兲 ␮ in共 x⬘ ,p⬘ 兲 . 2 ␲ 共兲 n⫺1

共5兲

Here we have defined for every horizontal wave vector k⬘ : k⫾ 共 k⬘ 兲 ⫽ 共 k⬘ ,k ⫾ n 兲,

2 2 兩 k⬘ 兩 2 ⫹ 共 k ⫾ n 兲 ⫽K ⫽

2 ␻⫹ . c2

共6兲

Notice that k⫹ corresponds to outgoing waves, and k⫺ corresponds to incoming waves. A detailed computation both for the Dirichlet and Neumann problems can be found, for instance, in Refs. 3–6. The first term in 共5兲 represents the specular reflection including a correction due to surface roughness. The second term is produced by the diffuse scattering from the rough boundary. The plane wave decomposition used in this calculation is not available in non homogeneous media. Thus one cannot directly generalize the above calculations to variable media. We have recently derived in Ref. 8 transport boundary conditions for the energy in inhomogeneous media with boundaries which vary on a large scale compared to the wavelength. We treat here the Dirichlet problem in an inhomogeneous domain with rough boundary for ␩ Ⰶ1 as a perturbation of the smooth boundary considered in Ref. 8. We show that the results of Ref. 8 allow us to compute the higher corrections in ␩ of the reflected energy, which coincide with 共5兲 in the case of a homogeneous medium. It is known that the Born expansion we use in this paper diverges for the Neumann and impedance problems at grazing angles and we do not consider them here. However, our method can be adapted to incorporate the smoothing method 共Refs. 7, 10, and 11兲. Then, we can reproduce the results obtained in Refs. 12 and 13 for uniform media including the coherent backscattering effect, now in the setup of a variable media.



Diffusive energy scattering from weakly . . .

4815

The paper is organized as follows. In Sec. II, we review the results of Ref. 8 and some basic facts from the Wigner distribution theory. We state our results for rough boundaries in Sec. III. The perturbation expansion is treated in Sec. IV up to second order in ␩. Finally, we investigate higher order corrections in Sec. V. II. ENERGY PROPAGATION IN A HALF SPACE A. The Wigner distribution

One way to describe scattering of phase space resolved energy is to consider the Wigner distribution matrix of the family w⑀ of the solutions of 共12兲. A detailed exposition to the theory of Wigner distributions can be found in Refs. 14 and 8. Given a family of vector-valued functions u⑀ (x) bounded in L 2 (R n ), its Wigner transform matrix is defined by W⑀ 共 x,k兲 ⫽

冕

冉冊冉冊

dy ik•y ⑀y ⑀y u⑀ x⫺ u* . ne ⑀ x⫹ 2 2 共2␲兲

共7兲

The family W⑀ , possibly after extracting a subsequence, has a weak limit as ⑀ →0 in the sense of Schwartz distributions. The limit matrix W共x,k兲, called the Wigner distribution, has a number of important properties, which we summarize in the following proposition. Proposition 1: The matrix W共x,k兲 is self-adjoint and non-negative. Given a bounded continuous function ␪共x兲, the Wigner distribution of the family g ⑀ (x)⫽ ␪ (x) f ⑀ (x) is W关 g ⑀ 兴共 x,k兲 ⫽ ␪ 共 x兲 W关 f ⑀ 兴共 x,k兲 ␪ * 共 x兲 .

共8兲

Given a differential operator L(x,D) with smooth coefficients, the Wigner matrix of the family p ⑀ (x)⫽L(x, ⑀ D) f ⑀ (x) is W关 p ⑀ 兴共 x,k兲 ⫽L 共 x,ik兲 W关 f ⑀ 兴共 x,k兲关 L 共 x,ik兲兴 * .

共9兲

If the family f ⑀ is ⑀-oscillatory,14 then Tr

冕

W共 x,dk兲 ⫽ lim 兩 u⑀ 共 x兲兩 2 . ⑀ →0

共10兲

The property 共8兲 allows one to consider the Wigner distributions of families bounded in 2 (R n ). This is important in dealing with time-harmonic solutions of the acoustic equations. The L loc property 共9兲 allows one to deal with high frequency waves in variable media. The last property shows that the Wigner matrix captures the energy of high frequency waves, which oscillate at a frequency of order ⑀ ⫺1 at most. One can also consider the Wigner matrix of two different families u⑀ ,v⑀ , defined as the limit of W关 u⑀ ,v⑀ 兴共 x,k兲 ⫽

冕

冉冊冉冊

dy ik•y ⑀y ⑀y u⑀ x⫺ v* . ne ⑀ x⫹ 2 2 共2␲兲

It has similar properties.14 B. Acoustic wave transport

The time harmonic acoustic equations for the acoustic velocity v⫽( v 1 ,..., v n ), n⫽2,3, and pressure p, in the high frequency regime, are

⑀ ⵜp ⑀ ⫽i ␻␳ 共 x兲 v⑀ , ⑀ ⵜ•v⑀ ⫽i ␻ ␬ 共 x兲 p ⑀ .

共11兲


4816



Here ␳ and ␬ are the density and compressibility of the medium, and the parameter ⑀ Ⰶ1 is the ratio of the wave length to the typical scale of the variations of ␳共x兲 and ␬共x兲. Equations 共11兲 can be rewritten as a reduced symmetric hyperbolic system n

⳵w

兺 ⑀ D j ⳵ x j⑀ ⫺i ␻ A 共 x兲 w⑀ ⫽0 j⫽1

共12兲

for the vector w⑀ ⫽(v⑀ , p ⑀ )苸C n⫹1 . Here the diagonal matrix A(x)⫽diag(␳,␳,␳,␬), for n⫽3, and the symmetric matrices D j are defined appropriately from 共11兲. 2 (R n ). Then the limit Let w⑀ be an ⑀-oscillatory family of solutions of 共12兲 bounded in L loc Wigner matrix W共x,k兲 is described as follows. The dispersion matrix of the system 共12兲 is 共in three dimensions兲

n

L⫽A ⫺1

兺 k jD j⫽ j⫽1

冉

0

0

0

k1 /␳

0

0

0

k2 /␳

0

0

0

k3 /␳

k1 /␬

k2 /␬

k3 /␬

0

冊

.

共13兲

The eigenvector b of the matrix L that corresponds to forward propagating waves is b⫽

冉

kˆ

1

,

冑2 ␳ 共 x 兲冑2 ␬ 共 x 兲

冊

共14兲

,

where kˆ⫽k/ 兩 k兩 . The corresponding eigenfrequency is given by

␻ ⫹ ⫽c 共 x兲兩 k兩 ,

c 共 x兲 ⫽

1

冑␬ 共 x 兲 ␳ 共 x 兲

.

共15兲

The solutions of 共12兲 have frequency ␻, thus we introduce the resonant wave number K(x) ⫽ ␻ 冑␬ (x) ␳ (x). Then, the positive definite matrix W共x,k兲 has the form W共 x,k兲 ⫽ ␮ 共 x,k兲 b共 x,k兲 b* 共 x,k兲 . Here the scalar measure ␮共x,k兲 is supported on the set S⫽ 兵共 x,k兲 : 兩 k兩 ⫽K 共 x兲其 .

共16兲

This statement is a generalization of the eikonal equation of geometrical optics. The measure ␮ satisfies the transport equation 共1兲.14,2 The scalar measure ␮共x,k兲 can be considered as the phase space resolved energy density of acoustic waves in the high frequency limit. Namely, the high frequency limit of the physical space energy density is given by2 lim E⑀ 共 x兲 ⫽

⑀ →0

冕␮

共 x,dk兲 .

Here the acoustic energy density is E⑀ 共 x兲 ⫽

␳ 兩 v⑀ 兩 2 ␬ 兩 p ⑀ 兩 2 ⫹ . 2 2

The limit energy flux F⫽ 21 (p ⑀¯v⑀ ⫹p ¯ ⑀ v⑀ ) is expressed via ␮ by



lim F⑀ 共 x兲 ⫽c 共 x兲

⑀ →0


冕

4817

kˆ␮ 共 x,dk兲 .

C. Acoustic energy transport in a half space

We review here briefly the results of Ref. 8. Let w⑀ be an ⑀-oscillatory family of solutions of the acoustic equations 共12兲 in the half space x n ⬎0 共without imposing any boundary conditions yet兲. We assume that the family w⑀ (x) and the family r⑀ 共 x⬘ 兲 ⫽w⑀ 共 x⬘ ,0兲

共17兲

2 2 of its boundary traces are bounded in L loc (R n ) and L loc (R n⫺1 ), respectively. Here and below we use the notation x⬘ ,k⬘ for the position on the boundary and the tangential component of the wave vector, respectively. In addition we assume that the measure of the resonant set B cr ⫽ 兵 (x⬘ ,k⬘ ) 苸R n⫺1 ⫻R n⫺1 :c(x⬘ ) 兩 k⬘ 兩 ⫽ ␻ 其 of grazing rays with respect to the Wigner measure ␯ (x⬘ ,k⬘ ) of the family r⑀ is zero. The grazing rays in the phase space energy context were studied recently in Refs. 15 and 16 and we avoid these technical complications. Physically, our assumption means that grazing rays are not charged. Then we have the following proposition. Proposition 2: Under the above-mentioned assumptions the following holds.

共i兲

共ii兲

The Wigner matrix ␯(x⬘ ,k) of the family r⑀ (x⬘ ) has the form ␯⫽ ␯ ␣ b共 k⫺ 兲 b* 共 k⫺ 兲 ⫹ ␯ ␤ b共 k⫹ 兲 b* 共 k⫹ 兲 ⫹ ␯ ␣␤ b共 k⫺ 兲 b* 共 k⫹ 兲 ⫹¯␯ ␣␤ b共 k⫹ 兲 b* 共 k⫺ 兲共18兲 ␯ ␯ with ␯ ␣ , ␤ , ␣␤ being distributions so that the matrix ( ¯␯ ␣ ␯ ␣␤ ) is positive definite. ␣␤

␤

The Wigner matrix W共x,k兲 of the family w⑀ (x) has the form W共 x,k兲 ⫽ ␮ 共 x,k兲 b共 x,k兲 b* 共 x,k兲 . 共19兲 The scalar measure ␮ is supported on the set (16) and satisfies weakly the transport equation ⫹ 共20兲 ⵜk␻ ⫹ •ⵜ x␮ ⫺ⵜ x␻ ⫹ •ⵜ k␮ ⫽ckˆ n 共 ␮ in␦ 共 k n ⫺k ⫺ n 兲 ⫹ ␮ out␦ 共 k n ⫺k n 兲兲 . Here the measures ␮ in and ␮ out are given by ␮in共 x⬘ ,k⬘ 兲 ⫽ ␯ ␣ 共 x⬘ ,k⬘ 兲 , ␮ out共 x⬘ ,k⬘ 兲 ⫽ ␯ ␤ 共 x⬘ ,k⬘ 兲共21兲 with ␯ ␣ , ␯ ␤ defined by (18), ␻ ⫹ (x,k)⫽c(x) 兩 k兩 is given by (2), and the wave vector k⫾ is defined by (6) with K⫽K(x⬘ )⫽ ␻ 冑␬ (x⬘ ) ␳ (x⬘ ).

Note also that if the measure ␮ is continuous up to the boundary x n ⫽0, then the weak form 共20兲 is equivalent to the boundary value problem: ⵜ k␻ ⫹ •ⵜ x␮ ⫺ⵜ x␻ ⫹ •ⵜ k␮ ⫽0,

共22兲

⫹ ␮ 共 x⬘ ,0,k兲 ⫽ ␮ in␦ 共 k n ⫺k ⫺ n 兲 ⫹ ␮ out␦ 共 k n ⫺k n 兲 .

共23兲

Thus we can interpret ␯ ␣ as the phase space resolved energy of the incoming waves at the boundary, and ␯ ␤ as the energy of the outgoing waves. D. Boundary conditions for the transport equation

The boundary conditions for the transport equation 共22兲 can be obtained from those for the acoustic equations 共12兲, by using the relations 共21兲 and 共23兲. Consider the Dirichlet boundary condition in the upper half space w ⑀ ,n⫹1 共 x⬘ ,0兲 ⫽0.

共24兲

Then the (n⫹1)-row and column of the matrix ␯ vanish. Using 共18兲, the explicit form 共14兲 of the eigenvectors b共x,k兲, and 共6兲, we get


4818



␯ ␣ ⫽ ␯ ␤ ⫽⫺ ␯ ␣␤ .

共25兲

␮ 共 x⬘ ,0,k⬘ ,k n 兲 ⫽ ␮ 共 x⬘ ,0,k⬘ ,⫺k n 兲 ,

共26兲

Then 共23兲 implies that

which is the boundary condition for 共22兲. The Neumann boundary condition w ⑀ ,n 共 x⬘ ,0兲 ⫽0

共27兲

implies that the nth row and column of the matrix ␯ vanish. Then we obtain, similarly to 共25兲, that

␯ ␣ ⫽ ␯ ␤ ⫽ ␯ ␣␤ ,

共28兲

so that 共26兲 still holds. A convenient way to rewrite 共26兲 is

␮ out共 x⬘ ,k⬘ 兲 ⫽ ␮ in共 x⬘ ,k⬘ 兲 , so that all the energy is reflected specularly as expected.

III. ENERGY REFLECTION AT A ROUGH SURFACE

We consider scattering of acoustic waves described by 共12兲 from a rough surface. The surface ⳵ H ⑀ is described by the equation x n ⫽ ⑀ ␩ h(x⬘ / ⑀ ). The small parameter ␩ is the ratio of the height of the surface to the wavelength. The height of the surface is varying on the scale of the wavelength ⑀. Recall that the random process h(y⬘ ) has mean zero, and is stationary, with covariance function R(y⬘ ) and with power spectrum Rˆ (k⬘ ), given by 共3兲 and 共4兲, respectively. We assume that h(y⬘ )苸C 1 (R n⫺1 ) a.s. so that solutions of the Dirichlet problem exist for every positive ⑀. The Dirichlet boundary condition for 共12兲, corresponding to a vanishing pressure, is given by w ⑀ ,n⫹1 共 x⬘ , ⑀ ␩ h 共 x⬘ / ⑀ 兲兲 ⫽0.

共29兲

We seek the solution w⑀ of 共12兲 as a power series in the parameter ␩: w⑀ 共 x兲 ⫽w0⑀ 共 x兲 ⫹ ␩ w1⑀ 共 x兲 ⫹ ␩ 2 w2⑀ 共 x兲 ⫹¯

共30兲

2 with all the terms bounded in L loc (R n ) and their boundary values r⑀j (x⬘ )⫽w⑀j (x⬘ ,0) bounded in 2 L loc(R n⫺1 ). Our main assumption is that such an expansion exists, i.e., that the rest in 共30兲 is bounded uniformly in ␩ and ⑀ by o( ␩ 2 ). We also assume that the medium is homogeneous above a certain height x n ⫽L with L arbitrarily large. This assumption is purely technical and allows us to formulate an outgoing condition at infinity. Namely, for x n ⬎L, the plane wave decomposition is valid, and we assume then that the amplitudes of the incoming waves ␣ ⑀ (k⬘ ) are deterministic and given. The correctors w⑀j are all outgoing in that region, so that ␣ ⑀j (k⬘ )⫽0 for j⭓1. The characteristics of the transport equation 共22兲 are given by

dx ⫽ⵜ k␻ ⫹ 共 x,k兲 , ds dk ⫽⫺ⵜ x␻ ⫹ 共 x,k兲 . ds We assume that the characteristics that leave the surface x n ⫽0 reach the level x n ⫽L and vice versa. Moreover, we assume that these characteristics do not come back. Then we have the following theorem.




4819

Theorem 1: Under the above assumptions, the Wigner matrix W共x,k兲 of the family w⑀ of solutions of the Dirichlet problem (29) is supported on the set S⫽ 兵 (x,k): c(x) 兩 k兩 ⫽ ␻ 其 , and has the form (19). The scalar measure ␮ has the form ␮ (x,k)⫽ ␮ ⬘ (x,k)⫹o( ␩ 2 ). The measure ⬘ (x⬘ ,k⬘ ) 典 is given ␮ ⬘ (x,k) is a solution of the transport equation (20). The average measure 具 ␮ out in terms of ␮ ⬘in(x⬘ ,k⬘ ) by

冋

⬘ 共 x⬘ ,k⬘ 兲典 ⫽ 1⫺4 ␩ 2 具 ␮ out ⫹4 ␩ 2

冕

冕

册

dp⬘ ⫹ ˆ 共 k⬘ ⫺p⬘ 兲 p ⫹ ⬘ 共 x⬘ ,k⬘ 兲 R n 共 x⬘ ,p 兲 k n 共 x⬘ ,k⬘ 兲 ␮ in 2 ␲ 共兲 n⫺1

dp⬘ 2 ˆ 共 k⬘ ⫺p⬘ 兲共 p ⫹ ⬘ 共 x⬘ ,p⬘ 兲 R n 共 p⬘ 兲兲 ␮ in 共 2 ␲ 兲 n⫺1

共31兲

with k⫹ and p⫹ given by (6) with K⫽K(x⬘ ). Here K(x⬘ ), determined by c(x⬘ )K(x⬘ )⫽ ␻ , is the radius of the sphere S of wave vectors, on which the measure ␮ is supported. The first term in 共31兲 corresponds to the specular reflection and provides the correction to the reflection coefficient. The second term is the result of the diffuse scattering at the surface. It appears because the boundary is varying on the scale of the wavelength. Note that the total mean flux across the boundary x n ⫽0 vanishes:

冕 ⬘ 冕

具 Fn 共 x⬘ ,0兲典 ⫽c 共 x⬘ 兲 ⫽c 共 x 兲

dk kˆ n 具 ␮ 共 x⬘ ,0,k兲典 2 ˆ⫺ dk⬘ 关 kˆ ⫹ n ⫹k n 兴 ␮ in共 x⬘ ,k⬘ 兲 ⫹4 ␩ c 共 x⬘ 兲

冕

dk⬘ dp⬘ Rˆ 共 k⬘ ⫺p⬘ 兲

⫹ ⫹ˆ⫹ 2ˆ⫹ ⫻兵共 p ⫹ n 兲 k n ␮ in共 x⬘ ,k⬘ 兲 ⫺ p n k n k n ␮ in共 x⬘ ,p⬘ 兲其 ⫽0.

The expression 共31兲 reduces in a homogeneous medium to 共5兲 as one would expect. The statement regarding the support and form of the matrix W共x,k兲 is proved exactly as in Ref. 8, so we will not repeat it here. We derive 共31兲 in the following sections. Note that we can treat more general interface problems in a similar way, at least for incident energy fluxes away from grazing angle. The result is then a direct generalization of the formulas given in Ref. 7 for homogeneous media.

IV. THE PERTURBATION ANALYSIS

Now, we show how the diffusive scattering is obtained when the wave equation satisfies Dirichlet boundary conditions. We derive 共31兲 in three steps. First we analyze the asymptotic expansion 共30兲 in Sec. IV A. Then the diffuse part of the scattered energy is computed in Sec. IV B. At last, we derive the correction to the reflection coefficient in Sec. IV C. A. The asymptotic expansion 2 We note that since the series 共30兲 is asymptotic in L loc (R n ), the Wigner matrix W is approximated by the Wigner matrix of the sum of the first N terms up to order o( ␩ N ). Thus we can compute the Wigner measure of the first three terms in the expansion 共30兲 in order to approximate W up to o( ␩ 2 ). The terms w⑀j , j⫽0,1,2 solve the acoustic equations 共12兲 in the upper half space with the following boundary conditions:

w ⑀0,n⫹1 共 x⬘ ,0兲 ⫽0,

共32兲

冉冊

共33兲

w ⑀1,n⫹1 共 x⬘ ,0兲 ⫽⫺h

x⬘ ⳵ w ⑀0,n⫹1 ⑀ 共 x⬘ ,0兲 , ⑀ ⳵xn


4820



冉冊

冉冊

x⬘ 2 ⳵ 2 w ⑀0,n⫹1 x⬘ ⳵ w ⑀1,n⫹1 1 w ⑀2,n⫹1 共 x⬘ ,0兲 ⫽⫺ h 2 ⑀ x ,0 ⫺h ⑀ 兲共共 x⬘ ,0兲 . ⬘ 2 ⑀ ⑀ ⳵xn ⳵ x 2n

共34兲

These boundary conditions are constructed so as to satisfy the Dirichlet boundary conditions 共29兲 on the rough surface ⳵ H ⑀ up to order ␩ 2 . The boundary conditions at infinity, where the medium is homogeneous above the level x n ⫽L, are as described above: w⑀0 has prescribed incoming flux, and w⑀1,2 are outgoing for x n ⬎L. The boundary conditions 共33兲 and 共34兲 can be rewritten using the acoustic equations for w0⑀ and w⑀1 and the Dirichlet boundary condition 共32兲 as r ⑀1,n⫹1 共 x⬘ 兲 ⫽⫺i ␻␳ 共 x⬘ 兲 h

冉冊

x⬘ 0 r 共 x⬘ 兲 ⑀ ⑀ ,n

共35兲

and r ⑀2,n⫹1 共 x⬘ 兲 ⫽⫺

冉冊

冉冊

i ⑀ ␻ ⳵␳ x⬘ 0 x⬘ 1 r ⑀ ,n 共 x⬘ 兲 ⫺i ␻␳ 共 x⬘ 兲 h r 共 x⬘ 兲 . 共 x⬘ ,0兲 h 2 2 ⳵xn ⑀ ⑀ ⑀ ,n

共36兲

Here r⑀j (x⬘ ) is the value of w⑀j at the boundary. The Wigner matrix of the sum of the first three terms in 共30兲 has the form W⫽W0 ⫹ ␩ 共 W01⫹W10兲 ⫹ ␩ 2 共 W1 ⫹W02⫹W20兲 ⫹o 共 ␩ 2 兲 ,

共37兲

where W0 ⫽W关 w0⑀ 兴 , W1 ⫽W关 w⑀1 兴 , W01⫽W关 w⑀0 ,w⑀1 兴 , etc. The leading order term in the expansion 共37兲 has the form W0 ⫽ ␮ 0 共 x,k兲 b共 x,k兲 b* 共 x,k兲

共38兲

since w0⑀ solves the acoustic equations. The Dirichlet boundary conditions 共32兲 for w0⑀ imply that the Wigner measure ␯0 of r⑀0 has the form 共18兲 with the coefficients ␯ ␣ , ␤ , ␣␤ related by 共25兲:

␯0 共 x⬘ ,k⬘ 兲 ⫽ ␯ ␣0 共 x⬘ ,k⬘ 兲关 b共 k⫹ 兲 b* 共 k⫹ 兲 ⫹b共 k⫺ 兲 b* 共 k⫺ 兲 ⫺b共 k⫹ 兲 b* 共 k⫺ 兲 ⫺b共 k⫺ 兲 b* 共 k⫹ 兲兴 .

共39兲

0 Then the outgoing Wigner measure ␮ out is 0 ␮ out ⫽ ␮ in ,

共40兲

where the measure ␮ in is known. Notice that 具 w1⑀ (x) 典 ⫽0, and since w⑀0 is deterministic, we get

具 W01典 ⫽ 具 W10典 ⫽0.

共41兲

Thus we have to compute only 具 W1 典 and 具 W02⫹W20典 . The first term gives rise to the diffuse scattering and is treated in the following section. The second term produces the correction to the energy reflection coefficient and is considered in Sec. IV C.

B. The diffuse reflected wave

The matrix W1 , being the Wigner matrix of a family of solutions w1⑀ of the acoustic equations, has the form W1 ⫽ ␮ 1 共 x,k兲 b共 x,k兲 b* 共 x,k兲 .




4821

The solution w⑀1 is outgoing at infinity. Since we assumed that characteristics do not come back to the boundary, it is equivalent to being outgoing at the boundary. Then we have ␯ ␣1 ⫽ ␮ 1in⫽0. Since ␯

1

the matrix ( ¯ 1␣

1

␯ ␣␤

1 ␯ ␣␤ ␯ ␤

1 ) is non-negative and definite, we also have ␯ ␣␤ ⫽0. Then the boundary Wigner

measure ␯1 of r⑀1 has the form

␯1 ⫽ ␯ ␤1 b共 k⫹ 兲 b* 共 k⫹ 兲

共42兲

and the boundary value of the measure ␮ 1 is

␮ 1 共 x⬘ ,0,k兲 ⫽ ␯ ␤1 共 x⬘ ,k⬘ 兲 ␦ 共 k n ⫺k ⫹ n 兲, 1 hence ␮ out ⫽ ␯ ␤1 (x⬘ ,k⬘ ). The measure ␯ ␤1 is determined as follows. We have from 共42兲, 1 ␯ n⫹1,n⫹1 ⫽

1 ␯1 , 2 ␬ 共 x⬘ 兲 ␤

共43兲

where the left side is the (n⫹1, n⫹1) entry of the matrix ␯1 (x⬘ ,x⬘ ). This entry is the Wigner measure of r ⑀1,n⫹1 (x⬘ ), which is explicitly given by the boundary condition 共35兲. Therefore,

冋冉冊册

1 ␯ n⫹1,n⫹1 ⫽ ␻ 2 ␳ 2 共 x⬘ 兲 ␯ h

x⬘ 0 r ⑀ ⑀ ,n

1 and so the average 具 ␯ n⫹1,n⫹1 典 is given by 1 具 ␯ n⫹1,n⫹1 典 ⫽ ␻ 2 ␳ 2 共 x⬘ 兲

冕

dp⬘ 0 ˆ 共 k⬘ ⫺p⬘ 兲 ␯ n,n R 共 x⬘ ,p⬘ 兲 . 共 2 ␲ 兲 n⫺1

共44兲

We use expression 共39兲 for ␯0 to get 1 具 ␯ n⫹1,n⫹1 典共 x⬘ ,k⬘ 兲 ⫽ ␻ 2 ␳ 2 共 x⬘ 兲

冕

4 ␯ ␣0 共 p⬘ 兲 p ⫹2 dp⬘ n ˆ R 共 k⬘ ⫺p⬘ 兲 . 2 ␳ 共 x⬘ 兲兩 p⫹ 兩 2 共 2 ␲ 兲 n⫺1

Insert this into 共43兲 and use the relations 兩 p⫹ 兩 2 ⫽ ␻ 2 /c 2 (x⬘ )⫽ ␻ 2 ␬ (x⬘ ) ␳ (x⬘ ) and ␮ in⫽ ␯ ␣0 yields 1 典 ⫽ 具 ␯ ␤1 典共 x⬘ ,k⬘ 兲 ⫽4 具 ␮ out

冕

dp⬘ 共 2 ␲ 兲 n⫺1

ˆ 共 k⬘ ⫺p⬘ 兲 p ⫹2 R n ␮ in共 x⬘ ,p⬘ 兲 .

共45兲

This is the second term in 共31兲.

C. Correction to the reflection coefficient

The correction to the coherent, or specular reflection coefficient arises from the term 具 W02 ⫹W20典 in 共37兲. Our analysis of this term proceeds in several steps. In Sec. IV C 1 we reduce the computation to evaluating the average 具 ␯关 r0 ,u1 兴典 of the cross Wigner distribution of r⑀0 and the conjugated wave function u1⑀ defined by 共51兲. This Wigner distribution is described by Lemma 1. We use this lemma in Sec. IV C 2 to derive 共31兲. Finally we prove Lemma 1 in Sec. IV C 3. 1. The reduction to the conjugated wave functions

The functions w0⑀ ,w2⑀ and w⑀0 ⫹w2⑀ are solutions of the acoustic equations, so the Wigner matrices W关 w0 兴 ,W关 w2 兴 and W关 w0 ⫹w2 兴 are all of the form 共19兲, and then


4822



W02⫹W20⫽W关 w0 ⫹w2 兴 ⫺W关 w0 兴 ⫺W关 w2 兴 ⫽ 共 ␮ 关 w0 ⫹w2 兴 ⫺ ␮ 关 w0 兴 ⫺ ␮ 关 w2 兴兲 b共 k兲 b* 共 k兲 ⫽ ␮ 02b共 k兲 b* 共 k兲 ,

共46兲

so that W02⫹W20 has the same form as W0 and W1 . The value of ␮ 02 on the boundary is

␮ 02共 x⬘ ,0,k兲 ⫽ 共 ␮ 关 w0 ⫹w2 兴 ⫺ ␮ 关 w0 兴 ⫺ ␮ 关 w2 兴兲共 x⬘ ,0,k兲 ⫽ 共 ␯ ␤ 关 r0 ⫹r2 兴 ⫺ ␯ ␤ 关 r0 兴 ⫺ ␯ ␤ 关 r2 兴兲共 x⬘ ,k⬘ 兲 ␦ 共 k n ⫺k ⫹ n 兲 ⫹ 02 ⫽ 共 ␯ ␤ 关 r0 ,r2 兴 ⫹ ␯ ␤ 关 r2 ,r0 兴兲 ␦ 共 k n ⫺k ⫹ n 兲 ⫽ ␯ ␤ ␦ 共 k n ⫺k n 兲 .

共47兲

The terms involving the incoming wave vector k⫺ cancel out since w2 is outgoing at the boundary and thus ␮ in关 w2 兴 ⫽0, and ␮ in关 w0 ⫹w2 兴 ⫽ ␮ in关 w0 兴 . Thus we have to find ␯ ␤02⫽ ␯ ␤ 关 r0 ,r2 兴 ⫹ ␯ ␤ 关 r2 ,r0 兴 to finish the computation. We note that the matrix Wigner measure ␯关 r0 ,r2 兴 ⫹ ␯关 r2 ,r0 兴 has the form

␯关 r0 ,r2 兴 ⫹ ␯关 r2 ,r0 兴 ⫽ 共 ␯ ␤ 关 r0 ,r2 兴 ⫹ ␯ ␤ 关 r2 ,r0 兴兲 b共 k⫹ 兲 b* 共 k⫹ 兲 02 02 ⫹ ␯ ␣␤ b共 k⫺ 兲 b* 共 k⫹ 兲 ⫹¯␯ ␣␤ b共 k⫹ 兲 b* 共 k⫺ 兲

共48兲

since r⑀2 is outgoing. Evaluating the entry (n⫹1,n⫹1) of both sides of 共48兲, we get that ␯ ␤02 is real. Moreover, evaluating the entry (n, n⫹1) of the two sides of 共48兲 we obtain 02 ␮ out ⫽ ␯ ␤02⫽

2 冑␬ ␳ kˆ ⫹ n

2 Re ␯ 关 r 0n ,r n⫹1 兴.

共49兲

2 Thus we have reduced our problem to computing ␯ 关 r 0n ,r n⫹1 兴 . Recall that r ⑀2,n⫹1 (x⬘ ) is given by 2 . Then we have 共36兲 and observe that the first term in 共36兲 vanishes in the limit ⑀ →0 in L loc

冋

2 ␯ 关 r 0n ,r n⫹1 兴 ⫽ ␯ r 0n 共 x⬘ 兲 ,⫺h

⫽i ␻␳ 共 x⬘ 兲

冕

冉冊

x⬘ i ␻␳ 共 x⬘ 兲 r ⑀1n 共 x⬘ 兲 ⑀

册

dp⬘ ˆh 共 p⬘ 兲 ␯ nn 关 r0 ,u1 兴共 x⬘ ,k⬘ ,p⬘ 兲 . 共 2 ␲ 兲 n⫺1

共50兲

Here we have introduced the conjugated wave functions u⑀1 : u⑀1 共 x⬘ ,p⬘ 兲 ⫽r⑀ 共 x⬘ 兲 e ⫺ip•x⬘ / ⑀ ,

共51兲

where the vector p⬘ plays the role of a parameter. The Wigner matrix ␯关 r0 ,u1 兴 is described by the following Lemma. Lemma 1: The Wigner distribution matrix ␯关 r0 ,u1 兴 has the form

␯关 r0 ,u1 兴 ⫽˜␯ ␣ b共 k⫹ 兲 b* 共共 k⫹p兲 ⫹ 兲 ⫺˜␯ ␣ b共 k⫺ 兲 b* 共共 k⫹p兲 ⫹ 兲 ,

共52兲

where ˜␯ ␣ is some distribution. We prove Lemma 1 in Sec. IV C 3. 2. The average reflection coefficient

We use Lemma 1 to evaluate ␯ nn 关 r0 ,u1 兴 in 共50兲 and get

冓冋冉冊册冔冕 ␯

r 0n ,h

x⬘ 1 r ⑀ ⑀n

⫽

⫹ k⫹ dp⬘ n 共 k⫹p 兲 ˆ ␯ ␣ 共 x⬘ ,k⬘ ,p⬘ 兲典 ⫹ . n⫺1 具 h 共 p⬘ 兲˜ ⫹ 兩 k 兩兩共 k⫹p兲兩 ␳ 共 x兲共2␲兲

共53兲




4823

The average 具 hˆ (p)˜␯ ␣ 典 is evaluated as follows. We have from the boundary conditions 共35兲, u ⑀1,n⫹1 共 x⬘ 兲 ⫽e ⫺ip⬘ •x⬘ / ⑀ r ⑀1,n⫹1 共 x⬘ 兲 ⫽⫺e ⫺ip⬘ •x⬘ / ⑀ i ␻␳ 共 x⬘ 兲 h

冉冊

x⬘ 0 r 共 x⬘ 兲 ⑀ ⑀ ,n

and hence 0 共 x⬘ ,k⬘ 兲 . 具 h 共 p⬘ 兲 ␯ n,n⫹1 关 r0 ,u1 兴典 ⫽i ␻␳ 共 x兲 Rˆ 共 p⬘ 兲 ␯ nn

共54兲

But we also have from the representation 共52兲,

具 hˆ 共 p⬘ 兲 ␯ n,n⫹1 关 r0 ,u1 兴典 ⫽ 具 hˆ 共 p⬘ 兲˜␯ ␣ 典

k⫹ n 兩 k⫹ 兩冑␳ ␬

,

so that 共54兲 implies

具 hˆ 共 p⬘ 兲˜␯ ␣ 典

k⫹ n ⫹

兩k

兩冑␳

0 ⫽ 冑␬ Rˆ 共 p⬘ 兲 i ␻␳ 共 x⬘ 兲 ␯ nn .

We insert this into 共53兲 and get

冓冋冉冊册冔冕 ␯

r ⑀0n ,h

x⬘ 1 r ⑀ ⑀n

⫽

⫽

冕

共 k⫹p兲 ⫹ dp⬘ n 0 ˆ 冑 R 共 p⬘ 兲 ␬ ␳ i ␻␯ nn 共 x⬘ ,k⬘ 兲兩共 k⫹p兲 ⫹ 兩共 2 ␲ 兲 n⫺1

dp⬘ 0 ˆ 共 k⬘ ⫺p⬘ 兲冑␬ ␳ i ␻ pˆ ⫹ R n ␯ nn 共 x⬘ ,k⬘ 兲 . 共 2 ␲ 兲 n⫺1

共55兲

Then we insert 共55兲 into 共50兲 and obtain 2 兴典 ⫽⫺ ␻ 2 ␳ 共 x⬘ 兲具 ␯ 关 r 0n ,r n⫹1

冑␬ 共 x⬘ 兲 ␳ 共 x⬘ 兲

冕

p⫹ dp⬘ n ˆ R 共 k⬘ ⫺p⬘ 兲 ␯ 0 共 x⬘ ,k⬘ 兲 . 兩 p⫹ 兩 nn 共 2 ␲ 兲 n⫺1

Recall that we have from 共39兲 0 ␯ nn 共 x⬘ ,k⬘ 兲 ⫽

2 4 ␯ ␣0 共 kˆ ⫹ n 兲

2␳

.

Then we get from 共49兲 and 共55兲 02 具 ␮ out 典 ⫽ 具 ␯ ␤02典 ⫽⫺

2 冑␬ ␳ ␻ 2 ␳ kˆ ⫹ n

c

⫽⫺4 ␯ ␣0 共 x⬘ ,k⬘ 兲

冕冕

dp⬘ 共2␲兲

ˆ 共 k⬘ ⫺p⬘ 兲 pˆ ⫹ R n n⫺1

dp⬘ 共 2 ␲ 兲 n⫺1

2 4 ␯ ␣0 共 kˆ ⫹ n 兲

⫹ ˆ 共 k⬘ ⫺p⬘ 兲 p ⫹ R n kn

2␳ 共56兲

because 兩 k⫹ 兩 ⫽ 兩 p⫹ 兩 ⫽ ␻ /c. Putting together 共40兲, 共45兲, and 共56兲, we get 共31兲.

3. Proof of Lemma 1

The proof of Lemma 1 is similar to that of the first part of Theorem 2, which is given in Ref. 8. The functions u⑀1 (x⬘ ) satisfy the system


4824

J. Math. Phys., Vol. 40, No. 10, October 1999 n⫺1

兺

j⫽1


n⫺1

⑀D j

⳵ u⑀1 ⳵ w⑀1 ⫹i p j D j u⑀1 ⫺i ␻ A 共 x⬘ 兲 u⑀1 ⫽⫺ ⑀ D n 共 x⬘ ,0兲 , ⳵x j ⳵xn j⫽1

兺

共57兲

while r⑀0 (x⬘ )⫽w0⑀ (x⬘ ,0) satisfy the system n⫺1

兺

j⫽1

⳵ r0⑀ ⳵ w0⑀ 0 n ⑀D ⫺i ␻ A 共 x⬘ 兲 r⑀ ⫽⫺ ⑀ D 共 x⬘ ,0兲 . ⳵x j ⳵xn j

共58兲

Let us define the reduced dispersion matrix n⫺1

L ⬘ 共 x,k兲 ⫽

兺

j⫽1

k j D j ⫺ ␻ A 共 x兲 .

Let P be any matrix such that PD n ⫽0, then 共57兲 implies that PL ⬘ 共 k⫹p兲 ␯关 u1 ,r0 兴 ⫽0 and 共58兲 implies that PL ⬘ 共 k兲 ␯关 r0 ,u1 兴 ⫽0. Then the Wigner matrix ␯关 r0 ,u1 兴 has the specific form

␯关 r0 ,u1 兴 ⫽˜␯ ␣ b共 k⫹ 兲 b* 共共 k⫹p兲 ⫹ 兲 ⫹˜␯ ␤ b共 k⫺1 兲 b* 共共 k⫹p兲 ⫺ 兲 ⫹˜␯ ␣␤ b共 k⫹ 兲 b* 共共 k⫹p兲 ⫺ 兲 ⫹˜␯ ␤␣ b共 k⫺ 兲 b* 共共 k⫹p兲 ⫹ 兲 .

共59兲

Recall that u⑀1 is outgoing, so 共59兲 reduces to

␯关 r0 ,u1 兴 ⫽˜␯ ␣ b共 k⫹ 兲 b* 共共 k⫹p兲 ⫹ 兲 ⫹˜␯ ␤␣ b共 k⫺ 兲 b* 共共 k⫹p兲 ⫹ 兲 .

共60兲

The Dirichlet boundary conditions for w⑀0 (x) imply that the fourth row of the matrix ␯关 r0 ,u1 兴 vanishes. Then ˜␯ ␣␤ ⫽⫺˜␯ ␣ , and 共60兲 becomes

␯关 r0 ,u1 兴 ⫽˜␯ ␣ b共 k⫹ 兲 b* 共共 k⫹p兲 ⫹ 兲 ⫺˜␯ ␣ b共 k⫺ 兲 b* 共共 k⫹p兲 ⫹ 兲 ,

共61兲

which is 共52兲. V. HIGHER ORDER ANISOTROPIC EFFECTS

The scattering cross-sections in the correctors of order ␩ 2 in 共31兲 depend upon the outgoing direction only through the power spectrum Rˆ (k⬘ ⫺p⬘ ). We show in this section that some anisotropy of the scattered field is captured in the higher order terms. We obtain, in particular, a peak in the backscattering direction. This peak is similar to the coherent backscattering effect in the Neumann and impedance problems in homogeneous media studied in Refs. 17, 12, 13, and 18, although much broader for Dirichlet boundary conditions. The nature of this peak is also the constructive interference between direct and reverse paths of scattered waves. The singular behavior of the reflection operator at grazing angles in the Neumann and impedance problems makes this effect much more pronounced. The study of the coherent backscattering effect for these cases in an inhomogeneous medium requires appropriate modifications and the smoothing method.19 A. The backscattering for the Dirichlet problem

The angularly resolved energy density of the second order corrector in the Dirichlet problem is described by the following proposition.




4825

Proposition 3: Assume that h(y) is a mean zero Gaussian process with covariance matrix R(y). Then the Wigner matrix W2 of the second order term w2⑀ in the asymptotic expansion (30) of the solution of the Dirichlet problem has the form W2 ⫽ ␮ 2 (x,k)b(x,k)b* (x,k). The scalar measure ␮ 2 (x,k) is supported on the sphere S(x)⫽ 兵 k: 兩 k兩 ⫽ ␻ /c(x) 其 and satisfies the transport equation (20) with ␮ in 2 ⫽0 and

具 ␮ out 2 共 x⬘ ,k⬘ 兲典 ⫽4

冕

⫹4

dp⬘ dq⬘ ˆ 共 k⬘ ⫺p⬘ 兲 Rˆ 共 p⬘ ⫺q⬘ 兲 q 2n p n 关 p n ⫹ 共 q⬘ ⫹k⬘ ⫺p⬘ 兲 n 兴 ␮ in ⬘ 共 q⬘ 兲 R 2 共 ␲ 兲 2d⫺2

冕

dp⬘ dq⬘ ˆ 共 k⬘ ⫺p⬘ 兲 Rˆ 共 k⬘ ⫺q⬘ 兲 p n q n k 2n ␮ in ⬘ 共 k⬘ 兲 . R 共 2 ␲ 兲 2d⫺2

共62兲

Here k n (x,k⬘ )⫽ 冑␻ 2 /c 2 (x) is the normal component of the outgoing wave vector in S, which has ⬘ is as in Theorem 1. horizontal component k⬘ and is pointing upwards, and ␮ in The first term in 共62兲 corresponds to the diffusive scattering. The second term provides a correction to the reflection coefficient. The differential scattering cross-section in 共62兲 is no longer isotropic, and is centered in the backscattered direction. This can be seen as follows. Let us ⬘ (q⬘ )⫽C(x⬘ ) ␦ (q⬘ ⫺q⬘0 ), so that waves are assume that the incident energy density has the form ␮ in coming from a single direction q0 . Then the diffusive scattering is maximal in the direction with tangential component k⬘ ⫽⫺q0⬘ because both terms in the diffusive scattering cross section are the same. The second term in this cross section is smaller in other directions, because when k⫹q0 ⫽0, then the integration in p⬘ in that term is carried over the region where both p⬘ and k⬘ ⫹q⬘0 ⫺p⬘ lie in the disk of radius K⫽ ␻ /c. This region is shrinking as k moves away from ⫺q0 , and so the contribution of this term diminishes. In particular, if q⬘0 is close to the boundary of the disk, and the incident wave is close to the grazing angle, then the contribution of this term in the forward direction k⬘ ⫽q0⬘ vanishes. This can be interpreted as an enhanced backscattering phenomenon, since the contribution of this term in 共62兲 corresponds to the interference of the direct and reverse paths, as will be seen in the derivation of 共62兲.

B. Derivation of the scattering cross-section

The statements regarding the form of the Wigner matrix W2 and the support of the measure ␮ 2 in Proposition 3 follow immediately from Proposition 2, and from the fact that w⑀2 is outgoing at infinity, together with our assumption that characteristics do not come back to the boundary. Thus, the only part we have to verify is the expression 共62兲 for the measure ␮ 2 at the boundary. Let r⑀2 (x⬘ )⫽w2⑀ (x⬘ ,0) be the boundary value of w2 . Then we have, using 共36兲 2 1 2 2 ␮ out 2 共 x⬘ ,k⬘ 兲 ⫽2 ␬ 共 x⬘ 兲 ␯ 关 r ⑀ ,n⫹1 兴 ⫽2 ␬ 共 x⬘ 兲 ␻ ␳ 共 x⬘ 兲 ␯ 关 h 共 x⬘ / ⑀ 兲 r ⑀ ,n 兴 .

共63兲

This can be rewritten with the help of the conjugated wave functions 共51兲:

␯ 关 h 共 x⬘ / ⑀ 兲 r ⑀1,n 兴 ⫽

冕

dp⬘ dq⬘ ˆh 共 p⬘ 兲 hˆ 共 q⬘ 兲 ␯ 关 u ⑀1,n 共 x⬘ ,⫺p⬘ 兲 ,u ⑀1,n 共 x⬘ ,q⬘ 兲兴 . 共 2 ␲ 兲 2d⫺2

It is easy to check that, similarly to Lemma 1 we have for any p⬘1 ,p2⬘ :

␯关 u1⑀ 共 p⬘1 兲 ,u⑀1 共 p2⬘ 兲兴共 x⬘ ,k⬘ 兲 ⫽ ␪ 共 x⬘ ,k⬘ ,p⬘1 ,p⬘2 兲 b共 k⬘ ⫹p1⬘ 兲 b* 共 k⬘ ⫹p⬘2 兲

共64兲

with ␪ being some unknown distribution. Thus we have

具 ␯ 关 h 共 x⬘ / ⑀ 兲 r ⑀1,n 兴典 ⫽

冕

dp⬘ dq⬘ 具 hˆ 共 p⬘ 兲 hˆ 共 q⬘ 兲 ␪ 共 x⬘ ,k⬘ ,⫺p⬘ ,q⬘ 兲典 b n 共 k⬘ ⫺p⬘ 兲 b n 共 k⬘ ⫹q⬘ 兲共65兲共 2 ␲ 兲 2d⫺2

and we need to evaluate the average inside the integral. Expression 共64兲 implies that


4826



具 hˆ 共 p⬘ 兲 hˆ 共 q⬘ 兲 ␪ 共 x⬘ ,k⬘ ,⫺p⬘ ,q⬘ 兲典 ⫽2 ␬ 具 hˆ 共 p⬘ 兲 hˆ 共 q⬘ 兲 ␯ 关 u ⑀1,n⫹1 共 ⫺p⬘ 兲 ,u ⑀1,n⫹1 共 q⬘ 兲兴共 x⬘ ,k⬘ 兲典 .

共66兲

The average on the right side can be computed using the expression 共35兲 for r ⑀1,n⫹1 in terms of r ⑀0,n , and the assumption that h(y) is a Gaussian random process so that

具 hˆ 共 p兲 hˆ 共 q兲 hˆ 共 p2 兲 hˆ 共 q2 兲典 ⫽ 具 hˆ 共 p兲 hˆ 共 q兲典具 hˆ 共 p2 兲 hˆ 共 q2 兲典 ⫹ 具 hˆ 共 p兲 hˆ 共 q2 兲典具 hˆ 共 q兲 hˆ 共 p2 兲典 ⫹ 具 hˆ 共 p兲 hˆ 共 p2 兲典具 hˆ 共 q兲 hˆ 共 q2 兲典 .

共67兲

具 hˆ 共 p⬘ 兲 hˆ 共 q⬘ 兲 ␯ 关 u ⑀1,n⫹1 共 ⫺p⬘ 兲 ,u ⑀1,n⫹1 共 q⬘ 兲兴共 x⬘ ,k⬘ 兲典 ⫽I⫹II⫹III,

共68兲

Then we get

where I⫽ ␻ 2 ␳ 2 共 x⬘ 兲 Rˆ 共 p⬘ 兲 ␦ 共 p⬘ ⫹q⬘ 兲

冕

0 dp1⬘ Rˆ 共 p1⬘ 兲 ␯ nn 共 k⬘ ⫺p⬘ ⫺p1⬘ 兲 ,

0 II⫽ ␻ 2 ␳ 2 共 x⬘ 兲 Rˆ 共 p⬘ 兲 Rˆ 共 q⬘ 兲 ␯ nn 共 k⬘ ⫹q⬘ ⫺p⬘ 兲 ,

and 0 III⫽ ␻ 2 ␳ 2 共 x⬘ 兲 Rˆ 共 p⬘ 兲 Rˆ 共 q⬘ 兲 ␯ nn 共 k⬘ 兲 .

The three terms in 共68兲 have a natural interpretation in terms of wave scattering from a collection of discrete random scatterers.18 The first term in 共68兲 comes from the first term in 共67兲 and corresponds to the interaction of a path with itself. It produces a scattering cross-section that is essentially isotropic. The second term in 共68兲 comes from the second term in 共67兲 and corresponds to the interaction of a path and its reverse one in the discrete picture. It has a peak in the backscattering direction as explained in the previous section. The last term arises from paths scattering twice on the same scatterer. It contributes to the specular reflection coefficient. 0 (k⬘ )⫽(k 2n /2␳ 兩 k兩 2 )4 ␮ ⬘in(x⬘ ,k⬘ ), and putting this together with 共63兲, Finally we note that ␯ nn 共65兲, 共66兲, we obtain 共62兲. This completes the proof of Proposition 3. VI. CONCLUSIONS

We have derived the boundary conditions for the transport equation for the phase space resolved energy density in an inhomogeneous medium. Our derivation is based on the assumption that the asymptotic expansion 共30兲 holds. These boundary conditions can be used for the radiative transport equation for acoustic waves1,2 when randomness of the medium is independent of the randomness of the surface. Moreover, one can use similar boundary conditions for more general radiative transport equations2 for electromagnetic, elastic and other waves in domains with rough boundaries. Our result may also be generalized to reflection and transmission at interfaces between two inhomogeneous media. The results are then a generalization of the diffuse energy reflection and transmission at a rough interface considered in Ref. 7. The analysis of the Neumann problem in an inhomogeneous medium with a rough boundary requires the smoothing method or any equivalent regularization technique. We plan to address this in a separate note.19 This allows one to incorporate the coherent backscattering effect into the boundary conditions for the radiative transport equation. ACKNOWLEDGMENTS

We are grateful to Joseph B. Keller for numerous valuable discussions. L.R. would like to thank Luc Miller for his hospitality and stimulating discussions. The work of Bal and Papanico-




4827

laou was supported by AFOSR Grant No. F49620-98-1-0211 and by NSF Grant No. DMS9622854. The work of Ryzhik was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-9400379. S. Chandrasekhar, Radiative Transfer 共Oxford University Press, Cambridge, 1960兲. L. Ryzhik, G. Papanicolaou, and J. B. Keller, ‘‘Transport equations for elastic and other waves in random media,’’ Wave Motion 24, 327–370 共1996兲. 3 J. A. DeSanto and G. S. Brown, ‘‘Analytical techniques for multiple scattering from rough surfaces,’’ in Progress in Optics XXIII, edited by E. Wolf 共Elsevier, Amsterdam, 1986兲, pp. 1–62. 4 I. M. Fuks and F. G. Bass, Wave Scattering from Statistically Rough Surfaces 共Oxford; Pergamon, New York, 1979兲. 5 M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics 共Wiley, New York, 1991兲. 6 A. Voronovich, Wave Scattering from Rough Surfaces 共Springer–Verlag, Berlin, 1991兲. 7 G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, ‘‘Transport theory for waves with reflection and transmission at interfaces,’’ Wave Motion 共in press兲. 8 L. Ryzhik, G. Papanicolaou, and J. B. Keller, ‘‘Transport equations in a half space,’’ Commun. Partial Diff. Eqns. 22, 1869–1910 共1997兲. 9 J. B. Keller and R. Lewis, ‘‘Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell’s equations,’’ in Surveys in Applied Mathematics, edited by J. B. Keller, D. McLaughlin, and G. Papanicolaou 共Plenum, New York, 1995兲. 10 V. Twersky, ‘‘On scattering and reflection of sound by rough surfaces,’’ J. Acoust. Soc. Am. 29, 209–225 共1957兲. 11 J. G. Watson and J. B. Keller, ‘‘Rough surface scattering via the smoothing method,’’ J. Acoust. Soc. Am. 75, 1705– 1708 共1984兲. 12 A. A. Maradudin and E. R. Mendez, ‘‘Enhanced backscattering of light from weakly rough, random metal surfaces,’’ Appl. Opt. 32, 3335–3343 共1993兲. 13 H. Ogura and N. Takahashi, ‘‘Wave scattering from a random rough surface: reciprocal theorem and backscattering enhancement,’’ Waves Random Media 5, 232–242 共1995兲. 14 P. Ge´rard, P. Markowich, N. Mauser, and F. Poupaud, ‘‘Homogenization limits and Wigner transforms,’’ Commun. Pure Appl. Math. 50, 323–380 共1997兲. 15 L. Miller, ‘‘Propagation d’ondes semi-classiques a travers une interface et measures 2-microlocales,’’ Doctorat de l’Ecole Polytechnique, Palaiseau. 16 L. Miller, ‘‘Refraction d’ondes semi-classiques par des interfaces franches,’’ C. R. Acad. Sci., Ser. I: Math. 325, 371–376 共1997兲. 17 R. M. Fitzgerald, A. A. Maradudin, and F. Pincemin, ‘‘Scattering of a scalar wave from a two-dimensional randomly rough Neumann surface,’’ Waves Random Media 5, 381–411 共1995兲. 18 P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena 共Academic, New York, 1995兲. 19 G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, ‘‘Coherent Backscattering from a Random Surface via the Smoothing Method,’’ preprint, 1999. 1 2


Using JMP format - Semantic Scholar

Using JMP format - Semantic Scholar

Suggest Documents

Using JMP format

Using JMP format

Using JMP format

Using JMP format - Jason Cantarella

Using JMP format - The Institute of Optics - University of Rochester

Using CIS format - Semantic Scholar

Using PHP format - Semantic Scholar

Using PHP format - Semantic Scholar

Using PHP format - Semantic Scholar

Using RSI format - Semantic Scholar

Using RSI format - Semantic Scholar

Using ASME format - Semantic Scholar

Using PHP format - Semantic Scholar

Using PHP format - Semantic Scholar

Using JEI format - Semantic Scholar

Using RSI format - Semantic Scholar

Using PHF format - Semantic Scholar

Using PHP format - Semantic Scholar

Using PHF format - Semantic Scholar

Using PHP format - Semantic Scholar

Using JEI format - Semantic Scholar

Using PHP format - Semantic Scholar

Using PHP format - Semantic Scholar