Reflection and transmission by randomly spaced elastic cylinders in a

Reflection and transmission by randomly spaced elastic cylinders in a fluid slab-like region Pierre-Yves Le Bas, Francine Luppe´,a) and Jean-Marc Conoir LAUE, UMR CNRS 6068, Universite´ du Havre, place R. Schuman, 76610 Le Havre, France

共Received 4 May 2004; revised 10 October 2004; accepted 16 November 2004兲 An extension of Fikioris and Waterman’s formalism is developed in order to describe both the reflection and transmission from a slab-like fluid region in which elastic cylindrical scatterers are randomly placed. The dispersion equation of the coherent wave inside the slab must be solved numerically. For solid cylinders, there is only one solution corresponding to a mean free path of the coherent wave larger than one wavelength. In that case, the slab region may be described as an effective dissipative fluid medium, and its reflection and transmission coefficients may be formally written as those of a fluid plate. For thin hollow shells, a second solution of the dispersion equation is found, at concentrations large enough for the shells to be coupled via the radiation of a circumferential Scholte–Stoneley A wave on each shell. This occurs at a few resonance frequencies of the shells. At those frequencies, then, two different coherent waves propagate in the slab, and it can no longer be considered a dissipating fluid slab. © 2005 Acoustical Society of America. 关DOI: 10.1121/1.1848174兴 PACS numbers: 43.35.Bf, 43.20.Bi, 43.20.Hq 关JJM兴

I. INTRODUCTION

Multiple scattering of waves by random distributions of scatterers is a problem that has received much attention since Foldy’s early work on isotropic point scatterers.1 Two different approaches are usually found in the litterature. In the first one, which is the subject of this paper, the study of the coherent wave leads to the description of the multiple scattering medium as an effective dissipative medium, in which incoherent scattering is considered only through the energy loss of the coherent wave. In the second approach, when the concentration of scatterers gets too high for a coherent wave to propagate, the propagation of the incoherent intensity is studied.2,3 Many models have been derived in order to get the coherent wave properties for nonisotropic scatterers. The most well-known may be, once again, divided into two groups. In the first one,1,4 –9 it is the effective wave number 共of the coherent wave兲 that is sought, usually in order to derive the mechanic constants 共bulk and shear moduli兲 of the effective medium. In the models of the second group,10–13 both the effective wave number and the amplitude of the coherent wave are sought. In this paper, the scatterers are infinitely long cylindrical elastic objects randomly placed in a slab-like region of a perfect fluid 共water兲, and we are interested in both the coherent wave properties and the reflection and transmission coefficients of the slab. The model used is derived from Fikioris and Waterman, and the same dispersion relation as theirs is found. The reflection and transmission coefficients of the slab are then determined the same way as they did for the interface. Throughout the paper, we shall often refer to the works of Foldy,1 Waterman and Truell,8 and Fikioris and Waterman.11 They may be briefly summarized as follows. For isotropic scatterers, namely point scatterers, and asa兲

Electronic mail: [email protected]

1088

Pages: 1088 –1097

suming double interactions between scatterers may be neglected,14 Foldy1 found an explicit dispersion equation for the coherent wave, relating the effective wave number to the concentration of scatterers and to the far-field scattered amplitude of each scatterer. However, scatterers of nonzero finite size are not isotropic scatterers, and this is the reason why Waterman and Truell8 later extended Foldy’s work to nonisotropic scatterers. In their work, multiple interactions between scatterers are taken into account, but it is assumed that the mean exciting field on a scatterer at a given position is the same as that when another scatterer is at a fixed position. This assumption is known as the quasicrystalline approximation.9 In order to derive an explicit dispersion equation for the coherent wave, they did take into account the nonisotropic character of the field scattered by each object, but they let the size of each tend towards zero. The dispersion equation they obtained relates the effective wave number to the concentration of scatterers and to both the forward and backward far-field scattered amplitudes of each scatterer. Whenever the backward amplitude may be neglected, the expression for the effective wave number reduces to that of Foldy, with Foldy’s far-field amplitude replaced by the forward far-field scattered amplitude. Taking into account both the nonisotropic characteristics of the scattered field of an object and its finite nonzero size leads to an implicit dispersion equation, as was shown by Fikioris and Waterman.11 The problem under study is presented in Sec. II, the general theory in Sec. III, and the numerical results, in both cases of aluminum cylinders and thin cylindrical shells, are discussed in Secs. IV and V. II. PRESENTATION OF THE PROBLEM

Let us suppose N elastic scatterers, all identical, cylindrical, of radius a, and infinitely long in the z direction of a Cartesian system of coordinates (O,x,y,z). All cylinders are

J. Acoust. Soc. Am. 117 (3), Pt. 1, March 2005 0001-4966/2005/117(3)/1088/10/$22.50

© 2005 Acoustical Society of America

FIG. 1. Geometry of the problem. Local and global polar coordinates of O j , center of cylinder j.

randomly distributed in a slab-like region, depth d, inside a perfect fluid that we shall suppose to be water. Figure 1 shows the different polar coordinates used. Cylinder j is centered at O j . P is the observation point; its coordinates, in the (O,x,y) system, are x⫽r cos ␪ and y⫽r sin ␪. The incident harmonic plane wave propagates in the fluid towards the slab, under angle ␣; it gives rise to a reflected wave and a transmitted wave outside the slab, as well as to a multiple scattering process inside the slab. Let p(dᐉ ) be the probability density of finding cylinder ᐉ at dᐉ ⫽(x ᐉ ⫽d ᐉ cos ␹ᐉ ,yᐉ⫽dᐉ sin ␹ᐉ), and p(dᐉ 兩 d j ) the conditional probability density of finding cylinder ᐉ at dᐉ provided cylinder j is at d j . The cylinders are supposed to be uniformly distributed in the slab, with concentration c⫽(N/S) ␲ a 2 ⯝ 关 (N ⫺1)/S 兴 ␲ a 2 . S is the surface of the slab, and

再

1 p 共 dᐉ 兲 ⫽ S 0

if dᐉ 苸S

共1兲

.

if dᐉ 苸S

Exclusion of interpenetration of the cylinders of radius a is ensured by the following conditional probability density:

再

1 共 1⫺ f 共 d ᐉ j 兲兲 S p 共 dᐉ 兩 d j 兲 ⫽ 0

if dᐉ 苸S

,

A. Multiple scattering equations and configurational averaging

d ᐉ j ⫽ 储 dᐉ ⫺d j 储 .

if dᐉ 苸S

共2兲

As in Refs. 6 and 11, the simplest ‘‘pair correlation function’’,5 corresponding to the ‘‘hole correction,’’ is chosen f 共 dᐉ j 兲⫽

再

1

if d ᐉ j ⭐ ␤ a

0

if d ᐉ j ⬎ ␤ a

,

␤ ⭓2.

with two main differences. First, they introduce the transition matrix of each scatterer, according to the T-matrix approach,6 which allows them to deal with cylinders of arbitrary cross sections. Second, they are not interested in the reflection and transmission coefficients of the half-space. In this paper, the transition matrix is used, and the notations follow those of Ref. 6, but the reflection and transmission coefficients of the fluid slab will also be sought. In a first step, then, the transition matrix formulation is used to express the different acoustic fields in the region containing the scatterers. These fields are the total pressure field exciting a given scatterer and the total field scattered by the same scatterer. They are related via the transition matrix. Each one of them is expanded as a modal series of cylindrical functions. In a second step, a configurational average1 of the modal coefficients of the exciting field is performed, under the assumption of a uniform distribution of identical scatterers, and using Fikioris and Waterman’s ‘‘hole correction’’11 as well as Lax’s quasicrystalline approximation.9 The averaged modal coefficients of the total exciting field on a given scatterer are then each decomposed into plane waves, with the amplitudes of the plane waves depending on the mode number, and with a complex wave number independent of the mode number. Hence, the equations describing the multiple scattering process lead to two different kinds of systems, with the amplitudes of the plane waves as unknowns, and the complex effective wave number as a parameter. The first sets of equations are homogeneous; equating to zero their common determinant gives the effective wave number. This process is often referred to as the ‘‘Lorentz–Lorenz law.’’ So far, our model is exactly the same as in Ref. 6. The second set of equations can then be solved and gives the unknown amplitudes. This process is called the ‘‘extinction Ewald–Oseen’s theorem;’’ this model differs from Varadan’s in the way the extinction theorem is used.

Generally speaking, the total exciting field on cylinder j can be expressed as a modal series

␸ 共 j 兲⫽

共3兲

兺n e 共nj 兲J n共 kr j 兲 e in ␪ , j

and the total scattered field by cylinder j as

In the local coordinates system of cylinder j, the incident harmonic pressure field on the slab is

␾ 共 j 兲 ⫽ 兺 C 共nj 兲 H 共n1 兲 共 kr j 兲 e in ␪ j . n

j兲 ␸ 共inc ⫽e ik"d j e ik"r j ⫽e ik"d j

共5兲

兺n i n J n共 kr j 兲 e in ␪ e ⫺in ␣ , j

共4兲

with an e ⫺i ␻ t time dependence that will be omitted through⫹⬁ , and k⫽(k x out the paper, 兺 n , standing for 兺 n⫽⫺⬁ ⫽k cos ␣, k y ⫽k sin ␣). III. THEORY

Varadan et al.6 have derived the implicit dispersion equation of the coherent wave propagating in a fluid halfspace with a random distribution of elastic cylinders inside. Their work is very similar to that of Fikioris and Waterman, J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

共6兲

First, the diagonal transition T matrix of each cylinder, assumed to be known 共cf. Appendix B兲, is used to link these two fields C 共nj 兲 ⫽T nn e 共nj 兲 .

共7兲

Next, the total exciting field ␸ ( j) is composed of the incident plane wave and of the waves scattered by all cylinders ᐉ ⫽j N

␸

共 j兲

j兲 ⫽ ␸ 共inc ⫹

兺

ᐉ⫽1,ᐉ⫽ j

␾ 共 ᐉ 兲.

Le Bas et al.: Reflection by randomly spaced cylinders

共8兲 1089

具 e 共nj 兲 典 j ⫽e ik"d j i n e ⫺in ␣ ⫹

c

␲a2

兺m T mm 冕S ⬘ ddᐉ G 共mnjᐉ 兲

⫻ 具 e 共mᐉ 兲 典 ᐉ .

共15兲

The averaged modal coefficient 具 e (nj) 典 j is decomposed into two refracted plane waves, one propagates, under angle ␣ r , in the direction of increasing x, and the other one propagates, under the same angle, in the direction of decreasing x iK 具 e 共nj 兲 典 j ⫽i n e ⫺in ␣ r X ⫹ n e

FIG. 2. Notations used in the Graf addition theorem.

⫹ "d

j

iK ⫹i ⫺n e ⫹in ␣ r X ⫺ n e

⫺ "d

j

,

共16兲

where K⫹ and K⫺ are effective wave vectors ( j) In Eq. 共8兲, ␸ inc is defined by Eq. 共4兲, and ␾ (ᐉ) by Eq. 共6兲. The Graf addition theorem15 is now introduced in order to express the waves scattered by cylinder ᐉ, ␾ (ᐉ) , as incident waves on the surface of cylinder j

H 共n1 兲 共 kr ᐉ 兲 e in ␪ ᐉ ⫽

兺m G 共nmjᐉ 兲J m共 kr j 兲 e im ␪ , j

r j ⬍d ᐉ j ,

共9兲

with 共see Fig. 2兲 G ( jᐉ) the Graf operator defined by jᐉ 兲 1兲 ⫽e i 共 m⫺n 兲 ␪ jᐉ H 共m⫺n G 共mn 共 kd ᐉ j 兲 .

共10兲

It follows that

␸ 共 j 兲 ⫽e ik"d j

兺n i n J n共 kr j 兲 e in ␪ e ⫺in ␣

K⫺ ⫽ 共 ⫺K x ,K y 兲 ,

兺 兺m 兺n G 共nmjᐉ 兲T 共nnᐉ 兲e 共nᐉ 兲J m共 kr j 兲 e im ␪ . j

ᐉ⫽1,ᐉ⫽ j

共11兲

Comparison of relations 共5兲 and 共11兲 gives the linear system of equations

which verify the Snell–Descartes law K y ⫽k y .

兺 兺 ᐉ⫽1,ᐉ⫽ j m

jᐉ 兲 G 共mn T mm e 共mᐉ 兲 .

Relations 共16兲 and 共17兲 are now introduced into system 共15兲, so that iK x x j ⫺iK x x j X⫹ ⫹ 共 ⫺1 兲 n e ⫹2in ␣ r X ⫺ n e n e

⫽e ik x x j e ⫹in 共 ␣ r ⫺ ␣ 兲

兺 兺m T mm 冕S ⬘ N

具

典

⫻具

1 S ᐉ⫽1,ᐉ⫽ j

典 .

共13兲

Relation 共13兲 is equivalent to relation 共23兲 of Ref. 6. S ⬘ is equal to the slab surface S, with exclusion of the surface of a (ᐉ) cylinder of radius ␤ a centered at d j . 具 e m 典 jᐉ stands for the (ᐉ) , over all possible positions of all average of quantity e m cylinders, except cylinders j and ᐉ, which are kept fixed. In order to break down the averaging process,9 the quasicrystalline approximation ᐉ⫽ j,

共14兲

is used, as well as the fact that all cylinders are identical, so that relation 共13兲 gives 1090

⫹

J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

c

␲a2

冋兺

⫹ T 共 p⫹n 兲共 p⫹n 兲 i p e ⫺ip ␣ r X ⫹ p⫹n I p

p

册

兺p T 共 p⫹n 兲共 p⫹n 兲i ⫺p共 ⫺1 兲 n e ip ␣ e 2in ␤ X ⫺p⫹n I ⫺p ,

共19兲

冕

共20兲

r

with I⫾ p⫽

jᐉ 兲 ddᐉ G 共mn

e 共mᐉ 兲 jᐉ

具 e 共mᐉ 兲 典 jᐉ ⫽ 具 e 共mᐉ 兲 典 ᐉ ,

⫹

共12兲

Equation 共12兲 is exact: no assumption has been made, except that the cylinders do not either touch or interpenetrate each other 共otherwise the Graf theorem would not be valid兲. For a given position of cylinder j, relation 共12兲 is averaged over all possible positions of cylinders ᐉ e 共nj 兲 j ⫽e ik"d j i n e ⫺in ␣ ⫹

共18兲

Both K⫹ and K⫺ , as well as the X ⫾ n coefficients, are unknowns in system 共15兲. Due to the invariance of the slab in the y direction, y j will be supposed equal to zero from now on, with no lack of generality.

N

e 共nj 兲 ⫽i n e ik"d j e ⫺in ␣ ⫹

共17兲

B. Dispersion equation of the coherent waves

j

N

⫹

K⫹ ⫽ 共 K x ⫽K cos ␣ r ,K y ⫽K sin ␣ r 兲 ,

S⬘

e ip ␪ jᐉ H 共p1 兲 共 kd ᐉ j 兲 e iK

⫾ "d

ᐉ

ddᐉ .

As in Ref. 6, the Green’s theorem is applied to the surface integrals in Eq. 共20兲 I⫾ p⫽

1 k ⫺K 2

⫺e iK

2

冕

⫾ "d

L⬘

ᐉ

e ip ␪ jᐉ 关 H 共p1 兲 共 kd ᐉ j 兲 nˆ •“ 共 e ik

nˆ •“ 共 H 共p1 兲 共 kd ᐉ j 兲兲兴 dl,

⫾ "d

ᐉ

兲

共21兲

with L ⬘ a closed curve, composed of the circumference L ␤ a of a circle of radius ␤ a centered at O j , the two straight lines that define the slab region (x ᐉ ⫽0 and x ᐉ ⫽d), and two arcs that connect the previous two straight lines at infinity. nˆ is the local unit vector normal to L ⬘ . The two integrals defined in Eq. 共21兲, equal to 共see Appendix A兲 Le Bas et al.: Reflection by randomly spaced cylinders

I⫹ p⫽

2i ⫺p 共 k ⫺K 兲 2

2

再

⫺

i i 共 K ⫹k 兲 e ik x x j e ip ␣ ⫹ 共 ⫺1 兲 p e ⫹i 共 K x ⫹k x 兲 d 共 K x ⫺k x 兲 e ⫺ip ␣ e ⫺ik x x j kx x x kx

冎

⫺ ␲ be ⫹iK x x j e ⫹ip ␣ r 关 KJ ⬘p 共 K ␤ a 兲 H 共p1 兲 共 k ␤ a 兲 ⫺kJ p 共 K ␤ a 兲 H 共p1 兲 ⬘ 共 k ␤ a 兲兴 , I⫺ p⫽

再

2i ⫺p

冎

⫹ 共 ⫺1 兲 p⫹1 ␲ be ⫺iK x x j e ⫺ip ␣ r 关 KJ ⬘p 共 K ␤ a 兲 H 共p1 兲 共 k ␤ a 兲 ⫺kJ p 共 K ␤ a 兲 H 共p1 兲 ⬘ 共 k ␤ a 兲兴

are introduced into system 共19兲. This system must be verified, whatever the position of cylinder j, i.e., for any x j . Equating separately the factors multiplying e iK x x j and e ⫺iK x x j yields the following linear homogeneous sytems known as Lorentz–Lorenz laws: X⫾ n ⫽⫺

2␤c

兺 T 共 p⫹n 兲共 p⫹n 兲X ⫾p⫹n 共 k 2 ⫺K 2 兲 a 2 p

⫻ 关 KaJ ⬘p 共 K ␤ a 兲 H 共p1 兲 共 k ␤ a 兲 ⫺kaJ p 共 K ␤ a 兲 H 共p1 兲 ⬘ 共 k ␤ a 兲兴 .

共23兲

Both systems have a determinant, depending on the effective wave number K, that must be equal to zero

冉

Det I⫹

2␤c 共 k ⫺K 2 兲 a 2 2

冊

AT ⫽0,

共24兲

with I the identity matrix, T the transition matrix, and A the square matrix defined by its elements 1兲 ⬘ 共 K ␤ a 兲 H 共n⫺m A nm ⫽ 关 KaJ n⫺m 共k␤a兲 1 兲⬘ ⫺kaJ n⫺m 共 K ␤ a 兲 H 共n⫺m 共 k ␤ a 兲兴 .

共25兲

Relation 共24兲 is the dispersion equation of the effective medium in the slab. As the transition matrix depends on frequency only via the reduced frequency ka, so does the effective wave number K. Contrary to the dispersion equation found by Waterman and Truell, Eq. 共24兲 is not an explicit relation on the effective wave number; it has to be numerically solved. It is exactly the same as that found by Fikioris and Waterman, as well as by Varadan et al. For cylindrical scatterers such as ours, the dispersion equation of Waterman and Truell8 gives

冉冊 冉 K k

共22兲

i ie ⫹i 共 k x ⫺K x 兲 d ik x x j ip ␣ p⫹1 K ⫺k e e ⫹ ⫺1 兲 兲 共 共 共 K x ⫹k x 兲 e ⫺ip ␣ e ⫺ik x x j x x kx 共 k 2 ⫺K 2 兲 k x

2

⫽ 1⫺

⫺

冉

2ic

␲ 共 ka 兲 2 2ic

兺n T nn

兺

␲ 共 ka 兲 2 n

冊

2

共 ⫺1 兲 p T nn

冊

ward scattering. If backward scattering is neglected, Eq. 共26兲 reduces to K 2ic ⫽1⫺ k ␲ 共 ka 兲 2

共26兲

The first infinite summation in Eq. 共26兲 corresponds to forward scattering by one cylinder, and the second one to backJ. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

共27兲

Equation 共27兲 is identical to Foldy’s dispersion equation for point scatterers that would radiate a far-field amplitude equal to the forward far-field amplitude radiated by a cylinder. It will be numerically observed, in Secs. IV and V, that Eq. 共24兲 has 共at least兲 one solution, which is very close to that given by Eq. 共26兲. As the resonance frequencies of one scatterer are linked to the poles of its transition matrix T 共cf. Appendix B兲, they correspond, on the plot of the forward共or backward兲 scattered amplitude versus frequency, to sharp variations. This will be, of course, also observed on the plot, versus frequency, of both the real and imaginary parts of the effective wave number.

C. Amplitudes of the coherent waves

Let us suppose that the equation of dispersion, Eq. 共24兲, has been solved, providing only one physically acceptable solution K. The X ⫾ n are still unknowns that will be determined in the following. Systems 共23兲 provide ᭙n⫽0,

⫽⫹

兺 m⫽0

冉

␦ nm ⫺

2␤c ⫺ ⫹ 2

␬ ␬ a

2␤c ⫺ ⫹ 2

␬ ␬ a

冊

⫾ A nm T mm X m

A n0 T 00X ⫾ 0 ,

共28兲

with

␬ ⫾ ⫽K x ⫾k x .

共29兲

Those two systems may be solved using Cramer’s method. Let D be the common determinant of the associated homogeneous systems and D m the same one, except that the elements of column m, row n, are replaced by A n0 , so that

2

.

兺n T nn .

᭙n⫽0,

X⫾ n ⫽

2␤c

Dn ⫾ T 00X ⫾ 0 ⫽ ␩ D n T 00X 0 , 共30兲 ␬ ␬ a D ⫺ ⫹ 2

with D n ⫽D ⫺n . The last step now consists of calculating X⫾ 0 . Le Bas et al.: Reflection by randomly spaced cylinders

1091

So far, we have equated separately the factors that multiply e iK x x j and e ⫺iK x x j in system 共19兲. Equating also those multiplying e ik x x j , on one hand, and those multiplying e ⫺ik x x j , on the other hand 共extinction theorem兲, leads to two equations

Once again, the last integral in Eq. 共38兲 is evaluated, as in Appendix A, by use of the Sommerfeld integral representation of the Hankel function. The result depends on the position of the observation point P 共via r j and ␪ j ), i.e., on whether the reflected or the transmitted field is sought.

1 2ic ␲ k xa ␬ ⫺a

1. The reflection coefficient R

冋

⫻

r

兺m T mm e im共 ␣ ⫹ ␣ 兲共 ⫺1 兲 m X m⫺ r

␬ ⫺ e ⫹iK x d ⫻

1

兺m T mm e ⫺im共 ␣ ⫺ ␣ 兲X m⫹ ⫺ ␬ ⫹ a

兺m

册

m ⫺im 共 ␣ ⫹ ␣ r 兲

T mm 共 ⫺1 兲 e

⫽⫺1,

共31兲

with

2iK x d ⫹ X⫺ X0 , 0 ⫽Qe

⫺

N0

1⫹ ␩ g ␬ , 1⫹ ␩ f ␬ ⫹

兺

n⫽⫺N 0 ,n⫽0

f⫽

兺

n⫽⫺N 0 ,n⫽0

D n T nn e in 共 ␣ r ⫺ ␣ 兲 ,

R⫽

共 ⫺1 兲 n D n T nn e ⫺in 共 ␣ r ⫹ ␣ 兲 .

冕

S

⫺1⫹e ⫺i ␬

␲a2 ⫹

c

␲a2

S

兺n

冕

⫹⬁

⫺⬁

T nn 具 e 共nj 兲 典 j H 共n1 兲 共 kr j 兲 e in ␪ j dd j .

共35兲

1 k ⫺K ⫻

1092

冕

2

兺n T nn i ⫺n e ⫹in ␣ X ⫺n L ⫺j , r

1⫺Q 2 e 2iK x d

⫺⬁

e

册

共41兲

.

共42兲

.

2i ⫺n ⫺ik x ik x ik y in ␣ e x je x e y e . kx 共43兲

Equation 共36兲 then reads

冋

2ic 1⫺e ⫹i ␬ ␲ k xa ␬ ⫺a ⫹

⫺1⫹e ⫺i ␬

共36兲

⫹d

␬ ⫹a

⫺d

兺n T nn e ⫺in共 ⫺ ␣ ⫹ ␣ 兲X ⫹n r

兺n T nn共 ⫺1 兲 n

册

ik x x ik y y ⫻e ⫹in 共 ␣ ⫹ ␣ r 兲 X ⫺ e . n e

冋冉

⫹⬁

r

e iK y y j 关 H 共n1 兲 共 kr j 兲 e in ␪ j 兴 dy j ⫽

⫹ T nn i n e ⫺in ␣ r X ⫹ n Lj

H 共n1 兲 共 kr j 兲 e in ␪ j e iK

2

兺n T nn e ⫹in共 ⫺ ␣ ⫹ ␣ 兲X ⫺n

⫺Q 共 1⫺e 2iK x d 兲

␾ diff共 x⬎d,y 兲 ⫽

⫾ "d

j

共37兲

dd j .

In the same way as before, the Green’s theorem is applied to the surface integral in Eq. 共37兲, so that L⫾ n ⫽

r

The observation point P corresponds to x⬎d, and the calculation of the integral in Eq. 共38兲 gives

with

冕

兺n T nn共 ⫺1 兲 n e ⫺in共 ␣ ⫹ ␣ 兲X ⫹n

2. The transmission coefficient T

p 共 dj 兲

兺n

⫺d

␬ ⫺a

Use of Eq. 共16兲 gives c

⫹d

共40兲

共34兲

The scattered field, outside the slab, is given by

L⫾ n ⫽

共39兲

Evaluation of the infinite series, with use of Eqs. 共30兲, 共33兲, and 共34兲, finally provides

D. The reflection and transmission coefficients of the slab

␾ diff⫽N

⫹

共33兲

with

N0

冋

2ic 1⫺e ⫹i ␬ R⫽ ␲ k xa ␬ ⫹a

i ␲ k xa ␬ ⫺a 1 , 2cT 00 共 1⫹ ␩ f 兲 1⫺Q 2 e 2iK x d

␾ diff⫽

2i n ik 共 x ⫺x 兲 ik y ⫺in ␣ e x j e y e . kx

␾ diff共 x⬍0, y 兲 ⫽Re ⫺ik x x e ⫹ik y y ,

共32兲

which, combined with Eq. 共30兲, gives

g⫽

e iK y y j 关 H 共n1 兲 共 kr j 兲 e in ␪ j 兴 dy j ⫽

Equation 共36兲 then reads r

Q⫽

冕

⫹⬁

⫺⬁

⫹ Xm ⫺ ␬ ⫹ e ⫺iK x d

兺m T mm e ⫹im共 ␣ ⫺ ␣ 兲X m⫺ ⫽0,

X⫹ 0 ⫽

The observation point P corresponds to x⬍0, and the calculation of the integral in Eq. 共38兲 gives

⫿iK x ⫹

iK y y j

冊

冉

⳵ ⳵ ⫹e ⫾iK x d ⫾iK x ⫺ ⳵x j ⳵x j

关 H 共n1 兲 共 kr j 兲 e in ␪ j 兴 dy j

.

J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

冊册

共38兲

共44兲

The transmission coefficient T, however, is related to the total field ␾ T at x⬎d, as

␾ T 共 x⬎d,y 兲 ⫽ ␾ diff共 x⬎d,y 兲 ⫹ ␸ inc共 x⬎d,y 兲 ⫽Te ik x x e ik y y .

共45兲

Evaluation of the infinite series in Eq. 共44兲, with use of Eqs. 共30兲, 共33兲, 共34兲, finally provides T⫽

共 1⫺Q 2 兲 e iK x d 2 2iK x d

1⫺Q e

e ⫺ik x d .

共46兲

Le Bas et al.: Reflection by randomly spaced cylinders

Now, let us define ␶ as

␶⫽

1⫺Q Z 2 cos ␣ ⫽ , 1⫹Q Z 1 cos ␣ r

共47兲

so that the expressions found for R and T are those of the reflection and transmission coefficient of a fluid slab 共medium 2兲,16 of acoustic impedance Z 2 , embedded in another fluid 共medium 1兲, of acoustic impedance Z 1 ⫽ ␳ 1 c 1 . Equation 共47兲 defines the acoustic impedance Z 2 of the effective medium in the slab. As the effective wave number K depends on frequency, so does the refraction angle ␣ r , and the acoustic impedance Z 2 . This, in turn, defines the frequency-dependent density of the effective medium, from

␻ , Z 2 ⫽ ␳ effc eff⫽ ␳ eff Re 共 K 兲

共48兲

with Re standing for ‘‘real part of.’’ The next section is devoted to the numerical results obtained when the scatterers are solid steel cylinders in water. IV. THE SCATTERERS ARE STEEL CYLINDERS: NUMERICAL RESULTS

Steel is characterized by its density ␳⫽7816 kg/m3, the velocity of the longitudinal waves c L ⫽6000 m/s, that of the shear waves c s ⫽3100 m/s, water by its density ␳ 1 ⫽1000 kg/m3 , and by the velocity of sound c 1 ⫽1480 m/s. In all cases considered, parameter ␤ is fixed to 2. The hole correction, Eq. 共3兲, is known17,18 to be less realistic than Percus–Yevick’s approximation, especially as the concentration increases. Like Tsang et al.,17 we found indeed that increasing concentration c could lead to obtain solutions of the dispersion equation that would correspond to an amplified coherent wave. This occurred, in the frequency range we considered, for concentration values greater than 0.25. This is the reason why concentration c is fixed to 0.1 in the following. A. The coherent wave properties

A numerical resolution of the dispersion equation has provided one 共and only one兲 solution K⫽K ⬘ ⫹iK ⬙ , with K ⬘ and K ⬙ positive real numbers such that K ⬙ /K ⬘ ⬍1/(2 ␲ ), i.e., the wavelength of the coherent wave is greater than its mean free path3 L s ⫽1/K ⬙ . The effective attenuation in the slab is then K ⬙ , and the effective velocity is given by c eff⫽

␻ . K⬘

ingly, the largest differences between c eff and c 0 occur at particular frequencies which are resonance frequencies of the cylinders. However, these differences never exceed 5%. The evolution with frequency of the reduced effective attenuation, or loss tangent, K ⬙ /K ⬘ , is shown in Fig. 4. At low frequency, the K ⬙ /K ⬘ ratio is roughly an increasing linear function of frequency, with a very slightly greater slope in Waterman and Truell’s case than in ours. The other two curves in Fig. 4 show the ratio of the mean elastic free path to the radius of the cylinders. At low frequency, as the loss tangent increases linearly, the elastic mean free path diminishes from infinity 共at zero frequency兲 to around 33a, at ka ⫽1. Then, as the frequency increases, the mean free path goes on decreasing, except at the first two resonance frequencies of the cylinders. Usually, a damped wave is considered to be propagating on distances no larger than three times its mean free path. The mean value of the elastic mean free path, here, is around 15a⫽7.5␤ a, which means that the coherent wave is about to collapse after having encountered, roughly, no more than 24 scatterers. In the following sections, we shall study the reflection and transmission coefficients of the slab. Considering our

共49兲

The solutions of the dispersion equations Eq. 共26兲 and Eq. 共27兲 are practically identical in the frequency range we consider. Both Fig. 3 and Fig. 4 show that the difference between the solution of Waterman and Truell’s Eq. 共26兲, plotted in thin lines, and that of Eq. 共24兲, plotted in thick lines, is also rather negligible. Figure 3 presents the evolution of the reduced effective velocity c eff /c0 with the reduced frequency ka, obtained from the resolution of Eq. 共24兲 and from that of Waterman and Truell’s dispersion equation. The presence of scatterers is seen to reduce slightly the propagation speed. Not surprisJ. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

FIG. 3. The scatterers are steel cylinders. Concentration c⫽0.1. Evolution, with reduced frequency ka, of the ratio of the effective velocity of the coherent wave to that of sound in water. The arrows point at the resonance frequencies of a steel cylinder. Thin line: Waterman and Truell’s solution. Thick line: solution of Eq. 共24兲.

FIG. 4. The scatterers are steel cylinders. Concentration c⫽0.1. Evolution, with reduced frequency ka, of the effective loss tangent of the coherent wave 共curves begining at zero frequency兲, and of the elastic mean free path normalized by the radius a of the cylindrical scatterers. The arrows point at the resonance frequencies of a steel cylinder. Thin line: Waterman and Truell’s solution. Thick line: solution of Eq. 共24兲. Le Bas et al.: Reflection by randomly spaced cylinders

1093

FIG. 5. The scatterers are steel cylinders. Concentration c⫽0.1. Normal incidence on the slab. Moduli of the reflection coefficient 共thin line兲 and of the transmission coefficient 共thick line兲 versus reduced frequency. The slab thickness d equals 10a. The arrows point at the resonance frequencies of a steel cylinder.

last remark on the mean free path, the slab thickness d we shall consider will be fixed to 10a. B. Reflection and transmission by the slab at normal incidence „ ␣ Ä ␣ r Ä0…

The moduli of the reflection coefficient R and of the transmission coefficient T are plotted in Fig. 5, versus 2 f d/c eff , with f the frequency. There is little reflection by the slab compared to the transmission. Energy is seen to be lost due to incoherent scattering. The curve of the modulus of the reflection coefficient presents damped periodic oscillations. The reflection is minimum at frequencies such that the coherent transmitted wave in the slab is a standing wave. As the reflection coefficient is roughly zero at 2 f d/c eff⭓12, i.e. ka⭓3.73, while the first resonance of a steel cylinder occurs at ka⯝5, no resonance of the cylinders may be observed on that curve. This is not the case of the transmission coefficient. Its modulus undergoes visible variations at each resonance frequency of the cylinders. The reflection and transmission coefficients, Eqs. 共42兲 and 共46兲, are those of a fluid slab of acoustic impedance Z 2 , defined in Eq. 共47兲. These coefficients may be written in an alternative way, using the Debye series,16 which allows the description of the fluid slab as a Fabry–Perot interferometer. These series are expressed via the local reflection and transmission coefficients R pq and T pq (p,q⫽1,2) at each local boundary between fluids p and q R pq ⫽

Z q ⫺Z p , Z q ⫹Z p

T pq ⫽

2Z p , Z q ⫹Z p

p,q⫽1,2,

共50兲

⫹⬁

R⫽R 12⫹T 12T 21R 21e 2iKd

兺

n⫽0

n 2niKd R 21 e ,

⫹⬁

T⫽T 12T 21e iKd

兺

n⫽0

2n 2niKd R 21 e .

共51兲

In the case of Fig. 5, the exact coefficients, determined after Eqs. 共42兲 and 共46兲, may be seen to be identical to those calculated from Eq. 共51兲 while keeping only, in the infinite series, the n⫽0 term. Moreover, the product T 12T 21 is merely equal to 1 on the whole frequency range investigated. This means that no slab effect appears on the transmission 1094

J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

coefficient: the coherent wave propagates through only once. This is due to the relatively low value of the ratio of the mean free path to the slab thickness. The slab effect, in contrast, appears on the reflection coefficient, as R⫽R 12 is not a valuable approximation of it. From a practical point of view, this means that the effective wave number K of the coherent wave may be determined experimentally by use of Sachse and Pao’s method19 on the transmitted field, as done, for example, in Ref. 20, in which no local transmission coefficients, such as T 12 , T 21 , are taken into account. In the next section, we are interested in the effect a resonant coupling between scatterers may have on the coherent wave. By resonant coupling, we mean that a circumferential wave that is responsible for a resonance of the scatterers has a skin depth, in the surrounding liquid, large enough to include some part of the closest neighboring cylinder. This gives rise to a new resonance, which is part of the signature of the scatterer composed of those cylinders that are included in the skin depth of the wave. Solid cylinders, even close to each other, are difficult to couple, as all the circumferential waves that are responsible for the resonances are internal waves, i.e., most of their energy is inside the solid. Thin empty elastic shells, on the contrary, may be coupled at low frequency,21 as the circumferential wave that is responsible for their resonances carries more energy in the fluid part of the shell outer interface than in its solid part. V. THE SCATTERERS ARE THIN EMPTY ALUMINUM SHELLS: NUMERICAL RESULTS

Aluminum is characterized by its density ␳⫽2790 kg/m3, the velocity of the longitudinal waves c L ⫽6120 m/s, and that of the shear waves c s ⫽3020 m/s. The shell inner radius is b⫽0.9a, with a the outer radius. Parameter ␤ is still fixed to 2, but the concentration c is now 0.25. The mean distance between the scatterers, then, is reduced, compared to the previous case. This increases the possibility for the shells to couple via the circumferential A wave 共Scholte–Stoneley wave兲.21 The frequency range investigated is the same as in the previous section; in this range, all resonances of the shells are due to the A wave. The dispersion equation, Eq. 共24兲, provides two solutions, and the corresponding loss tangents are plotted, versus ka, in Fig. 6, along with that corresponding to Waterman and Truell’s solution. The sharp variations observed on each curve occur at a resonance frequency of the shells. The plot range of the K ⬙ /K ⬘ ratio extends from zero 共no damping兲 to 0.2: when K ⬙ /K ⬘ ⫽1/(2 ␲ )⯝0.16, the mean free path equals the wavelength, and the wave is 共quasi兲 nonpropagative. Both the first solution of Eq. 共24兲 and Waterman and Truell’s solution are seen to be propagative in the whole frequency range, except at two resonance frequencies (ka around 0.3 and 1.2兲, while the second solution of Eq. 共24兲 is nonpropagative for ka values less than 3.5, except at those same particular resonance frequencies. Figure 7 shows the velocity dispersion curves of the same three solutions. Both the first solution of Eq. 共24兲 and Waterman and Truell’s solution correspond, this time, to an effective velocity greater than that of the incident wave. Le Bas et al.: Reflection by randomly spaced cylinders

FIG. 6. The scatterers are thin empty aluminum shells, b/a⫽0.9, concentration c⫽0.25. Evolution, with reduced frequency ka, of the effective loss tangent of the coherent wave. Thin line: Waterman and Truell’s solution. Thick line: second solution of Eq. 共24兲. Very thick line: first solution of Eq. 共24兲.

When propagative (ka⭓3.5), the second solution of Eq. 共24兲 propagates with a lower velocity than that of sound in water. These results are similar to those found by Jing et al.22,23 on colloidal suspensions of spheres, in which they observed two quasimodes of finite life time. The one they call ‘‘the high-frequency mode’’ corresponds to our first solution of Eq. 共24兲, which has the closest dispersion curve to that of sound in the homogeneous fluid. They observe gaps in its dispersion curve, corresponding to a single sphere resonance. In our case, indeed, the mean free path of the first solution of Eq. 共24兲 is seen, in Fig. 8, to decrease at each resonance frequency of a single shell. The low-frequency mode of Jing et al. corresponds to our second solution of Eq. 共24兲. Its velocity is lower than that of sound in the homogeneous fluid, and its mean free path, while quite low in our case, increases at each resonance frequency of a shell. This mode is due to the coupling of neighboring shells, via the Scholte–Stoneley A wave, just as in the case considered by Jing et al. The mean free path of both solutions is lower than ten

FIG. 7. The scatterers are thin empty aluminum shells, b/a⫽0.9, concentration c⫽0.25. Evolution, with reduced frequency ka, of the ratio of the effective velocity of the coherent wave to that of sound in water. Thin line: Waterman and Truell’s solution. Thick line: second solution of Eq. 共24兲. Very thick line: first solution of Eq. 共24兲. J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

FIG. 8. The scatterers are thin empty aluminum shells, b/a⫽0.9, concentration c⫽0.25. Evolution, with reduced frequency ka, of the elastic mean free path normalized by the radius a of the shells. Thin line: second solution of Eq. 共24兲. Thick line: first solution of Eq. 共24兲.

times the radius of the shells, for reduced frequencies ka greater than 2.5. At such frequencies, then, the study of the reflection/transmission by the slab is no longer interesting. At lower frequencies, however, and particularly at resonance frequencies of the shells, the analysis of the reflection and transmission coefficients of the slab should take into account the presence of two coherent waves, which is not done in the present study. This work is currently in progress.

VI. CONCLUSION

We have presented an extension of Fikioris and Waterman’s formalism to describe the reflection/transmission process from a slab-like region in which multiple scattering occurs. The dispersion equation of the coherent wave inside the slab must be solved numerically, unlike that found by Waterman and Truell. The hole correction has been used, so that there is no possibility for interpenetration of scatterers. The dispersion equation may have, a priori, more than one solution. If only one corresponds to a mean free path of the coherent wave larger than one wavelength, the slab region may be described, from the coherent wave point of view, as an effective dissipative fluid medium, and its reflection and transmission coefficients have been formally written as those of a fluid plate. For solid elastic cylinders such as scatterers, both coefficients have been shown to be numerically equivalent to the first term共s兲 of their Debye series, as the damping process of the coherent wave kills the multiple reflections in the slab. For thin hollow shells, a second solution of the dispersion equation may be found, at concentrations large enough for the shells to be coupled via the radiation of a circumferential Scholte–Stoneley A wave on each shell. This process gives rise to a propagating coherent wave, with a velocity slightly lower than that of the homogeneous incident wave, and a mean free path rather low, compared to the radius of the shells, except at a few resonance frequencies of the shells. At those frequencies, then, two different coherent waves propagate in the slab, and the theoretical derivation of the reflection and transmission of the slab, presented in Sec. III, is no longer valid. Le Bas et al.: Reflection by randomly spaced cylinders

1095

APPENDIX A: DETERMINATION OF

I⫾ p⫽

1 k ⫺K 2

⫺e

2

冕

L⬘

iK⫾ "d

ᐉ

J p 共 x ᐉ ⬍x j 兲 ⫽

e ip ␪ jᐉ 关 H 共p1 兲 共 kd ᐉ j 兲 nˆ •“ 共 e iK

⫾ "d

ᐉ

兲

⫻

nˆ •“ 共 H 共p1 兲 共 kd ᐉ j 兲兲兴 dl

L ⬘ is a closed curve composed of the circumference L ␤ a of a circle of radius ␤ a centered at O j , the two straight lines that define the slab region (x ᐉ ⫽0 and x ᐉ ⫽d), and two arcs that connect the previous two straight lines at infinity ⫾ ⫾ ⫾ ⫾ ⫾ I⫾ p ⫽I L ␤ a ⫹I x⫽0 ⫹I x⫽d ⫹I arcsup⫹I arcinf .

共A1兲

J p 共 x ᐉ ⬍x j 兲 ⫽

␤a K ⫺k 2

⫺kJ p 共 K ␤ a 兲 H 共p1 兲 ⬘ 共 k ␤ a 兲兴 ,

J p 共 x ᐉ ⬍x j 兲 ⫽

⫾

⫾ ⫽ I x⫽0

⫾ I x⫽d ⫽

1 k ⫺K 2

2

e ⫾iK x d k ⫺K 2

2

冋

⫿iK x ⫹

冋

⫾iK x ⫺

⳵ ⳵xᐉ

册

⳵ ⳵xᐉ

册

J p 共 x ᐉ ⬍x j 兲 ⫽

共A2兲 ⫾

as well as to show that both integrals, I arcsup and I arcinf , tend towards 0 as the arc radius tends to infinity. The last two integrals, on x⫽0 and x⫽d, may be written as

x ᐉ ⫽0

x ᐉ ⫽d

⫽

J p共 x ᐉ 兲 ⫽

冕

⫺⬁

冕

1 ⫽ ␲

⫹⬁

⫺⬁

共A3兲

J p共 x ᐉ 兲 ,

W

⫺⬁

e

e

共A4兲

e

dy ᐉ d ␥ ,

with W the Sommerfeld’s integration path. J p (x ᐉ ) depends on whether x ᐉ is greater or less than x j . 1. First case: x 艎 Ë x j

As y ᐉ varies from ⫺⬁ to ⫹⬁, so does ␪ jᐉ from ⫹␲/2 to ⫺␲/2, through 0. Letting ␥ 1 ⫽ ␥ ⫹ ␪ jᐉ , with ⫺ ␲ ⬍ ␪ jᐉ ⬍ ␲ , one gets 1 ␲

冕冕 W

⫹⬁

⫺⬁

e ik sin ␣ y ᐉ e ikd ᐉ j cos ␥ 1 cos ␪ jᐉ

⫻e ikd ᐉ j sin ␥ 1 sin ␪ jᐉ e ip 关 ␥ 1 ⫺ 共 ␲ /2兲兴 dy ᐉ d ␥ 1 , 1096

e ik 共 x j ⫺x ᐉ 兲 cos ␥ 1 e ip 关 ␥ 1 ⫺ 共 ␲ /2兲兴

W

2 k

冕

共A6兲

e ik 共 x j ⫺x ᐉ 兲 cos ␥ 1 e ip 关 ␥ 1 ⫺ 共 ␲ /2兲兴

W

1 ␦共 ␣⫺␥1兲d␥1 , cos ␣

2 e ik 共 x j ⫺x ᐉ 兲 cos ␣ e ip 关 ␣ ⫺ 共 ␲ /2兲兴 k cos ␣ 2 ⫺p ik 共 x ⫺x 兲 ip ␣ i e x j ᐉe . kx

⫾ ⫽ I x⫽0

2i ⫺p⫹1 k x 共 k 2 ⫺K 2 兲

共 ⫿K x ⫺k x 兲 e ik x x j e ip ␣ .

共A7兲

As y ᐉ varies from ⫺⬁ to ⫹⬁, so does ␪ jᐉ from ⫹␲/2 to ⫺␲/2, through ␲. Letting ␥ 1 ⫽ ␥ ⫹ ␪ ᐉ j , with ⫺ ␲ ⬍ ␪ ᐉ j ⬍ ␲ , and e ip ␪ jᐉ ⫽(⫺1) p e ip ␪ ᐉ j , one gets

冕冕 W

⫹⬁

⫺⬁

e ik sin ␣ y ᐉ e ikd ᐉ j cos ␥ 1 cos ␪ ᐉ j

which, in the same way as in the previous case, finally gives 2 ⫺ik 共 x ⫺x 兲 ⫺ip ␣ e x j ᐉe , kx

共A8兲

and introduction of Eq. 共A8兲 into the second Eq. 共A3兲 provides

共A5兲

J p 共 x ᐉ ⬍x j 兲 ⫽

冕

J p 共 x ᐉ ⬎x j 兲 ⫽ 共 ⫺1 兲 p i ⫺p

ik y y ᐉ ikd ᐉ j cos ␥ ip 关 ␥ ⫺ 共 ␲ /2兲兴 ip ␪ jᐉ

e

e ik sin ␣ y ᐉ e ⫺ik sin ␥ 1 y ᐉ dy ᐉ d ␥ 1 ,

⫻e ikd ᐉ j sin ␥ 1 sin ␪ ᐉ j e ip 关 ␥ 1 ⫺ 共 ␲ /2兲兴 dy ᐉ d ␥ 1 ,

e ik sin ␣ y ᐉ 关 H 共p1 兲 共 kd ᐉ j 兲 e ip ␪ jᐉ 兴 dy ᐉ

冕冕

1 2␲ ␲ k

J p 共 x ᐉ 兲 ⫽ 共 ⫺1 兲 p / ␲

e iK y y ᐉ 关 H 共p1 兲 共 kd ᐉ j 兲 e ip ␪ jᐉ 兴 dy ᐉ .

⫹⬁

⫺⬁

2. Second case: x 艎 Ì x j

J p共 x ᐉ 兲 ,

Use of the Snell–Descartes law gives J p共 x ᐉ 兲 ⫽

⫹⬁

Introduction of Eq. 共A6兲 into the first Eq. 共A3兲 gives

with ⫹⬁

e ip 关 ␥ 1 ⫺ 共 ␲ /2兲兴 e ik 共 x j ⫺x ᐉ 兲 cos ␥ 1

W

⫻

e ⫾iK x x j i ⫿p e ⫾ip ␣ r 关 KJ ⬘p 共 K ␤ a 兲 H 共p1 兲 共 k ␤ a 兲

2

冕 冕

⫻ ␦ 共 sin ␣ ⫺sin ␥ 1 兲 d ␥ 1 ,

It is rather easy to find that I L⫾␤ a ⫽2 ␲

1 ␲

J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

⫾ ⫽ 共 ⫺1 兲 p I x⫽d

2i ⫺p⫹1 e ⫾iK x d 共 ⫾K x ⫺k x 兲 e ⫺ip ␣ e ⫺ik x 共 x j ⫺d 兲 . kx k 2 ⫺K 2 共A9兲

Now, Eq. 共A1兲 together with Eq. 共A2兲 and either Eq. 共A7兲 or Eq. 共A9兲, leads to the final result, Eq. 共21兲, in the main text. APPENDIX B: THE TRANSITION MATRIX OF A CYLINDRICAL SCATTERER

For a vacuum-filled shell, use of the boundary conditions at r⫽a and r⫽b leads to the following linear system of equations M X⫽S,

共B1兲

with the M i j elements of matrix M given, for each mode n(n苸Z), by Le Bas et al.: Reflection by randomly spaced cylinders

M 11⫽ 共 ␳ 0 / ␳ 兲 kaH 共n1 兲 ⬘ 共 ka 兲 , M 13⫽k L aN n 共 k L a 兲 , M 15⫽nN n 共 k T a 兲 ,

⫹

2 c T2

1

2 c T2

␳ 20 共 k L a 兲 2 c 20 ␳ 2

H 共n1 兲 共 k L a 兲 ,

D n 共 ka 兲 ⫽0. k aJ n 共 k L a 兲 2 L

c L2 ⫺2c T

1 c L2 ⫺2c T2

k L aN n 共 k L a 兲

关共 n 2 ⫺ 共 k L a 兲 2 兲 N n 共 k L a 兲 ⫺k L aN n⬘ 共 k L a 兲兴 ,

M 24⫽u 1 共 a 兲 ⫽

M 25⫽ v 1 共 a 兲 ⫽

2 c T2 2 c T2

n 关 k T aJ n⬘ 共 k T a 兲 ⫺J n⬘ 共 k T a 兲兴 , n 关 k T aN n⬘ 共 k T a 兲 ⫺N n⬘ 共 k T a 兲兴

M 31⫽M 41⫽M 51⫽0, M 33⫽h 1 共 b 兲 ,

M 32⫽m 1 共 b 兲 ,

M 34⫽u 1 共 b 兲 ,

共B2兲

M 35⫽ v 1 共 b 兲 ,

M 42⫽m 2 共 a 兲 ⫽2n 关 1⫺k L aJ n⬘ 共 k L a 兲兴 , M 43⫽h 2 共 a 兲 ⫽2n 关 1⫺k L aN n⬘ 共 k L a 兲兴 , M 44⫽u 2 共 a 兲 ⫽⫺ 共 n 2 ⫺ 共 k T a 兲 2 兲 J n 共 k T a 兲 ⫹2J n⬘ 共 k T a 兲 , M 45⫽ v 2 共 a 兲 ⫽⫺ 共 n 2 ⫺ 共 k T a 兲 2 兲 N n 共 k T a 兲 ⫹2N n⬘ 共 k T a 兲 , M 52⫽m 2 共 b 兲 ,

M 53⫽h 2 共 b 兲 ,

M 54⫽u 2 共 b 兲 ,

M 55⫽ v 2 共 b 兲 , with k L 共respectively, k T ) the wave number of the longitudinal 共respectively, shear兲 waves in the solid, ␳ 0 and ␳ the densities of the surrounding fluid and of the solid, a the outer radius of the shell, and b its inner one. If the components S j of the source vector in Eq. 共B1兲 are S j ⫽⫺Re 共 M j1 兲 ,

共 j⫽1,5兲 ,

共B3兲

the nth element T nn of the diagonal transition matrix is the first component of the unknown vector X; as such, it is obtained from system 共B1兲 by use of the Cramer’s rule T nn ⫽

D 关n1 兴 Dn

,

共B4兲

with D n the determinant of matrix M, and D 关n1 兴 the same one, except that column 1 is replaced by vector S.

J. Acoust. Soc. Am., Vol. 117, No. 3, Pt. 1, March 2005

共B5兲

1

关共 n 2 ⫺ 共 k L a 兲 2 兲 J n 共 k L a 兲 ⫺k L aJ n⬘ 共 k L a 兲兴 ,

M 23⫽h 1 共 a 兲 ⫽⫺

⫹

M 14⫽nJ n 共 k L a 兲 , M 21⫽⫺

M 22⫽m 1 共 a 兲 ⫽⫺

In the case of a cylinder, rather than a shell, system 共B1兲 reduces to a linear system of three equations and three unknowns, obtained by suppressing rows 3 and 5, in the whole system, together with columns 3 and 5 in matrix M. The reduced resonance frequencies (ka) res are the real parts of the complex roots of the following equation:

M 12⫽k L aJ n 共 k L a 兲 ,

L. L. Foldy, ‘‘The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,’’ Phys. Rev. 67共3,4兲, 107–119 共1945兲. 2 D. Sornette, ‘‘Acoustic waves in random media,’’ Acustica 67, 199–215 共1989兲; 67, 251–265 共1989兲; 68, 15–25 共1989兲. 3 A. Tourin, M. Fink, and A. Derode, ‘‘Multiple scattering of sound,’’ Waves Random Media 10, R31–R60 共2000兲. 4 M. Lax, ‘‘Multiple scattering of waves,’’ Rev. Mod. Phys. 23共4兲, 287–310 共1951兲. 5 S. K. Bose and A. K. Mal, ‘‘Longitudinal shear waves in a fiber-reinforced composite,’’ Int. J. Solids Struct. 9, 1075–1085 共1973兲. 6 V. K. Varadan, V. V. Varadan, and Y.-H. Pao, ‘‘Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves,’’ J. Acoust. Soc. Am. 63共5兲, 1310–1319 共1978兲. 7 R. B. Yang and A. K. Mal, ‘‘Multiple scattering of elastic waves in a fiber-reinforced composite,’’ J. Mech. Phys. Solids 42共12兲, 1945–1968 共1994兲. 8 P. C. Waterman and R. Truell, ‘‘Multiple scattering of waves,’’ J. Math. Phys. 2共4兲, 512–537 共1961兲. 9 M. Lax, ‘‘Multiple scattering of waves. II. The effective field in dense systems,’’ Phys. Rev. 85共4兲, 621– 629 共1952兲. 10 V. Twersky, ‘‘On scattering of waves by random distributions. I. Freespace scatterer formalism,’’ J. Math. Phys. 3共4兲, 700–715 共1962兲. 11 J. G. Fikioris and P. C. Waterman, ‘‘Multiple scattering of waves. II. ‘Hole corrections’ in the scalar case,’’ J. Math. Phys. 5共10兲, 1413–1420 共1964兲. 12 C. Ariste´gui and Y. C. Angel, ‘‘New results for isotropic point scatterers: Foldy revisited,’’ Wave Motion 36共4兲, 383–399 共2002兲. 13 Y. C. Angel, C. Ariste´gui, and J. Y. Chapelon, ‘‘Reflection and transmission of plane waves by anisotropic line scatterers,’’ Poromechanics II, edited by Auriault, Geindreau, Royer, Bloch, Boutin, and Lewandowska 共Sweets & Zeitlinger, Lisse, 2002兲, pp. 607– 618. 14 A. Ishimaru, Wave Propagation and Scattering in Random Media 共Academic, New York, 1978兲. 15 M. Abramovitz and I. Stegun, Handbook of Mathematical Functions 共Dover, New York, 1964兲. 16 J. M. Conoir, in La Diffusion Acoustique par des Cibles E´lastiques de Forme Ge´ome´trique Simple. The´ories et Expe´riences 共Acoustic Scattering by Elastic Targets of Simple Geometric Form兲, edited by N. Gespa 共CEDOCAR, Paris, 1987兲, Chap. 5. 17 L. Tsang and J. Au Kong, Scattering of Electromagnetic Waves— Advanced Topics, Wiley Series in Remote Sensing 共Wiley, New York, 2001兲. 18 S. Torquato, Random Heterogeneous Materials—Microstructure and Macroscopic Properties, Interdisciplinary Applied Mathematics Volume 16 共Springer, New York, 2002兲. 19 W. Sachse and Y.-H. Pao, ‘‘On the determination of phase and group velocities of dispersive waves in solids,’’ J. Appl. Phys. 49共8兲, 4320– 4327 共1978兲. 20 F. Vander Meulen, G. Feuillard, O. Bou Matar, F. Levassort, and M. Lethiecq, ‘‘Theoretical and experimental study of the influence of the particle size distribution on acoustic wave properties of strongly inhomogeneous media,’’ J. Acoust. Soc. Am. 110共5兲, 2301–2307 共2001兲. 21 P. Y. Le Bas, F. Luppe´, J. M. Conoir, and H. Franklin, ‘‘N-shell cluster in water: Multiple scattering and splitting of resonances,’’ J. Acoust. Soc. Am. 115共4兲, 1460–1467 共2004兲. 22 X. Jing, P. Sheng, and M. Zhou, ‘‘Theory of acoustic excitations in colloidal suspensions,’’ Phys. Rev. Lett. 66共9兲, 1240–1243 共1991兲. 23 X. Jing, P. Sheng, and M. Zhou, ‘‘Acoustic and electromagnetic quasimodes in dispersed random media,’’ Phys. Rev. A 46共10兲, 6513– 6534 共1992兲.

Le Bas et al.: Reflection by randomly spaced cylinders

1097