A T matrix for scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness

A T matrix for scattering from a doubly infinite fluid-solid interface with doubly periodic surface roughness

Garner C. Bishop and Judy Smith Naval Undersea Warfare Center Division Newport, Newport, Rhode [s/and 02840

(Received 14 December 1992; accepted for publication 1 June 1993)

The T-matrix formalism is used to calculate scattering of a plane wave from a doubly infinite fluid-solid interface with doubly periodic surface roughness. The Helmholtz-Kirchhoff integral equations are used to represent the scattered pressure field in the fluid and the displacement field in the solid. The boundary conditions are applied and a system of four coupled integral equations is obtained. The incident and scattered pressure fields in the fluid, as well as the surface pressure field, are represented by infinite series of scalar Floquet plane waves, while the scattered displacement field in the solid and the surface displacement field are represented by infinite series of rectangular vector basis functions constructed from Floquet plane waves. This process discretizes the integral equations and transforms them into a system of four coupled doubly infinite linear equations. The extended boundary condition is applied and the T matrix that relates the spectral amplitudes of the incident field to the spectral amplitudes of the scattered fields is constructed. An exact analytic solution and numerical results are obtained for plane wave scattering from doubly periodic sinusoidal and triangular surface roughness.

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PACS numbers: 43.30.Hw

INTRODUCTION

The T-matrix formalism has been used to calculate

scattering from a doubly infinite surface with doubly periodic roughness on which the total fields satisfy Neumann and Dirichlet boundary conditions • and from a singly infinite surface with periodic roughness on which the total fields satisfy the elastic tensor boundary conditions. 2 Al- though it is a relatively simple matter to construct a T matrix for scattering from a doubly infinite surface with doubly periodic surface roughness on which the total fields satisfy the elastic tensor boundary conditions, since the computational effort required to obtain a numerical solution to the problem is formidable, the problem has not received as much attention.

In this paper, plane wave scattering from the doubly infinite interface between an ideal fluid and an elastic solid

on which there is doubly periodic surface roughness and on which the total fields satisfy the exact tensor elastic boundary conditions is calculated. A T-matrix scattering formalism is developed that is similar to that developed by Waterman. 3-5 However, rather than spherical or cylindrical vector basis functions, rectangular vector basis functions that are more appropriate for scattering from periodic surface roughness are constructed from scalar Floquet plane waves and used to represent vector and tensor fields. These vector functions admit a representation of the scattered displacement field in the solid which is the sum of three vector mode fields; •'/• longitudinally polarized p-wave field and two transversely polarized s-wave fields, i.e., $H and $V s-wave fields. Once the formalism for the rectan-

gular vector functions is developed, this representation is considerably more convenie•f•than a representation of the

displacement field in terms of its rectangular components or in terms of spherical or cylindrical vector basis functions. The problem is reduced to determining the scattering amplitudes for these vector mode fields in the solid and the scattering amplitudes for the scalar Floquet modes in the fluid.

It is assumed that a pressure wave is propagating in an infinite, homogeneous, and isotropic fluid half-space and is incident on and is scattered from the doubly infinite surface of an infinite, homogeneous, and isotropic elastic half- space on which there is doubly periodic roughness. Fur- ther, it is assumed that the surface roughness is periodic along the positive and negative x and y axes, has an amplitude z=•(x,y), and along the x axis, has a maximum amplitude ax, period Ax, and wave number K•=2rr/A•, and similarly along the y axis, has a maximum amplitude ay, period Ay, and wave number Ky=2rc/Ay.

The incident wave p(i)(r,t) is a time harmonic plane wave and is given by

p(i) (r,t) =po(i)ea•i-•'re -iøt. ( 1 ) The time-dependent factor e -iø't and explicit dependence on time is suppressed in all subsequent equations. For con- venience, the plane wave amplitude p0 (i) is chosen to satisfy the relationship given by po•i•/,•k•= 1, where ,• is the Lame parameter of the fluid. The superscript ( 1 ) is used to designate quantities in the fluid and a superscript (2) will be used to designate quantities in the solid. The wave vector for the incident plane wave k• i- • is given by

kp(i-) (i). kp(})• kp?œ (2a) = kp•.a, + -- where

1560 J. Acoust. Soc. Am. 94 (3), Pt. 1, Sept. 1993 1560

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•(•): •(1 ) sin 0 ½ 0 cos •b ½0 = •(1 ) O•o,

k•(•=k• (1• sin 0 •0 sin •b½ø =k•(l• o,

(2b)

(2c)

and

( i) = kp 1) COS o( i) = kp(1) •/0 (2d) ß

In Eqs. (2b)-(2d), 0 (i) and •b(ø are, respectively, the polar and azimuthal angles of incidence, kp(l) •(o/½p (l) and is the p-wave wave number in the fluid with ½p(1) • •(1)//9(1) and is the p-wave wave speed in the fluid, and/9 (1) is the fluid mass density.

The boundary conditions imposed by the fluid-elastic interface couple the total pressure field p(1)(r) and displacement field u • 1)(r) in the fluid and the displacement field in the solid u•2• (r). The boundary conditions at the fluid-elastic interface are; (1) the normal component of the displacement is continuous which requires

t•(r'). u(l) (r ' ) = t•(r'). u (2) (r')

=•(r')' •7'p(1) (r')/,;[ (1) k• (1)2' (3a) (2) the normal component of the traction is continuous which requires

r(i) (r') ß t•(r')= r(2)(r') ß t•(r')= -p(1)(r')•(r'); (3b)

and (3) the tangential component of the traction vanishes Which requires

•(r') X½:• (r') = •(r') X•-• (r') =0. (3c)

The tensors •'• 1 • (r) and •-•2• (r) are, respectively, the stress tensors in the fluid and the solid. The vector t• (r') is the unit vector normal to the surface.

I. INTEGRAL REPRESENTATION OF FIELDS SCATTERED FROM A FLUID-ELASTIC INTERFACE WITH PERIODIC SURFACE ROUGHNESS

A. Integral representation of the displacement field in the solid

The following discussion applies to an arbitrary solid and the superscript (2) that designates the solid medium in this paper is suppressed temporarily. In addition, as in the previous discussion it is assumed that all fields are steady state and time harmonic with time dependence exp(- iot) and explicit dependence on time is suppressed.

The Helmholtz-Kirchhoff integral equation representation of the displacement field in an elastic solid produced by scattering from a surface s of an incident field u •ø (r) that originates from a source located in the region z < z' is given by •'7

sdS'([•(r ') ß r(r') ] ß Go(r',r;kp,ks) --u(r') ß [•(r').l;o(r',r;k/•,ks)]•

u(r)--u(0 (r) (z<z'), = --u(i) (r) (z>z'). (4)

The vectors r=x•+y.•+zœ and r' =x'•' +y'j•' +•(x',y')œ' are, respectively, radius vectors to a field point and to a point on the surface. The integral is over the surface s and ds' is an infinitesimal arc on s and is given by

ds'=dx' dy' •l-F [Ox,•(x',y') ]2-F [Oy,•(x',y') ] 2. (5) The vector ti(r') is a unit vector normal to the scattering surface and is directed out of the volume that bounds the

source. The quantity ti(r'). V'=O/On' that appears when the scalar products are evaluated, is the derivative normal to the surface and is given by

-- •/1 + [ax,•(x',y')]2+ [Oy,•(x',y')]2' (6)

The tensor r(r) is the stress tensor and when expressed in terms of the displacement field u(r) is given by

r(r) =/LIV. u(r) +p[Vu(r) +u(r)V], (7)

where ;t and • are the Lame parameters. The dyadic G0(r,r';k•,,ks) is the free space Green dyadic for the displacement field for a unit impulsive source. The subscript 0 on G0(r,r';k•,,k s) is used to distinguish between quantities related to a single unit impulsive source from quantities related to other sources that will be considered. The vec-

tors k•, and k s are, respectively, the p-wave and s-wave wave vectors in the solid with c•= (/t +2/•)/p kp 2 = co2/%2 with 2

2

and ks2=Co2/Cs 2 with Cs=la/p. The triadic Zo(r,r';k•,,k s) is the free space stress triadic and is given by

I•o(r,r' ;k• ,ks) = ;LIV. Go( r,r' ;k• ,ks)

+/• [VGo(r,r';kp,ks)

+ Go(r,r';kp,ks)V]

or in indicial form is given by

(8)

Eoaor ( r,r' ;k•, ,ks) = 26 at•OnGonr( r,r' ;k•, ,ks)

+ tt [ a•Goor( r,r' ;k•, ,ks)

+Ot•G%r(r,r';kp,ks) ]. (9) It is important to note that the indicial form of the stress triadic defines the manner in which the differential opera- tions are performed and that this definition differs from conventional definitions. However, when the stress triadic

is defined in this manner, it is related to G0(r,r';kp,ks) in the same manner the stress tensor r(r) is related to the displacement field u (r).

The representation for the displacement field given by Eq. (4) is valid for an arbitrary surface. However, when the surface has double periodic roughness, the double infinite integral over the surface may be expressed as a double infinite sum of two finite integrals over the periods of the surface roughness. When Eq. (4) is specialized to a surface with double periodic roughness, the displacement field is represented by

1561 J. Acoust. Soc. Am., Vol. 94, No. 3, Pt. 1, Sept. 1993 G.C. Bishop and J. Smith: T matrix for scattering 1561


mx---- oo my---- oo { [n(r'). •-(r' +m•AxR' +myA•f')].ao(r' +m•A•.•' +myAyy ,r,k•,ks)

[u(r) --u(O (r) (z <z') --u(r'+m•A•'+myAj•')'[n(r').J;o(r'+m•Ax•'+myA• ,r,k•,ks)]}=l_u(O(r) (z>z'). (lO)

In writing Eq. (10), the double finite integrals and the double infinite sums have been commuted. The Floquet theorem may be used to write Eq. (10) in the form

dx' dy'{ [n(r'). •'(r') ] ß •(r',r;kp,k s)

--u(r') ß [n(r') ß •(r',r;kp,ks) ])

[u_(r)- u(i)(r)(z <z'), = u(i) (r) (z > z'), (11) where

• (r',r;kp,ks)

m x----oo my------oo

-n t- IHyA)•',r;kp ,ks) e i( mxAxkx + myAyky) ( 12 ) and

X (r',r;k•,ks)

have been included and it has been assumed that all

sources are in the fluid so that u (i) (r)=0. The wave vector k• (2) is the p wave vector for the solid and is given by

(2) (2)• ,_(2) ^-- ,_(2) ,, where

(15a)

and

kpy(2) =k(•)=k(i) (15c) py "py ,

kp(z 2)= !imp(2)2 -- kp(x2) 2 -- k(2) 2 (15d) --py ß

The wave vector k32) is the s-wave wave vector for the solid and is given by

sx (2)•__ L(2) ,, k• 2) =k(2).•+ksy y-t-tCsz z, (16a)

where

= Z Z Xo(r'+mxAx•' mx-- -- oo my-- -- oo

q_ rnyA•,,r;kp,ks)ei(mx^xkx+ myny•y). ( 13 ) Therefore, when calculating scattering from a periodic surface it is more convenient to consider the Green dyadic 6 (r',r;k•,k s) rather than 60(r',r;kp,k s) and the stress triadic X (r',r;k•,k s) rather than X0(r',r;k•,ks). Although this result is not new, sometimes these functions are intro- duced a priori and their origin is obscured. However, the discussion given here indicates exactly the manner in which these functions arise from the periodicity of the surface. Representations of these functions are developed in Sec. III.

An integral representation of the displacement field of interest in this paper is obtained by incorporating the boundary conditions given by Eqs. (3a)-(3c) in Eq. (11 ), and is given by

-- dx' dy'{p(1)(r')n(r ' ) ß G (r',r;k•(2) ,k• 2) )

k•x2) =k (1) (16b) px ,

k (2) =k (1) (16c) sy 'py ,

and

k•: 2) •/k• 2)2 k(2) 2 k(2) 2 = ---sx -- -sy ß (16d)

The quantities k(1) and k(•) are, respectively, the x and y ß 'px ß 'py

components of the p-wave wave vector in the fluid kp which is given by

kp(1) = kp(xl )2• q- k(1)j•q- k(1)z? py pz

with

(17a)

k(1) •/kp( 1 )2 2 k(1)2 (17b) pz = - k}} ) -- ' -py ' The equalities given by Eqs. (15b), (15c), (16b), and (16c) are necessary to enforce the boundary conditions.

+u(2) (r') ß [n(r'). X (r',r;kp(2),kJ 2)) 1} u(2) (r) (z<z'), (14a)

0 (z>z'). (14b)

In writing Eqs. (14a) and (14b), the superscripts ( 1 ) and (2) referring, respectively, to the fluid and solid medium

B. Integral representation of the scattered pressure field in the fluid

The Helmholtz-Kirchhoff integral equation representation of the pressure field in a fluid produced by scattering from a surface of a pressure wave from a source located in the region z > z' is given by



; ds' [p(r')•q(r'). V'go(r',r;kp) -/•k•o(r',r;kp) • (r') ß u(r' )]

[p(r)--p(i)(r) (z> z'), = l--p(i)(r) (z<z'). (18)

In writing Eq. (18), the relationship given by Vp (r) = pco2u (r) for time harmonic fields has been used. The quantity go(r',r;k•) is the scalar Green function for and unbounded region.

To obtain a representation that is convenient for scattering from a doubly infinite surface with doubly periodic surface roughness, the same procedure used to obtain Eq. ( 11 ) from Eq. (4) is applied to Eq. (18), and the result is given by

;o'X 2 dx' dy'[p(r')n(r') . V'g(r',r;k•)

--/•k•(r',r;k•)n(r') ß u(r')]

p(r)--p•i•(r) (z> z'), = --p•i)(r) (z<z'), where

(19)

g(r',r;k•) = • • go(r' +mxA•' mx= -- oo my-- -- oo

+ myA•', r; kp) e i( mxAxkx + myAyky). ( 20 ) Once again, it is evident that the Green function that is most useful for scattering from surfaces with periodic roughness is not the Green function for a single point source go (r',r;kp), but rather a Green function of the form given by g(r',r;k•,). To obtain a representation of the pressure field of interest in this paper, the boundary conditions given by Eqs. (3a)-(3c) are applied and the total pressure field in the fluid is given by

fo'f: dx' dy' [ p(i)(r')n (r ' ) ß V'g(r',r;kp (1)) 2 1

-/],(1)kp(1) g(r',r;kp ( ))n(r').u(2)(r')] p(r) --p(i) (r) (z> z'),

= --p(i)(r) (z<z'). (21) In writing Eq. (21 ), the superscripts ( 1 ) and (2) denoting, respectively, the fluid and solid have been reinserted.

II. SCALAR AND VECTOR BASIS FUNCTIONS

Fundamental to the T-matrix formalism is the representation of unknown fields and the T matrix in terms of

eigenfunction solutions to the scalar, or when appropriate, to the vector wave equation. Eigenfunction solutions to the scalar wave equation in spherical, cylindrical, and even prolate spheroidM s coordinates have been used to represent scalar fields and to construct vector basis functions to represent vector and tensor fields. However, to calculate scattering from an infinite fluid-solid interface with periodic roughness, the most appropriate scalar basis functions are

Floquet plane waves and it is convenient to represent vector and tensor fields in terms of rectangular vector basis functions constructed from the scalar Floquet plane waves. Although it is standard practice to use scalar Floquet plane waves as a basis to represent fields scattered from surfaces with periodic roughness, their use in the construction of vector basis functions has not received as much attention.

Therefore, the following is a brief description of the construction and properties of rectangular vector basis functions constructed from scalar Floquet plane waves.

A. Scalar basis functions: Floquet plane waves

The properties of the scalar Floquet plane wave modes are well known. However, since many of their properties are used to construct the rectangular vector functions, in the following some of these properties are catalogued. Since both incoming and outgoing wave modes are required to represent wave fields, in the following both types of modes are considered explicitly. The ruth-order Floquet plane wave mode is denoted x(k(m +)' r), is given by

x(k(m +). r) =exp(tl•(m +). r) (22) and satisfies the homogeneous scalar wave equation

(7 2 + k2)x (k(m+) ß r) =0. (23) The wave vector k(• +) is given by

k(m+ ) = kxmx• + kym/rl: kzm •, (24a) with

kx•= k sin Om cos •bm

=k(sin 0 (i) cos •b (i) +mxK•/k) =ka%,, (24b)

kx%= k sin Om sin •bm

=k(sin 0 (i) sin •b (i) +myKy/k)=kfi%, (24c)

and

•/k 2 2 _k 2 =kl'm • k•cm x ymy

i !/ k2xmx+ k•ymy- k2= k'}"m

( 2 2 kxmx'-ll-k•my<k2), (24d)

+ > (24e)

COS 0 m = kzm/k •/1 -- a 2 2 = mx'-•-fimy , (24f)

tan •m=fim/Cgmx. (24g) The symbol m is used to denote the ordered pair of indices (rex,my), where m• and my are positive or negative inte- gers that specify the order of the mode. The Floquet modes satisfy the periodic condition given by

x(k(m +)' r + Ax-• + A•0) =X (k(m +)' r) ei(/½x^•x+/½•A• v). (25)



The scalar product of two modes is defined and is given by

r)Ix(k r))

fo'afo,ayx , = •xx •yy (am 'r)x(•)'r) ß •) (•) =•mxnx•mxn f •( --k• )z, (26)

where * denotes the complex conjugate.

B. Hectan•ular vector basis functions

Rectangular vector basis functions are defined by

1

•l(r;k•)) = k• VXVX•(k•) ß r), (27a) 1

•2(r;k•+)) = k-• VX•x(k• +) .r), (27b) and

1

•3 (r;k•n•) ) = • VX (k•n•) ß r). (27c) The wave number kstm = •/•xm + • and is the magnitude of the transverse wave vector ksm = k•X•, so that k•. ksm = •' ksm = 0. The basis functions ß 1 (r;k• • ) ) and

r'k (•)) are solenoidal and may be used as a basis to •2( , • rfpresent SH and S V waves, and the basis function ß 3(r;k• •) is irrotational and may be used as a basis to represent p waves.

The inner product of two rectangular vector basis functions is defined by

(•a(r;km) I •(r;•n) )

fo'axf = (28) The inner products of the rectangular vector basis functions are given by

(•l(r;k(sm •)) I •l(r;k(sn•)))

(•2(r;k(•+)) i ̂ (+) = q•2(r;ksn ))

•mxnx •myny ( k=m real), •mxnx•myny exp( •: 21 I z)

ß (+) (•3(r;k•)) 1•3(r,kpn ))

(k=m imag. ); (29)

• rnx nx • myny ( k•=m real ), •mxnx•myny exp( •: 21 --•m

< •t 1 (r;k• + )) I •2 (r;k(s. + )) ) = 0;

(kp•m imag. ); (30)

(31)

(32)

and

ß (+) {•2(r;k• +)) 1•3(r,ken )) ^( • ^( • ) ( • ) _ k(sm • . =•mxn•mynykstm)*økpn exp[i(kp, )*) .z]

(33)

It is important to note that the inner products of the rectangular basis functions given by Eqs. (29)-(33) apply only to those functions that have the same 'sign for the z component of the wave vector and only to wave vectors whose x and y components are equal, i.e., the constraints imposed by the boundary conditions. The rectangular vector basis functions for s waves (p waves) are orthonormal among themselves and carry unit energy for propagating s-wave (p-wave) modes. The transverse and longitudinal basis functions are not generally orthonormal when their arguments differ, but, when the arguments are the same, they form a triad of vector basis functions that is orthonormal in the subspace spanned by the propagating wave modes.

III. REPRESENTATION OF SCALAR AND TENSOR GREEN'S FUNCTIONS FOR SCATTERING FROM PERIODIC SURFACE ROUGHNESS

by 9 The Green displacement dyadic G0(r ,r,kp,ks) is given

1

Go (r',r;kp ,ks) = pco• [ V'g0 (r',r;k•) V -- V'g0 (r',r;ks) V --IV2go(r',r;ks) ]. (34)

For a wave vector k, g0(r',r;k) is the scalar Green function for an unbounded region and satisfies the equation given by

(V '2 q- k2)go (r',r;k) = --g(r'--r). (35)

It is evident that if a function g(r',r;k) satisfies the equation given by

(V'2 + k2)g(r',r;k)=- • • tS(x'--I-mxAx--x) mx= -- oo my=-- oo

X 6(y' +rnyAy--y)6(z'-z)

X e i(mxAxkx + myAyky), (36)

then g0(r',r;k) may be used to construct g(r',r;k), which is given by

g(r',r;k) = -- • • go(r'+mA•fi' mx= -- oo my= -- oo

+ myAj•',r;k)ei(mxAxkx+myAyky ). (37) Therefore g(r',r;k) is the scalar Green function for a double periodic array of phased point sources. Since the functions G (r',r;k• ,ks) and • (r',r;k• ,ks) are constructed from g(r',r;k•) and g(r',r;ks), these functions are, respectively, the Green displacement dyadic and stress triadic for a double periodic array of phased unit impulse body forces.

The scalar Green function g0(r',r;k) for an unbounded region is well known and has an integral representation given by lø



1 ;_• exp•i[K•,(x--x') +Ky(y--y') +Kz(z> --z< ) ]• go(r',r;k) = (2;r) 3 dK K•_+_K•_+_K•_k 2 , (38) where z< (z>) refers to the lesser (larger) of z and z'. When this integral representation of g0(r',r;k) is used in Eq. (37), the integration and summation are commuted, and the summation is performed, g(r',r;k) is given by

1 ;• exp{i[Kx(x--x') +Ky(y--y') +Kz(z> --z< ) ]• g(r',r;k) = (27r) 3 ark K•+Ky2+Kz2_k2 1

X 1 --exp{--i[ nx(kx--Kx) + ny(ky--Ky) ]}' (39)

The Cauchy integral theorem may be used to perform the integration. The poles of the integrand are Kx = kx% , Ky = ky%, and K• = k•m ( - k•m) for z > z' (z < z' ). After the integration is performed, g(r',r;k) is given by

g(r',r;k) =g( + ) (r',r;k)©(z--z') +g(-• (r',r;k)©(z'--z), (40a)

where gt + • (r',r;k) is given by

i •] •] 1 g• + • (r',r;k) -- 2AxAy mx= - oo my= - oo kzm Xx[k(m+• ß (r--r') ]. (40b)

This Green function allows the infinite integral over the surface to be reduced to an integral over a period of an infinite sum. Fokkema • gave a derivation of this Green function starting from Eq. (36) and using finite Fourier transforms. Tong and Senior • derived the same Green function using a procedure similar to that used here.

To obtain a representation of G( ' ß r ,r, kp,ks) in terms of the rectangular vector basis functions, G ' ß (r ,r, kp,ks) is represented as a sum of dyadics constructed from the rectangular vector basis functions and unknown scalar expansion coefficients. To determine the unknown expansion coefficients, a representation of G ' ß (r ,r,k•,k•) is obtained from Eq. (34) by replacing g0(r',r;k•) with r ,r,k•) and g0(r',r;k•) with g(r',r;k•) and using the identity given by

V' X Ig(r',r;ks) X V = -- IV':g(r',r;ks) + V'g(r',r;ks) V. (41)

Then the two representations for 6(r',r;k•,,ks) are equated and the orthogonality relations are used to determine the expansion coefficients. The Green displacement dyadic expressed in terms of the rectangular vector basis functions is given by

6 (r',r;kp,ks) =6 ½ +•(r',r;kp,ks)O(z--z')

+6•-•(r',r;kp,ks)©(z'--z), (42a) where ©(z) is the Heaviside step function and G ½ + •(r',r;kp,k s) is given by

, . (+) G (•) (r ,r,k•,k s )

i • • k•[•(_r,.k(•) ) -- 2poCA•Ay m•= - • %= - • X • (r;k(•)) + •:( -- r';k(• )) • (r;k(•)) ]

+k• •3(--r';k•')•3(r;k•') ' (42b) The stress triadic ß (r',r;kp,ks) is derived kom Eq. (8) by interchanging r' and r and replacing •o(r,r';kp,ks) by •(r',r;kp,ks) and is given by

ß (r',r;k•,ks) = •½ + •(r',r;k•,ks)O(z--z')

+ •- ) (r',r;kp ,ks)O(z'--z), (43a)

where •(•)(r',r;kp,ks) is given by

• • ( pks• -- ;k(sm •) r;k(sm •) V'•: r';k(sm •) •2(r;k• +) i Z Z k--• {[V'•,( r' )•,( )+ (- ) )] I• ( + ) (r',r;kp ,ks) = 2 po•-A•Ay mx= - oo my= - oo

+ [• (- r';k(sm • ))V'• (r;k(sm + )) + •(- r';k(sm • ))V'•2 (r;k(sm • )) ]} +k-• {AIV' ß •3( -r';k•(m • ))

--r'.k(•) r;k•(m • ) •3 --r"k( + ) ^ ( + ) ) X•3(r;kp(m•))+/3[V'•3( -,--pm )•3( )+ ( )V'q/3(r;kpm )1} - ,--pm ß (43b)



IV. A T MATRIX FOR SCATTERING FROM A FLUID-SOLID INTERFACE WITH DOUBLE PERIODIC SURFACE ROUGHNESS

To begin construction of the T matrix, the representations of the scalar and tensor Green's functions are used

in the Helmholtz-Kirchhoff integral representation of the displacement field in the solid given by Eqs. (14a) and (14b) and the total pressure field in the fluid given by Eq. (21). The resulting system of integral equations for the unknown fields is given by

u(2)(r) =-- dx' dy'{u(2) (r') ß [n(r')

ß •;(-)(r,,r;k•2),k•2)) ] q_p(1)(r' )n(r')

ß G(-) (r',r;k•2) ,k•2) ) } (z <z'), (44a)

p(1) (r) --p(i) (r) = dx' dy'[p(1)(r')n(r ' )

ß V'g( + ) (r',r;k• 1 ) )

__ •, (1)kp( 1 )2g( + ) (r',r;k•( 1 ) ) n (r ' ) ß u(2) (r') ] (z>z'), (44b)

fo'X 0= dx' dy'{u(2) (r') ß [n(r ' )

ß •;(+)(r,,r;k•2),kj2)) ] _•_•(1)(r')n(r')

ß G (+) (r',r;k•2) ,kJ2)) } (z> z'), (44c)

and

--pti) (r) = dx' dy'[p(1)(r')n(r') ß V'

Xg(-) (r',r;k• (1)) _2,(1)k• 1): Xg(-)(r',r;k•(1))n(r ').u(2)(r')] (z<z').

(44d)

Since there are no sources in the solid, the scattered displacement field in the solid is u (s2) ( r ) -- u (2) ( r ) which is represented by a series of rectangular vector basis functions given by

u(S2) (r) • • [•(•:-)•l(r;k• -) mx= -- oo my---- -- o•

. •:-•:(r;k •:-•)

-•:- • (r:k •:-• ) ] (45)

The scattered pressure field in the fluid is p•sl• (r)=p•l• (r) _p(i) (r), and is represented by a series of Floquet plane wave modes given by

p(sl)(r)=Z(1)kp (1) E E /•(n•l+)x(k/•a +)'r) mx= -- oo my---- -- o•

=--Z(1) E E A (sl +)V ø •3(r;k/•a+)) mx-- • • my • • •

(46)

To obtain integral representations for the un•own expansion coe•cients u• 2-) , Eq. (45) is used in Eq. (44a), then •s. (29)-(33) are used to evaluate the inner products of the basis functions • 1 (r;k•-) ) •2 (r'k (2-) ) and •3 (r;k(:-) ) with the result. This process produces a system of equations for the coe•cients •:-) that can be solved. Similarly, to obtain an integral representation for the un•own expansion coefficient• •+) , •. (46) is used in Eq. (44b) and Eq. (26) is used to evaluate the inner product of x(k•+).r) and the result. Integral representations of the coefficients •:-) and •+) are given by

(s2-) --i k•2)2 fo'• dx' ffy dy' •lm --2p(2)• 2 k• Xi • •(2•u(2•(r') ß [n(r'). [V'•i( --r';k(• -) )

-•- •1( --r';k•-•)V' ] ] +P •1• (r')n(r')

ß [n(r'). [V'•2( --r';k(• -) )

+ •2(--r';k•-) )V' ]'l-'•-• (1) (r') n(r')

and

(47b)

(47c)

•'(1) kp(1 •1 + ) -- 2k•lz•) Ax •y tP ½1' (r')n(r') ß V' _•(1)kp(1)2n(r'). u(2)(r') ] XX(--k(l+).r ') (47d)

Since the integral representations of the expansion coefficients contain the unknown surface pressure and displacement fields, the coefficients remain unknown. How- ever, when the same calculations are performed on the



equations that follow from the extended boundary conditions, i.e., Eqs. (44c) and (44d), a system of equations for the unknown surface fields is obtained which is given by

--i k•2)• fo'• dx' ;:• dy'(jt(2)u(2) r, Tyy ß [n(r')- [V'•l(--r';k• +)) +•l(--r';k(•+))V']]

+p(1) (r')n(r')- •(--r';k•2+•)}, (4ga)

o=2p(•)•a k(• A• • •(•)u(•)(r ') ß [n(r') ß [v'• (- r';k(• +)) + •(- r';k• +))v' ] l

+p(•) (r')n (r') ß •2( --r';k• +)) }, (48b)

--i k•2)2 foX• dx' 0=20(2)• 2 • A• • (r) - -p•

ß [n(r')- [•(2)lV' ß •3(--r';k•+) )

+•(2)V,•3 ( ,. (2+) +•(2) _ ;k•+) --r,kpm ) •3( r' )V']] +p(•) (r')n(r') ß •3 (--r"k(2+)) ] (48c)

and

--2• • • [P(1)(r')n(r')'V' --•(1)k•l)2n(r•) ß u(2) (r •) ]x(--k (1-) -r•), (48d)

where

fO•X dx f•Y dye(i) •(1)k•l• '-)= • • (r)x(k• -•'r)*' (49)

To obtain a solution to Eqs. (48a)-(48d) for the surface displacement and pressure fields, the surface displacement field is represented by an infinite series of rectangular vector functions given by

(2+)•1 (r•;k•+) u•:•r'• = E E [•ln •X • • •y• • •

•:n )• 3n )]

and the surface pressure field is represented by

p(1) (r,) __--•(1)k31) Z Z' /•(nl+)x(kp(n l+)'r') /•X • -- O0 /'/y• -- O0

= _•(1) Z r,.k(l+) Z /•(nl+)v''•3( , p n )'

(5O)

nx•-- • oo ny •-- -- oo

(51)

It is important to note that the representations of the surface fields given by Eqs. (50) and (51 ) depend explicitly on the surface roughness and although this representation is frequently used, it is not necessary. Alternate representations of the surface fields that do not depend explicitly on the surface roughness may be obtained by evaluating Eqs. (50) and (51 ) at z' = 0. When scattering of a plane wave from sinusoidal roughness is calculated, the former representation introduces Bessel functions with arguments that are in general complex, while the latter introduces Bessel functions with arguments that are either pure real or pure imaginary. In principle, the specific representation of the surface fields cannot effect the solution, however as a practical matter, it is simpler to evaluate a Bessel function with a pure real or pure imaginary argument than with a complex argument. Therefore, for numerical purposes, representations of the surface fields that do not depend explicitly on surface roughness may be more accurate and preferable. In this paper, both representations were used to obtain numerical results and in Sec. VI, some of the numerical results are shown and compared.

When Eqs. (50) and (51) are used in Eqs. (48a)- (48d), the resulting system of equations is given by

0 Z Z [f•(+) (2+) f)(+)A(I+) = gaBmn•Bn -•- kSa4mW•n ]

(a,/•= 1,2,3 ), (52a)

(2+) •}(--) z(l+) •4Bmn•Bn + Z44ma•n ],

(52b)

($2--)__ •am •

f}(--)•(l+) -•- •a4mn• n ] (a,/5 = 1,2,3 ), (52c)

and

(sl+) •_ (2+) •}(+) A(I+) •4Bmn•Bn + Z44mu•n ].

(52d)

To simplify Eqs. (52a)-(52b), the summation convention that repeated indices are to be summed has been used to express a sum over the/• index. The elements of the matrix Qa( • ) gm• expressed in terms of the scalar and vector basis functions are given by

O( + ) --i/L(2) k32)2 fO •'x •afimn -- 2 p ( 2 ) O)2 •ss( 22 dx' f :y dy' Ax •-f{•fi (r';k(sn2+)) ß ß __ '.k( 2:• )) [n(r') [V'•a( r,--•m

-I-•a(--r';k(2+))V']]) (a,]•-- 1,2) (53a)



a3mn--2pf2)w2 k(2s• Ax •y {•3 +)) ß [n(r') ß [V'•a(--r';k(• •))

'.k(2•))V']]} (a 1,2), (53b) +•af--r ,__•

Q•) il •1) kJ 2 ) f dx' f ;y dY' v, •4m•= 2pf2)• 2 k•

ß ') ß •(--r';k• •)) (a=l,2), (53c)

ß [n(r')' _• •.(2• ) +•2)V'•3(-, ;• )

+•(2)•3(___,.•.(2•) •, , ) ]1} (53d)

33= • (2• ß [n(r') ß

_,.•.•2•) •3(_r,.k•2•))V,l ]}, X •3( --- ,%m ) +•2) ,--pro (53e)

34•n_2p(2)o 2 k• Ax • n(r') "!•(2•)), (53f) '•3( --r ,--œm

r}( • ) - ikp( • ) f o•'•' dx' f ••' dY' *4t•mn-- •kp-•) Ax • n(r') --k(•)-r') (fi= 1,2,3) ' •fi(r';k•+))Z(

and

x(k (•+) ß r')n(r') pn

(53g)

k(l+)-r ') (53h) ß V';y(-__• . To obtain representations of the matrix elements that are convenient for numerical computations, the representations of the scalar basis functions and the vector

basis functions are used in Eqs. (53a)-(53h). The results of this calculation are given in the Appendix.

Equations (52a)-(52d) may now be written in the following manner:

! r•( + ) r•( + ) r•( + ) r•( + ) \ l•;13ran l•;14mn / / (2+) •12mn •1 lmn /•]n 0

I n( + ) n( + ) n( + ) n( + ) / • / g2n •22mn •23• •24mn// (2+ / 0 / n( + ) n( + ) n( + ) n( + ) // (2 + = 0

•33mn •32mn

/ (-) (-) (_) (_) 1+ (•-) e,mn ehmn emn/ --Am •2mn (54a)

1568 J. Acoust. Soc. Am., Vol. 94, No. 3, Pt. 1, Sept. 1993

and

(--) (--) (--) (--) / ] (s2- Qllmn Q12mn Q13mn Q14mn / (2+) )

(-) (-) (-) (-) • / (s2 Q•3mn Q•4mn / (2+) ) Q•2mn Q[lm• /•2n /•2m

(_) (_) (_) (_)•/ (2+) =/ (s2) ß Q31mn Q32mn Q33mn Q34mn ! •Z3n [ •Z3m

k•41mn k•42mn •43mn

(54b)

It is assumed that the inverse of the matrix in Eq. (54a) exists and is nonsingular, and that a solution to Eq. (54a) exists and is unique. Then the spectral coefficients of the scattered fields are related to the spectral coefficients of the incident field by

•I•-)/ /Tllmn T12mn T13mn T14mn (s2-)l / T31mn T32m •3m ] T33nm T34mn

'4• l+)] \T41m. T42m T43m. To, m. 0

0

X 0 ' (55)

where Tijmn are the elements of the T matrix given by

Tin, = Q•,)[ Q(m+. ) ] -', (56a)

where

(56b)

and

o2.)= (56c)

V. PLANE WAVE SCATTERING FROM A DOUBLY INFINITE FLUID-ELASTIC INTERFACE WITH DOUBLY PERIODIC SURFACE ROUGHNESS

A. Energy conservation

Since conservation of energy provides a necessary although not sufficient condition that the numerical solution to the scattering problem must satisfy, an expression for the conservation of energy is derived. The power density d• (r) for a time-harmonic displacement u (r) is giYen by

1

•'(r) =-5 Re[iwu(r)*-r(r)]. (57)

G. C. Bishop and J. Smith: T matrix for scattering 1568


The power • produced by a time harmonic displacement u (r) through a surface whose normal is • (r) and whose area is AxAy is given by

•= • • •(r) .•(r). (58) When Eq. (68) is used to calculate the power of the incident and scattered displacement fields in the fluid and the scattered displacement field in the solid, an equation for the conservation of energy is obtained and is given by

= a;•(2) n_ •(2) • 1 • (•'SOmT Srm,+ ks• real k• real

with

and

•- Z •(1) •, (59a) kp(lz•) real

k(1)

•(•1) --pzm 1,6(sl +) 2, =kT v•m [ (59b) ---pan (s2--) 2 •)__ (t(2) + 2•(2) •k (2) t(1)k(1) I I (59c)

pz

•(:) • (:)k(:) ß -s• (s2-) 2

z) I"lm I,

o7• (2) ] -/'(2)k(2) ß -san (s2-) 2+ (s2-) 2 •lm ß --SVm--•l,(1)k(1) [I I I I ] •'•2m pz

(59e)

The coefficients given by Eqs. (59a)-(59e) will be referred to as power spectral coefficients.

B. Analytic expressions for plane wave scattering from sinusoidal and triangular periodic surface roughness

In the following, the quantities necessary to calculate plane wave scattering from two specific types of periodic surface roughness are catalogued. When the incident field is a plane wave described by Eq. (1), the expansion coefficient,A•'-) that is obtained from the inner product of the plane wave and the Floquet plane wave x(k•-).r), i.e., Eq. (49), is given by

,/•'-- ) = •mx ,O•my ,O . (60) Two roughness profiles of the form • (x,y) = •'; (x) + •'.v (Y) are considered; The first type of roughness is sinusoidal roughness with

and

•(x,y) =a• sin K•+ay sin Ky, (61)

•rnxrny=•z(i/2)(ax•rnx •l•my o-•ay•m O•m •1) (62) ' ' X' y' '

Cm•my(k)=(-1)(m•+my)Jm•(a•k)Jmy(ayk), (63) where Jm(at) is the Bessel function of the first kind. The

coefficients •mx% and Cmx%(k) are, respectively, the two- dimensional finite Fourier transform of the rough surface and the rough surface phase factor. These coefficients ap- pear in the matrix elements and are defined in the Appen- dix by Eqs. (A3) and (A4). The second type of surface roughness is triangular roughness with

g(x,y) =

ax ay -•--'l-•- (mx=O, my=O),

a x ay

(taxer)2 [1--(--1)m•]6my, O--(my•r)2 [1--(--1)my]6m•,O ( m•---•O, my---•O ),

(64)

(65)

and

Cm,%(k) = - 2axk 2ayk

(2axk) 2- (mx2•) 2 (2ayk) 2- (my2•) 2

)< [e -i(axk+mxrr)-- 1]

X [e--i(a)+m/r)-- l ]. (66)

Figures 1 and 2 show, respectively, a finite portion of a doubly infinite surface with doubly periodic sinusoidal and triangular roughness. Both types of surface roughness have the same period and maximum amplitude and A•= Ay and ax=ay.



VI. NUMERICAL CONSIDERATIONS AND RESULTS

A. Numerical procedures

To obtain a numerical solution for the spectral coefficients of the scattered fields, the infinite matrices O(ag ) and Oim +) were truncated and expanded as finite rectangular matrices for all m and n such that I m•l <m, I myl <m, I <m, and !nyl <m for some integer rn. For a given value of rn, this process produced matrices Q•) and Q(m+n ) of order 4(2rn+ 1)2. To avoid having to invert a large matrix, the system of equations given by Eq. (55) was written in the form

O(=+. )•m =•r• (67) where •n is the column vector of the spectral coefficients of the incident field and the column vector o9• m is defined by

Q(m•).•m•- ø•'m, (68) where •'m is the column vector of the spectral coefficients of the scattered fields. Then a numerical solution for the

column vector •m was obtained by solving the system of linear equations given by Eq. (67), and the spectral coefficients of the scattered fields were obtained by the matrix multiplication indicated by Eq. (68). The subroutine LFCCG from the International Mathematical and Statis-

tical Library (IMSL)13 was used to compute the LU factorization of the coefficient matrix and estimate its L 1 con-

dition number. Then the LU factorization was used in the

IMSL subroutine LFICG to obtain a solution to the system of linear equations using Gaussian elimination with partial pivoting and iterative refinement. The integer rn that parametrizes the order of the system of linear equations was increased until three conditions were met; (1) the real and imaginary parts of the spectral amplitudes > 10 -5 converged to within a specified infinitesimal (E =0.01) for all [rn•l <3 and Irnyl <3, (2) energy was conserved to within a specified infinitesimal (E-0.01), and ( 3 ) an estimate of the condition number was less than 10 •ø. If the condition number exceeded 10 •ø, the calculation was terminated.

To be able to determine the effects on scattering of various parameters, a basis set of parameters was chosen and is given by k(i)=10 m -1, 0(i)=•b(i)=45 ø, ax=a. =0.01 m, and Ax=Ay=2 m, p(•)=1000 kg/m( c•1)=1493 m/s, cp(2)=2400 m/s, c32)=900 m/s, p(2) ---- 2000 kg/m3.14 In the following discussions, these parameters remain fixed unless specifically noted otherwise.

B. Program validation and surface field representations

Since it appears that there are no previous numerical or experimental results for scattering from the types of surfaces considered in this paper, it is somewhat problem- atic to check the correctness and accuracy of the computer code beyond the numerical constraints noted previously. However, three calculations were performed to verify the correctness and determine the accuracy of the computer

code. It is important to note that in all three calculations, scattering from a doubly infinite surface with doubly periodic sinusoidal surface roughness was calculated and the surface fields were •epresented by infinite series in which explicit dependence on the roughness profile •'(x',y') was not included. ( 1 ) For each of the incident azimuthal angles 0 ø, 90 ø, 180 ø, and 270 ø, the scatter of a plane wave was calculated and the pressure field in the fluid and the displacement field in the solid were checked for isotropy. For the p wave in the fluid, the p wave, SH wave, and SV wave in the solid, the amplitude of all power spectral coefficients corresponding to modes scattered with equal polar angles and with azimuthal angles shifted from the value at 0 ø incident azimuthal angle by an angle equal to the incident azimuthal angle, should be equal. (2) The scatter of a plane wave with an incident polar angle of 0 ø was calculated and the expansion coefficients and the power spectral coefficients of the pressure field in the fluid and the displacement field in the solid were checked for symmetry. The fields should be symmetric about the z axis and modes of order m,,,my and -- m,,,-- my should be scattered with equal polar angles and spectral amplitudes, but with azimuthal angles that differ by 180 ø. For several modes of the p wave in the fluid, and the p wave, $H wave and $V wave in the solid, Table I (a)-(d) shows the polar and azimuthal angles of scattering and the amplitude of the power spectral coefficient for each of the incident azimuthal angles 0 ø, 90 ø, 180 ø, and 270 ø and Table II (a)-(d) shows the polar and azimuthal angles of scattering and the amplitude of the power spectral coefficient for several pairs of orders rn,,, my and - tax, -- rny.• The results in these tables indicate that the scattering was isotropic and symmetric to a high degree of accuracy. (3) Reciprocity was checked in the following manner: The scatter of a plane wave for the standard parameters was calculated. Then the polar and azimuthal angles of the incident plane wave were replaced by those for a given scattered p-wave mode in the fluid, the scatter of the resultant plane wave calculated, and the amplitude of power spectral coefficient for the mode scattered in the direction of the original plane wave was determined. Rec- iprocity is obtained if the amplitude of the power spectral coefficient for the given mode scattered by the original plane wave and for the mode scattered in the original direction by the reciprocal plane wave are equal. Table III shows incident and scattered polar and azimuthal angles and power spectral coefficients for reciprocal pairs of plane wave modes and that reciprocity was obeyed to a high degree of accuracy. Based on the results of these three calculations, and the fact that in all calculations the numerical constraints mentioned previously were obeyed, it was assumed that the analytic and numerical solutions to the scattering problem were correct, accurate, and could be used to perform numerical calculations and investigate scattering from a doubly infinite fluid-elastic interface with doubly periodic surface roughness.

In the calculations described above, it was noted that the surface fields were represented by infinite series in which explicit dependence on the roughness profile •(x',y') was not included; however, this representation is



TABLE I. Results of isotropy calculations for some p-wave modes in the fluid and in the solid, and $H-wave and $ V-wave modes in the solid

p-wave modes in the fluid (deg) 0 90 180

(deg) m x my (deg) •(l) •(l) •(•) /•pm mx my (deg) mx (deg) m• / •pm my /•pm my

270

( deg ) 6p 1)

50.6923

45.0000

50.6923

23.1380

0 1 23.9550 1.7071X10 -3 --1 0 113.9550 1.7071X10 -3 0 0 0.0000 5.7368X10 -• 0 0 90.0000 5.7368X10 -• 0 --1 336.0450 1.7071X10 -3 1 0 66.0450 1.7071X10 -3

--1 0 0.0000 2.2511X10 -3 0 --1 90.0000 2.2511X10 -3

0 --1 203.9550 1.7071X10 -3 1 0 0 180.0000 5.7368X10 -• 0 0 1 156.0450 1.7071X10 -3 --1 1 0 180.0000 2.2511X10 -3 0

0 293.9550 1.7071X10 -3 0 270.0000 5.7368X10 -• 0 246.0450 1.7071X10 -3 1 270.0000 2.2511X 10 -3

(b) p-wave modes in the solid

. (.2)_ •:(2) (deg) mx my (deg) /z•b m m x my (deg) /*pm m x my (deg) •(2) /•pln mx % (deg) •(2) /•pln

53.9715 --1 --1 321.3579 3.0045X10 -8 1 --1 51.3579 3.0045X10 -8 39.1730 --1 0 0.0000 4.5160X 10 -4 0 --1 90.0000 4.5160X10 -4 53.9715 --1 1 38.6421 3.0045X 10 -8 --1 --1 128.6421 3.0045X10 -8

,

(c) SH-wave modes in the solid OSHm 2) .t, (2) .t,(2)

•SHm (2) •SHm (2) (deg) m• my (deg) ,ASH m mx my (deg) ,ASH m

1 1 141.3579 3.0045X 10 -8 --1 1 0 180.0000 4.5160X 10 -4 0 1 1 218.6421 3.0045X 10 -8 1

1 321.3579 3.0045X 10 -8 1 270.0000 4.5160X 10 -4 1 308.6421 3.0045 X 10 -8

2) .•(2) SHIn (2) tP SHm (2)

m• my (deg) ,ASH m m• my ( deg) ,ASHm

27.8028

25.2300

27.8028

13.7021

37.9979

0 --1 336.0450 4.9025X10 -4 1 0 66.0450 4.9025X10 -4 0 1 156.0450 4.9025X 10 -4 --1 0 246.8028 4.9025X10 -4 0 0 0.0000 4.1228X10 -• 0 0 90.0000 4.1228X10 -• 0 0 180.0000 4.1228X10 -• 0 0 270.0000 4.1228X10 -• 0 1 23.9550 4.9025X10 -4 --1 0 113.9550 4.9025X10 -4 0 --1 203.9550 4.9025X10 -4 1 0 293.9550 4.9025X10 -4

--1 0 0.0000 3.0265X10 -3 0 --1 90.0000 3.0265X10 -3 1 0 180.0000 3.0265X10 -3 0 1 270.0000 3.0265X10 -3 1 0 0.0000 2.8747X10 -3 0 1 90.0000 2.8747X10 -3 --1 0 180.0000 2.8747X10 -3 0 --1 270.0000 2.8747X10 -3

(d) SV-wave modes in the solid s(2) •(2) .•(2) •(2) .•(2)

Vm WSVm (2) tPSVm (2) WSVm (2) tPSVm (2) (deg) • m x -my-- (deg) •ASV m -- m x my (deg) /4SV m m x my (deg) /4SV m m x my (deg) /4SV m

27.8028 0 --1 336.0450 4.7928X10 -4 1 27.8028 0 1 23.9550 4.7928X10 -4 --1 40.0984 1 --1 342.9012 2.0317X10 -6 1 40.0984 1 1 17.0988 2.0318X 10 -6 --1

0 66.0450 4.7928X10 -4 0 1 156.0450 4.7928X 10 -4 --1 0 246.8028 4.7928X10 -4 0 113.9550 4.7928X 10 -4 0 --1 203.9550 4.7928X 10 -4 1 0 293.9550 4.7927X 10 -4 1 72.9012 2.0318X10 -6 --1 1 162.9012 2.0317X10 -6 --1 --1 252.9012 2.0318X10 -6 1 107.0988 2.0318X 10 -6 --1 --1 197.0988 2.0318X10 -6 1 --1 287.0988 2.0318X 10 -6

not unique and as indicated in Eqs. (50) and (51) the surface fields may be represented by infinite series that include explicit dependence on the surface roughness. Be- fore proceeding with extensive numerical calculations, calculations were performed to determine the most convenient representation of the surface fields. To determine the most convenient representation of the surface fields, both

surface field representations were used to calculate scattering from a doubly infinite fluid-solid interface with doubly periodic sinusoidal surface roughness as a function of the polar angle of incidence. Then the spectral amplitudes of the scattered fields, energy sum, number of equations, and condition number for the solutions based on both surface

field representations were examined and compared. It was

FIG. 1. A finite section of a doubly infinite surface with doubly periodic sinusoidal roughness.

FIG. 2. A finite section of a doubly infinite surface with doubly periodic triangular roughness.

1571 d. Acoust. Soc. Am., Vol. 94, No. 3, Pt. 1, Sept. 1993 G.C. Bishop and d. Smith: T matrix for scattering 1571


TABLE II. Results of symmetry calculations for some p-wave modes in the fluid and in the solid and $H-wave and $V-wave modes in the solid.

(a) p-wave modes in the fluid

mx my (deg) (deg) ,A• ) mx my (deg) (deg) ,A• ) 0 1 18.3101 90.0000 2.7636X 10 -3 0 --1 18.3101 270.0000 2.7636X 10 -3 1 --1 26.3778 315.0000 2.6129X 10 -5 - 1 1 26.3778 135.0000 2.6129X 10 -5 1 0 18.3101 0.0000 2.7636X 10 -3 -1 0 18.3101 180.0000 2.7636X 10 -3 1 1 26.3778 45.0000 2.6129X 10 -5 -1 --1 26.3778 225.0000 2.6129X 10 -5

(b) p-wave modes in the solid

rn• my (deg) (deg) ,A (•a• 2) rn• my (deg) (deg) ,A(,• 2• 0 1 30.3321 90.0000 2.2521X 10 -4 0 -- 1 30.3321 270.0000 2.2521X 10 -4 1 -- 1 45.5772 315.0000 1.5395X 10 -7 - 1 1 45.5772 135.0000 1.5395X 10 -7 1 0 30.3321 0.0000 2.2521X 10 -4 - 1 0 30.3321 180.0000 2.2521X 10 -4 1 1 45.5772 45.0000 1.5394X 10 -7 - 1 -- 1 45.5772 225.0000 1.5395X 10 -7

(c) $H-wave modes in the solid (2) d•(2) •(2) d•(2) SHm •SI-Im •'SI-Im 't'SHm

m x my (deg) (deg) • (2) • (2) /t• $Hm m x my (deg) (deg) / t• SHin

0 1 10.9166 90.0(X)0 9.9453X 10 -5 0 --I 10.9166 270.0(X)0 9.9453X 10 -5 1 - 1 15.5348 315.0(X)0 1.8081X 10 -7 --1 1 15.5348 135.0000 1.8081X 10 -7 1 0 10.9166 0.0(X)0 9.9453X 10 -5 -1 0 10.9166 180.0(X)0 9.9454X 10 -5 1 1 15.5348 45.0(X)0 1.8081X 10 -7 -- 1 -- 1 15.5348 225.0000 1.8081X 10 -7

(d) $V-wave modes in the solid 5(2) ,/,(2) •(2) ,/,(2)

Vm • $ Vm "' $ Vm • $ Vm

mx my (deg) (deg) •(2) •(2) /•sgm m x my (deg) (deg) /•$Vm

1 --2 25.0535 296.5651 3.3083 X 10-• - 1 2 25.0535 116.5651 3.3083 X 10-• 1 2 25.0535 63.4350 3.3083 X 10- TM -- 1 -- 2 25.0535 243.4350 3.3083 X 10- • 2 -- 1 25.0535 333.4349 3.3083X 10 -• --2 1 25.0535 153.4349 3.3083X 10 -• 2 1 25.0535 26.5651 3.3083 X 10- • -- 2 -- 1 25.0535 206.5651 3.3083 X 10- •

found that neither representation was consistently more convenient than the other. However, since the surface field representation that does not depend explicitly on the surface roughness tended to have a slightly lower condition number and in a few cases satisfied the convergence criteria with fewer equations, this representation was used in all subsequent calculations.

C. Characteristics of plane wave scattering from doubly periodic sinusoidal and triangular surface roughness

Plane wave scattering was calculated from doubly infinite surfaces with doubly periodic sinusoidal and triangu-

TABLE III. Reciprocity calculations.

lar surface roughness as functions of ( 1 ) the polar angle of incidence, (2) azimuthal angle of incidence, (3) the ratio a,,/A,,=a A.=a/A, a varied, and (4) the ratio y/.. as was k(i)/g'--k(t)/•"--k(i)/K, as kp (i) ..•, ,..,,--..•, ,..y--..•, was varied. To investigate scattering anisotropy, Ay was increased to 3 m and scattering was calculated as a function of (1) the polar angle of incidence and (2) the azimuthal angle of incidence. Some of the results of these calculations are shown

in Figs. 3-15. All figures use the nomenclature given in Fig. 3 to refer to the modes of the four wave fields.

Figure 3 shows the amplitude of the power spectral coefficient versus polar angle of incidence for the rnx=0,--1 and my=O,--1,--2 modes of the p wave in the

0 (i) qb(i) (deg) (deg) m• my

(1) mxmy

(deg) • (•) Energy sum

45.00000 45.00000 0 1

72.83030 58.44465 0 -- 1

45.00000 45.00000 0 -- 1

32.23681 20.38917 0 1

45.00000 45.00000 1 0

72.83030 31.55535 - 1 0

45.00000 45.00000 1 -- 1

56.62626 12.85808 --1 1

45.00000 45.00000 -- 1 1

56.62626 77.14191 1 - 1

45.00000 45.00000 -- 1 0

32.23681 69.61083 1 0

72.83030

45.00000

32.23681

45.00000

72.83030

45.00000

56.62626

45.00000

56.62626

45.00000

32.23681

44.99998

58.44465 4.7088• 10 -4 45.00000 4.7088X 10 -4 0.9999987 20.38917 2.4039X 10 -3 1.(XXXX)00 45.0(X)01 2.4039X 10 -3 0.9999995 31.55535 4.7088X 10 -4 1.(X)(X)(X)0 45.00000 4.7088X 10 -4 0.9999992 12.85808 2.7214X 10 -6 44.99999 2.7214X 10 -6 1.(X)O(X)(O 77.14191 2.7214X 10 -6 1.(X)O(X)(O 45.00000 2.7214X 10 -6 1.0000001 69.61083 2.4039X 10 -3 1.• 45.00000 2.4039X 10 -3 0.9999988



m x=O, my=O 1.

0.1- :o

0.01 .' /,'

0.001

0.0001

0.00001'

rex=O, my=-I 1

o.1

O.Ol

0.001 • ............ ',, 0.0001 .... 7,, • - 00001 ß I .... I .... I .... I .... I .... I'""1 .... l .... I .... I

rex=O, my=-2 0.01

o.ool.

03001.

0.00001.

o.ooooo•. .,. ......... 0.0000001 ?,'rh .... I ';'1 t' 0 10 20 30 40 50 60 70 80 90

POLAR ANGLE ('DEG.)

m x=-l, my=O

tax=-1, my=-1

0 10 20 50 40 50 60 70 80 90

POLAR ANGLE ('DEC.)

0.01

0.001'

0.0001'

0.00001

0.001

0.0001'

0.00001'

0.000001'

0.0000001 .... • .... I .... I .... I .... I .... I .... I '•" ' ' I .... I 0 10 20 50 40 50 60 70 80 90

POLAR ANGLE •DEC.)

mx=-l, my=O

0 10 20 50 40 50 60 70 80 90

POLAR ANGLE (DEC.)

FIG. 3. Amplitude of the power spectral coefficients for the mx=O, -- 1 and my=O,-- 1,--2 modes of the scattered p wave in the fluid ( ) and the p wave ('-'), the SH wave ( .... ), and the SV wave (-----) in the solid versus the polar angle of incidence for plane wave scattering from a doubly infinite surface with doubly periodic sinusoidal surface roughness.

fluid, and the p wave, SH wave, and S V wave in the solid for scattering from doubly periodic sinusoidal surface roughness and Fig. 4 shows the same quantities for scattering from doubly periodic triangular surface roughness. Plots were omitted for my=O,--1 when the amplitude of all spectral coefficients was less than 10-5 and for my------2 when the amplitude of all spectral coefficients was less than 10 -7 '

These figures show that the power spectral coefficients of all waves exhibit scattering anomalies, i.e., cusps or dis- continuities as the polar angle of incidence changes. Three types of scattering anomalies have been identified and described:•5'•6 ( 1 ) A Brewster angle anomaly occurs when the specular scattering coefficient vanishes as a parameter is changed, (2) A Wood P anomaly occurs when energy is rapidly exchanged between the specular and backscattered orders as a parameter is changed. (3) A Rayleigh anomaly occurs when a higher propagating mode becomes evanescent or when an evanescent mode becomes a propagating mode which occurs in the former case when the wavenum-

ber for the mode becomes imaginary and in the latter case when the wavenumber for the mode becomes real. These

anomalies have been used to describe scattering of a p wave from the surface of a solid on which there is periodic roughness and the total field satisfies a hard or soft boundary condition. Since mode conversion does not occur on such surfaces, scattering anomalies are caused by the

FIG. 4. Amplitude of the power spectral coefficients for the mx--0,--1 and my=O,--1,--2 modes of the scattered p wave in the fluid ( ) and the p wave (.'-), the SH wave ( .... ), and the SV wave (-----) in the solid versus the polar angle of incidence for plane wave scattering from a doubly infinite surface with doubly periodic triangular surface roughness.

modes of the scattered p-wave field. However, when scattering from the surface of an elastic solid on which mode conversion occurs, p-wave, SH-wave, and S V-wave modes in the solid as well as p-wave modes in the fluid can pro- duce scattering anomalies in the power spectral coefficients of the scattered p-wave field in the fluid. For example, Fig. 3 shows that when the polar angle of incidence is in the vicinity of 39', the mx=O, my=0,-1,-2 and mx=--l, my=0 power spectral coefficients increase sharply and exhibit a cusp discontinuity characteristic of a scattering anomaly. In fact, in this region there is very little energy coupled into the solid and the scatter is dominated by the specular return in the fluid. Examination of the fluid p-wave modes shown in Fig. 3 shows that this anomalous behavior is not due to the exchange of energy between the specular mode and any other fluid p-wave modes, nor does it occur when any fluid p-wave mode changes from propagating to evanescent or vice versa. Therefore, this behavior must be caused by an exchange of energy between the fluid p-wave modes and the solid p-wave and SH-wave modes. In this region, the mx=O, my=0 solid p-wave mode becomes evanescent and exhibits what might be termed an solid p-wave Brewster angle anomaly and the mx=O, my-- - 1 and mx = -- 1, my=0 solid p-wave modes increase and exhibit a cusp. The power spectral coefficient for the mx=O, my--0 SH-wave mode decreases in an anomalous manner and exhibits what might be called an SH-wave Rayleigh anomaly since the mx=O, my-- - 1 and mx= -- 1,



1

(b) sou9 P-WAv• (d) S0U3 SV. WAV •

FIG. 5. Power spectra for (a) the scattered p wave in the fluid and (b) the p wave, (c) the $H wave, and (d) the $V wave in the solid for a plane wave with a 45* polar angle of incidence scattered from a doubly infinite surface with doubly periodic sinusoidal surface roughness.

rny=O $H-wave modes begin to propagate. However, since the rnx=0, my=-- 1 and rnx= -- 1, rny=O $V-wave modes also begin to propagate in this region, it is not evident that the anomalous behavior of the rn•=0, rny=O $H-wave power spectral coefficient is due solely to the behavior of the m==0, my= -- 1 and m== -- 1, my=O SH-wave modes. From these observations, it is evident that when the mx=O, my=O solid p-wave mode becomes evanescent, energy is made available to the other wave modes and although the manner in which this energy is partitioned between the various modes is not clear from the results shown in Fig. 3, the anomalous behavior of the fluid p wave can only be a consequence of an exchange of energy from the solid wave modes and that the energy exchange must occur principly from the solid mx=O, my=O p-wave mode.

Figures 3 and 4 indicate that the location of scattering anomalies is independent of the type of surface roughness. Since the wave number for a given mode depends on the order of the mode, the period of the surface roughness, and the polar and azimuthal angles of incidence and not on the type of surface roughness, Rayleigh and Brewster angle scattering anomalies occur at the same polar incidence angles for both types of surface roughness. The location of Wood P anomalies is more difficult to calculate and from

the results presented in this paper it is not clear why it is also independent of the type of surface roughness. It also appears that the amplitudes of the power spectral coefficients for the mx=0, my=O fluid p-wave mode and the solid p-wave and $H-wave modes are independent of the type of surface roughness. However, Eqs. (A2), (63), and (66) show that, in general, the amplitude of the power

spectral coefficients for these modes is not independent of the type of surface roughness and examination of the numerical results showed that for the parameters used in these calculations, the amplitudes of the power spectral coefficients for these modes are very weakly dependent on the type of surface roughness. These figures also indicate that for both types of surface roughness, the the amplitudes of the power spectral coefficients for the rn•=0, my=--1 and rnx---- 1, my=O modes are identical. This is simply a consequence of the fact that the surface heights and periods in the x and y directions are the same.

Also, Figs. 3 and 4 show that the although the amplitude of the power spectral coefficients of the modes of all fields depends on the type of surface roughness, the dependence of the amplitudes of the power spectral coefficients on the polar angle of incidence is independent of the type of surface roughness.

It is important to note that for both types of surface roughness, energy was conserved to a high degree of accuracy, but that scattering from triangular surface roughness was more difficult to calculate than scattering from sinusoidal roughness since more equations were necessary to satisfy the numerical constraints. This result tended to be characteristic of all calculations.

Figure 5 shows the amplitude of all power spectral coefficients greater than 10-5 as a function of the polar and azimuthal scattering angles for the fluid p wave, and the solid p, $H, and $V waves for scattering from sinusoidal surface roughness for a polar angle of incidence of 45 ø. Figure 6 shows the same quantities for scattering from triangular surface roughness. The arrow on the fluid



1 1

0.1 (a) FLUID P"WAVE 0.1 (c) SOL/D SH--WAV•

ß

' 0.1 (b) $OL/D P"WAV[ • 0.1 i (d) $OUD SV..WAV E

ø.ool t

FIG. 6. Power spectra for (a) the scattered p wave in the fluid and (b) the p wave, (c) the $Hwave, and (d) the $Vwave in the solid for a plane wave with a 45* polar angle of incidence scattered from a doubly infinite surface with doubly periodic triangular surface roughness.

p-wave plots indicates the specular mode and on the solid wave plots it indicates the m•,=rny=O mode. These plots show that the power spectrum of all fields, except the solid p-wave field, produced by scattering from sinusoidal surface roughness exhibit more spectral components than the fields produced by scattering from triangular surface roughness. However the scatter of all fields, except the solid p-wave field, from both types of surface roughness is dominated by the mx=my=O mode, i.e., the specular mode for the scattered fluid p wave. In addition, these figures show that the amplitude of a mode common to the power spectra of fields produced by scattering from both types of surface roughness, is greater for scattering from sinusoidal surface roughness than for scattering from triangular surface roughness.

Figure 7 shows the amplitude of the power spectral coefficients versus azimuthal angle of incidence for the tax=0,--1 and my=O,-1,-2 modes of the fluid p wave and the solid p, $H, and $V waves produced by scattering from sinusoidal surface roughness and Fig. 8 shows the same quantities produced by scattering from triangular surface roughness. These figures show the specular mode of the fluid p wave and the m•,=my=O $H-wave mode are independent of the azimuthal angle of incidence for both types of surface roughness and that the amplitude of these two modes appears to be independent of the type of surface roughness. However, examination of the numerical results indicates that although the amplitude of these coefficients is independent of the azimuthal angle of incidence, there appears to be a very weak dependence on the type of surface roughness. For example when the azimuthal angle of incidence is -- 180 ø, the amplitudes of the specular mode of

rex-O, my=O mx--1, my-O 1:.-

0.1-

0.01

0.001

0.0001-

0.00001 .... • .... • .... • .... • .... • .... { .... • .... • i

mx=O, my=-1 tax=-1, my=-1 1

o.1

o.o1'

0.001 "'"

0 0001 I .... •,l,' •,[ ,,,•,:,,,• ,-1,;•',,>,• ....

mx=O, my=-2 -186.135-90-45 0 45 90 155 180 AZIMUTHAL ANGLE ('DEC.)

0.011 0'0011

ooooooo,,,,,,:, ,/1,,,, ........ , ....... -186.155-90-45 0 4,5 90 155 180

AZIMUTHAL ANGLE ('DEG.•

FIG. 7. Amplitude of the power spectral coefficients for the rn•,=0,--1 and my=O,--1,--2 modes of the scattered p wave in the fluid ( ) and the p wave ("'), the $H wave ( .... ), and the $V wave (-----) in the solid versus the azimuthal angle of incidence for plane wave scattering from a doubly infinite surface with doubly periodic sinusoidal surface roughness.



rex-O, my-O mx--1, my-O

0.1'

0.01'

0.001'

0.00001 ....

mx=0, my=-1 -186.155-90-45 0 45 90 155 180 AZIMUTHAL ANGLE (DEC.) 1

o.1

:::) O.Ol

C3_ O.001-=J

o-ooo,1.i :.,.."'.,,-; "- 0.00001 ,,• ....

rex=O, my=-2 0.011

0,001 l

0.00011 0.000011 0.000001• o.ooooool/ .... 1'::',':•:'::'• .... ,, ,,,,, ,",•,,

-186-155-90-45 0 45 90 155 180

AZIMUTHAL ANGLE (DEG.)

FIG. 8. Amplitude of the power spectral coefficients for the mx=0,--1 and my--O,-- 1,--2 modes of the scattered p wave in the fluid ( ) and the p wave (...), the SH wave ( .... ), and the SV wave (-----) in the solid versus the azimuthal angle of incidence for plane wave scattering from a doubly infinite surface with doubly periodic triangular surface roughness.

the fluid p wave and mx=my=O SH-wave mode are, respectively, 0.573 6704 and 0.412 2827 when scattering from sinusoidal surface roughness and when scattering from triangular surface roughness the amplitudes of these modes are, respectively, 0.579 2143 and 0.418 4053. This behavior is a consequence of the fact that for these modes the surface dependent quantity 11m(k) defined by Eq. (A2), depends only on z •' Coo(k) when rnx=0, my=O, k--•0, and œ' when rn•=0, my=O, k=0. Therefore the fact that the amplitudes of power spectral coefficients for the specular mode of the fluid p wave and rn•=my=0 SH- wave mode are independent of the azimuthal angle is a general property of scattering from doubly periodic rough surface roughness and does not depend on the types of surface roughness considered in this paper nor is it a consequence of the fact that the surface maximum height and period along the x and y directions are equal. However, the fact that the amplitudes of power spectral coefficients of these modes depend weakly on the type of surface roughness is a consequence of the types of surface roughness considered and for different types of surface roughness, the amplitudes of power spectral coefficients of these modes might differ significantly from those shown in Figs. 7 and 8.

In addition, these figures show that the amplitudes of power spectral coefficients for the m•=0, my=--1 and m•= -- 1, my=O modes of the fluid p wave and of the solid p, SH, and $V waves depend strongly on the azimuthal

angle of incidence and on the type of surface roughness and also exhibit scattering anomalies. The amplitudes of power spectral coefficients for the mx=O, my------1 modes are symmetric about 0 ø azimuthal angle, i.e., the x axis, and for the m•=- 1, my=O modes they are symmetric about, 90 ø azimuthal angle, i.e., the y axis. This is a consequence of the fact that when rn•=0 (my=O), the component of fire(k) along the x axis (y axis) is zero, and the fact that the surface maximum height and period along the x and y .axes are the same. These figures indicate that the scattering anomalies of the fluid p wave can only be a result of energy exchange between the fluid p wave and solid p, $H, and $V wave modes. For example, when the azimuthal angle of incidence is in the vicinity of --117 ø, the amplitude of the power spectral coefficients of the m•=0, my=-- 1 mode of the fluid p-wave and the m•=0, my= -- 1,- 2 modes of the SV wave essentially vanish and the mx=0, my=--1,--2 modes of the $H wave increase sharply and exhibit cusps. This suggests that energy is exchanged between these modes and that the $H modes gain energy at the expense of the fluid p-wave and the $V-wave modes. However, when the azimuthal angle of incidence is in the vicinity of 63 ø , the exchange of energy appears to be in the opposite direction and the power spectral coefficients for the m•=0, my=--I mode of the fluid p wave and the m•=0, my=--1,--2 modes of the $V gain energy at the expense of the m•=0, my=--1,--2 modes of the SH wave. Al- though there appears to be energy exchange between the m•=0, my.=--1,--2 fluid and elastic modes, it appears that energy is not exchanged between the modes of different symmetry, i.e. the mx=- 1,--2, my=O modes. These figures also show that the amplitudes of the power spectral coefficients of the modes of all fields depend on the type of surface roughness, but that the dependence of the amplitudes of the power spectral coefficients on the azimuthal angle of incidence is independent of the type of surface roughness.

Figure 9 shows the amplitude of the power spectral coefficients versus a/A --a•/A•=ay/Ay for the rn•=0,--1,--2 and my=O,--1,--2 modes of the fluid and solid wave fields for scattering from sinusoidal surface roughness and Fig. 10 shows the same quantities for scattering from triangular surface roughness. To obtain these results the value of the surface maximum amplitude a was increased until a numerical constraint could not be satisfied

or a computational limit was reached. Scattering from the roughest surfaces was performed on a CRAY X-MP computer and for this computer, the maximum number of equations for which computer memory could be allocated was 1156 which occurs when rn= 8. This limit was principly a consequence of the memory required to store the O(m• ) matrices and solve the system of linear equations. Therefore, if the system of equations was numerically. sta- ble, but other numerical constraints were not satisfied for m= 8, the maximum computer memory allocation limited the maximum surface roughness that could be considered. It is important to note that for these calculations energy conservation was not a sufficient numerical constraint on

the calculation. Typically, energy conservation was satis-



0.1

0.01

0.001

0.0001

0.00001

m x=O, my=O

''''l' ''' I .... I .... I .... I

mx=O, my--1 1

o.1

• :.:.:.:.::.':.-'-•':- ..... .-.:_..? I--- '

el_ 0.001

0.0001

0.00001 ',..,1 .... • .... • .... • .... •

tax=-1, my=O

0.01

0.001

0.0001-

0.00001'

0.000001'

0.0000001

mx=-l, my=-1

mx=0 , my=-2

m x=-2, my=O

mx=-2, my--1

mx=-2, my=-2 mx=-l, my=-2

0.00 0.01 0.02 0.0,.3 0.04 0.050.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03.3 0.04 0.05

a/A a/A a/A

FIG. 9. Amplitude of the power spectral coefficients for the mx=O,- 1 and my=O,- 1,--2 modes of the scattered p wave in the fluid ( ) and the p wave (-"), the SH wave ( .... ), and the SF' wave (-- - --) in the solid versus a?A for plane wave scattering from a doubly infinite surface with doubly periodic sinusoidal surface roughness.

0.01

0.001-

0.0001'

0.00001

m x=0, my=0 m x=- 2, my =0

'''[l''''l''''l .... I .... I

mx=O, my=-1

0.1

::• 0.01 ..............

a. 0.0011 .. ..../• ....... 0-00011

0.000011 • • .... •

mx=O, my=-2 0.01'

o.ool

0.0001

0.00001

0.000001

0.0000001

mx=-l, my=0

I .... I

tax=-1, my=-1

mx=-l, my=-2 mx=-2, my=-2

i,, , i, ,, , i .......... I

0.00 0.01 0.02 0.03 0.04 0.050.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03 0.04 0.05

a/A a/A o/A

FIG. 10. Amplitude of the power spectral coefficients for the mx=0, - 1 and my=O,- 1,-2 modes of the scattered p wave in the fluid ( ) and the p wave ('"), the SH wave ( .... ), and the SF' wave (-- - --) in the solid versus a?A for plane wave scattering from a doubly infinite surface with doubly periodic triangular surface roughness.



0.030

FIG. 11. Power spectra for (a) the scattered p wave in the fluid and (b) the p wave, (c) the SH wave, and (d) the SV wave in the solid for a plane wave scattered from a doubly infinite surface with doubly periodic sinusoidal surface roughness with a/A =0.030.

a/A = 0.030

FIG. 12. Power spectra for (a) the scattered p wave in the fluid and (b) the p wave, (c) the $H wave, and (d) the $V wave in the solid for a plane wave scattered from a doubly infinite surface with doubly periodic triangular surface roughness with a/A =0.030.



mx=O, my-O mx--1, my-O mx--2, my=O 1

0.01 "% ....,/• .• 0.0001-

0.00001 .... • '• •

mx=O , my=-1

o.1

::> O.Ol

Q- 0.001

000001l .... , .... • .... ,,. •. ii• .... , rex=O, my=-2

0.1'

0.01'

0.001 •

0.0001 •

0.00001

0.000001

0 25 50 75

k(•/K 100 125 0

mx=-l, my=-1 mx:-2, my:-1

tax=-1, my=-2

, ' i

25 50 75 100 125 0

' ' i

mx=-2, my--2

25 50 75 100 125

k½•/K FIG. 13. Amplitude of the power spectral coefficients for the mx=0,-- 1 and my----O,-- 1,--2 modes of the scattered p wave in the fluid ( p wave ('"), the $H wave ( .... ), and the $V wave (--- doubly periodic sinusoidal surface roughness.

) and the

k(i)/K for plane wave scattering from a doubly infinite surface with --) in the solid versus

Ld

0.01

0.001

0.0001-

0.00001

0.1'

0.01

0.001

0.0001

0.00001

0.1'

0.01

0.00!

0.0001

0.00001

0.000001

mx=O, my=O

mx=O, my=-1

m x-O, my:-2

mx--1, my=O mx=-2, my=O

,''' ' .... ' ' ' i

mx=-l, my=-I mx=-2, my=-1

mx=-l, my=-2

_

0 50 75 100 125 0 25 50 75 100 125 0

,,, i

m x=-2, my=-2

25 50 75 100 125

kO)' p/K

FIG. 14. Amplitude of the power spectral coefficients for the tax=0,-- 1 and rn =0,-- 1,--2 modes of the scattered p wave in the fluid ( )and the k•)/A for plane wave scattering from a doubly infinite surface with doubly p wave (' "), the $H wave ( .... ), and the SVwave (-- -) in the solid versus ..p

periodic triangular surface roughness.



Ld

0.1

0.01

0.001

0.0001'

0.00001

mx=O, my=-1 1

0.1

0.01

0.001

o.ooo1•'"',, /"-.%, o.oooo,. .... ....

mx=O, my=O m x=-l, my=O

,,',1 .... i,,,,i .... i .... i,,,,i,,,,i,,,,i

tax=-1, my=-1

mx=O, my--2 0.011 0'0011

0.0001•

0.00001 •

ooooooo, i.i,., ' .... i ..... ,.,i.,, ...; ,,, ,.,..,, -188-155-90-45 0 45 90 155 180

AZIMUTHAL ANGLE •DEG.)

. ,'r,i .... • .... •,, , i, ,,;•,., i.r,, ,1 -186-155-90-45 0 45 90 155 180

AZIMUTHAL ANGLE ('DEG.)

FIG. 15. Amplitude of the power spectral coefficients for the mx=0,- 1 and my=O,-1,-2 modes of the scattered p wave in the fluid ( ) and the p wave (-.-), the SH wave ( .... ), and the SV wave (-----) in the solid versus the azimuthal angle of incidence for plane wave scattering from sinusoidal surface roughness with a period along the x axis of 2 m and a period along the y axis of 3 m.

fled to within the desired accuracy with relatively few equations, however, to satisfy the convergence criteria for the spectral coefficients of the various fields considerably more equations were required. Energy conservation was satisfied for larger values of a/A than are shown in Figs. 9 and 10. However, since the convergence criteria for the spectral coefficients could not be satisfied and since the value of rn could not be increased beyond 8, these results are not shown.

Figures 9 and 10 show that the amplitude of the power spectral coefficient for the specular mode of the fluid p-wave mode and the mx=my=O mode of the $H-wave field tends to decrease for both types of surface roughness as a/A increases and that the decrease is greater for sinusoidal surface roughness than for triangular surface roughness. These figures also show that for large values of a/A, the amplitudes of the power spectral coefficients of the higher order modes depend on the type of surface roughness. This is particularly evident in the power spectral plots shown in Figs. 11 and 12 for a/A =0.030. When the results shown in these figures are compared with the results shown in Figs. 5 and 6 where a/A=0.005, it is evident that as surface roughness increases, the number of modes that contribute to the scattered fields increases significantly, that the power spectral content of fields scattered from sinusoidal surface roughness differs significantly from the power spectral content of fields scattered from triangular

surface roughness, and also indicates the very complex na- ture of scattering from very rough surfaces.

Figure 13 (or 14) shows the amplitude of the power spectral coefficients versus k• I '/K= k• l '/Kx= k• '/Ky for scattering from sinusoidal(triangular) surface roughness. These calculations were performed in a manner similar to that used to calculate scattering as a function of a/A. The value of k• 1) was increased until a numerical constraint or computational limit was reached. In these calculations, as in the surface roughness calculations, the number of modes necessary to satisfy conservation of energy was considerably fewer than the number of modes required to satisfy the convergence criteria for the field spectral amplitudes and the calculations were limited by the same maximum memory allocation of the computer. The difference in the frequency dependence of the two types of surface roughness is quite striking. Perhaps, the most striking difference between the scatter from sinusoidal and triangular surface roughness is that all modes of the fluid p wave scattered from sinusoidal surface roughness exhibit scattering anomalies whereas none of the modes of the fluid p wave scattered from triangular surface roughness exhibit scattering anomalies. This result suggests that the inverse problem, i.e., characterization of the scatterer or surface roughness, might be more readily solved by observation of the field scattered by a broadband pulse rather than by spatial sam- pling of the field scattered by a cw pulse. However, the difference between the scatter from sinusoidal and triangular surface roughness is evident not only in the frequency dependence of the power spectral amplitudes of the scattered fields, but also in the high frequency power spectra of the scattered fields.

To investigate scattering anisotropy, the period of the surface roughness along the y axis was increased to 3 m and scattering from sinusoidal surface roughness was calculated as a function of the azimuthal angle of incidence. Some results of this calculation are shown in Fig. 15. Com- parison of the results shown in this figure with those shown in Figs. 10 shows that when taytO, the amplitude of the power spectral coefficients of these modes are identical for all fields. This is simply a consequence of the fact that when my=O, the surface-dependent quantity fin,(k) is independent of Ky. However, when my=fi:O, the amplitude of the power spectral coefficients for scattering from sinusoidal surface with Ay=3 m and Ay=2 m differ for all modes for all fields. The power spectral coefficients for the mx=O my=--1 modes and those for the m•=--1 my=O modes are not simply translated replicates, the symmetry between these modes is broken. To see the asymmetry, it is necessary to "look" in the direction in which the my-•:O modes are scattered.

VII. CONCLUSIONS

It has been shown that a T-matrix formalism can pro- vide an analytic and numerical solution for the scatter of a plane wave from a doubly infinite fluid-solid interface with doubly periodic surface roughness. The formalism provides an extremely accurate numerical solution, however since it



is necessity to store large dense matrices, some calculations are limited by the maximum memory allocation of the computer and not because the formalism becomes numerically unstable.

It has been demonstrated that the scattering anomalies are present in the various modes of the fluid p field and the solid p, SH, and S V fields and that these anomalies arise from an exchange of energy not only between higher modes of the same field, but also from an exchange of energy between modes of different fields. It has been shown that

although the scatter from different types of surface roughness exhibits spatial dependence, this dependence is often weakly dependent on the type of surface roughness particularly when the maximum surface amplitude is small: The difference between the various types of surface roughness is most readily observed in the broadband frequency re- sponse. It also has been shown that when the scattering surface is not symmetric, the asymmetry is not apparent in all modes of the scattered fluid p field.

ACKNOWLEDGMENTS

This work was funded by the Naval Undersea Warfare Center Division Newport Independent Research program and the Advanced Scientific and Engineering Computing Center. We would also like to acknowledge Dr. R. P. Rad- linsky who kindly reviewed the manuscript and provided several useful comments.

APPENDIX

When the representations of the scalar basis functions given by Eq. (22) and the representations of the vector basis functions given by Eqs. (27a)-(27½) are used in Eqs.

r3( + ) (53a)-(53h), the matrix elements •at•mn are given as fol- lows:

+ (f,(=2+)x ) ß •st. ' Xkstm) ]

k (2) k(2)) 'll=--n(•.-=m--.-=n , (Ala)

ikJ 2) QI +) f•(2) *(2. f•(2) *(2.) f•(2) •(2.) __ . )ks= + 2,n 2k(2) [ (ks=)X ß • '•stn '•Stnl '•stn ' -s=

szrrl

X (f(:(s•2-4-) X f•(2) ) ] o f.•. (_1_ k(2)k(2)) ß VS•!11 • n ' -szrrl • ' 'S•!l '

ik(2) _ s f,?) . •13mn__2k(2sA [( X ) --pn '"Still

'"Still

k (2) k(2)), ø •m-n ( •--sz=--"vzn

(Alb)

(Alc)

Q(+) _i•(1) kp(1) 14mn--'• 2k(2) (f(:(2-4-)xf"(2)) $zln

k(2) k(1)), ø l'•m- n ( -I-..szm --' -pzn (Aid)

Q( 4_ ) --ik• 2) 21ran-- 2k(2s• [ (f(:(sn2+)X f..(2))o f..(2) f•(2-4- ) ß •stn ' •st!11 '

q_ (f(:(sn2+) Xf•(2)) o •(2+)f•(2) ] '•stn ' -s= ' •st!11

k(2) k(2)), ø •'•.--n( =E'-szm--"szn (Ale)

Q(+) -k• 2) 22ran-- 2k(2) [ ['(2)'fc(2)•(2+) fc(2)'•(2+)•(2)] ß •stn ' •st!11 •vs= -J" '•stn ß 'sin ß •st!11

k(2) k(2)) ' l'•m_ n ( -I- .-szan-- .-szn , (Alf)

k(2) k(2)), ø l'•m- n ( -I-..szm --' -pzn (Alg)

_•(1) kp(1) f•( q- ) f•(2). l'•m_n ( _l_k (2) k(1) ) k•24mn = /1(2) 2k(2) '"stm '-szm--"pzn ' $z=

(Alh)

-ik•(2) ( ,;1. (2) f•(+) f•(2) • • '•$tn •31=n 2k•()z= •l.(2)q-2/z(2) (fC(sn 2+)x )

•(2) ) +2 )1.(2) +2•(2) (•(sn2+)X fc(2)) . •(2+)•(2+) ß '$tn --pm --jan

k(2) _ k(2)) ø l'•m-n ( -I-..Vz m '-szn , (Ali)

t q- 2 •1.(2) q- 2/z(2) "sin '-tan '-tan k(2) k(2)), ø l'•m- n ( -I-..Vz m --..szn (Alj)

k•:33=n-- 2kp(2z•) ,)1,(2) + 2p(2) k• +)

•(2) + 2 )1.(2) q- 2p(2 ) fc• +) ̂ (2+ •(2) _ k(2)), (Alk) ø l'•m--n( +'-pzm '-pzn

,

Q(+) _ --')1'(1) kp (1) •.(24_) o l.•=_n ( _l_k(2) __k (1)) 34=n--)1.(2 ) q- 2/Z(2) 2k/,(2z•) '-/an '-/,zm '-/,m ,

(All)

/3(q-) -- ikp(1) f(:(sn 2+) f•(2) ß k(1) _k(2) ß 'stn '-pzm --szn , •41mn__2kp(lz•) ( X ) f'•m-n( -I- ) (Aim)



and

+) 2mn =

+) 3mn =

(Aln)

(Alo)

kp(1) fc(l+•'l•m n( :i:kp(lz• )--•(1)) Q(14+dn = 2kp(lz•)--/an - --p2n , (Alp)

where for arbitrary indices m and wave number k the quantity 11 m (k) is given by

( mx=O, my=O,k-•O ) , ( mx-•O,m•O,k=O ), (mx=O, my--O,k=O), (A2)

where •m and Cm (k) are, respectively, the two-dimensional finite Fourier transforms of the rough surface • (x',y') and the rough surface phase e -ikg(x"y') and given by

•m=•mxmy

dx' Ax (x',y')exp[--i(mdrxx' + myKyy') ] (A3)

and

( k ) = k )

dx' f dy' = A• • exp{ -- i[ m•drxx' + myKyy' + kg(x',y' ) ] ). (A4)

To obtain the expression for fire(k), the following results were used:

Oy,•(x',y')e-ikg(x',Y ' )

mx-- -- • my= -- oo

myKy k Cmxmy (k)exp[i(mxKxx'

and

+ myK•' ) ], (A7)

myKy C%my (k) k

dx' f cly' &, • 0y,• (x',y') exp(- i[ mxKxx' + myKyy' +k•(x',y')]). (A8)

These expressions are generalizations of similar results derived previously for two dimensional surfaces. 17 It is important to note that Eqs. (A4)-(A7) can be used for triangular surface roughness, even though •'(x',y') is not differentiable everywhere.

mx= - oo my- -- oo

Xexp[i(mxKxx' +myK•') ], (A5)

k Cmm(k)

= Ax • O•,•(x',y')exp(--i[m•Jixx' + myKyy' +k•(x',y')]). (A6)

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R. H. Hackman, "The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates," J. Acoust. Soc. Am. 75, 35-45 (1984).

9p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1783.

•0p. M. Morse and K. U. Ingard, Theoretical •lcoustics (Princeton U.P., Princeton, NJ, 1986), p. 364. J. T. Fokkema, "Reflection and transmission of time-harmonic elastic waves by the periodic interface between two elastic media," Department of Electrical Engineering, Delf University of Technology, Delft, The Netherlands (1979), pp. 28-33.

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A T matrix for scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness