On the scattering amplitudes for elastic waves

Journal of Applied Mathematics and Physics (ZAMP) Vol. 38, November 1987 0044-2275/87/006856-18 $ 5.10/0 9 Birkh/iuser Verlag Basel, 1987 On the scattering amplitudes for elastic waves By George Dassios*), Dept. of Mathematics, University of Patras, Kiriakie Kiriaki, Dept. of Mathematics, National Technical University of Athens, and Demosthenes Polyzos, Dept. of Applied Mechanics, University of Patras, Patras, Greece 1. Introduction In the framework of the direct theory of elastic scattering, there are three kinds of boundary value problems for the Navier equation, which characterize the scattering region from a physical point of view. They correspond to the rigid scatterer where no displacement is allowed on the surface of the body, the cavity where no surface traction appears on the boundary, and the penetrable scatterer where continuity of both the displacement and the surface traction across the surface of the scatterer is imposed. All three scattering problems of the linearized theory of elasticity become well-posed whenever appropriate radiation conditions are imposed at infinity, securing the radiative nature of the scattered field as well as its radial rate of decay. The elastic radiation conditions have been introduced by Kupradze [12]. The linearity of the problem allows the analysis of the general time dependent problem into a family of stationary problems parameterized by the angular frequency, which is the Fourier variable dual to the variable of time. The introduction of a time harmonic dependence reduces the original hyperbolic problem in space-time to an elliptic problem in space. The incident wave could be either a longitudinal plane wave, or a transverse plane wave propagating in a particular direction. To our knowledge, the first successful attempt to solve analytically the elastic scattering problem for a sphere has been carried out by Ying and Truell [17] for longitudinal incidence and by Einspruch, Witterholt and Truell [9] for transverse incidence. The low-frequency theory of elastic scattering for all three types of boundary conditions as well as corresponding applications to the general triaxial ellipsoid and its degenerate forms are given in [6, 7, 8, 11]. Jones [10] has investigated the low-frequency behaviour of elastic waves for a general scatterer. He gave estimates for the low-frequency field both for physical and *) Part of this work was done during the time that the first author was visiting the Department of Mathematics of The University of Tennessee at Knoxville. The authors want to thank the Greek Ministry of Research and Technology for partially supporting the present work. Vol. 38, 1987 On the scattering amplitudesfor elastic waves 857 geometrical variations of the scatterer. Using a regularized integral equation method, Ahner and Hsiao reduced the low-frequency elastic scattering problem to a Neumann series, both in two [2] and three [1] space dimensions. Scattering amplitudes for elastic waves are introduced by Barratt and Collins [3] in a well written paper, more that twenty years ago. They also established relations between the forward scattering amplitudes and the scattering cross-section. Rigorous proofs for integral representations of exterior boundary value problems in the theory of homogeneous and isotropic linear elasticity can be found in [18]. For the case of acoustic and electromagnetic scattering the basic reciprocity and scattering theorems are treated in [14, 15, 16]. The present work deals with the reciprocity relations and the general scattering theorems connecting the three normalized spherical scattering amplitudes [6] of the elastic theory of scattering. More specifically, we prove that the radial scattering amplitude for incidence along k, as it is observed in the direction of ?, coincides with the radial scattering amplitude for incidence along - ?, as it is observed in the direction - k . Similar relations between the angular scattering amplitudes, as well as between the radial and the angular scattering amplitudes, hold true with specific modifications. We also state and prove all possible scattering theorems that come out of longitudinal-longitudinal, transverse-transverse and longitudinaltransverse incidence. They express particular combinations of the scattering amplitudes in specific directions of incidence and observation, in terms of an integral over the unit sphere involving all the spherical scattering amplitudes. Specializing to the same direction of incidence and observation, we obtained closed form expressions for the scattering cross-section for longitudinal incidence in terms of the forward radial amplitude, and for transverse incidence in terms of the forward angular amplitudes. Detailed physical interpretation of all reciprocity and scattering relations are given throughout this work. A brief statement of the elastic scattering problem is given in Section 2. For more detailed information the reader is referred to [6]. Section 3 contains all the main results of this work as they are indicated above. The implications of the results of Section 3 for the scattering cross-section are discussed in Section 4. Finally, in Section 5, we specialize to the case of acoustic scattering and we recover all the basic reciprocity relations as well as scattering theorems for the amplitudes and the cross-section. The general approach followed in the present paper is heavily based upon analytical techniques introduced by Victor Twersky [16]. 2. The scattering problems in elasticity Let V- be a bounded, convex and closed subset of IR3 having a smooth boundary S. The set V- defines the scatterer, while its boundary S supports the discontinuities of the physical parameters of the medium, which in this case are 858 G. Dassios, K. Kiriaki and D. Polyzos ZAMP the Lam6 constants 2,/t, and the mass density ~. Focusing attention to the time independent wave problem corresponding to the Fourier component e -i'~ scaling out the mass density and in the absence of body forces, the displacement field u satisfies the stationary Navier equation [6] A<*)u(r) + co2u(r) = O, r c V= (V-) c (1) where A(*) = c2 A + (c2 - c2) VV. (2) cv = V/2 + 2# ~s = ~/~ (3) are the phase velocities for the longitudinal and the transverse waves, respectively, and A stands for the Laplacian in ~,3. Helmholtz's decomposition theorem Splits the displacement field into the sum of a longitudinal (irrotational) and a transverse (solenoidal) wave, i.e. (4) u(r) = u"(r) + ~'(r). The wavenumbers k v of the longitudinal wave and ks of the transverse wave, are related to the corresponding phase velocities c v and c~ and the angular frequency co, via the formulae co = cpkp = c~ k~. (5) The incident wave could be either the longitudinal time independent plane wave (6) #"(r) = ke ik,~'r, or the transverse time independent plane wave ,~S(r) = ~ e iks~r, ~ . r, = O, (7) where k*) is the direction of propagation and ~ is the direction of polarization of the transverse wave ~ . The total wave ~ is the sum of the incident wave and the scattered wave u which is assumed to satisfy the Kupradze radiation conditions ((15) and (16) in [6]). The physical characteristics of the scatterer enter the mathematical formulation of the problem through the boundary conditions that are described on the surface of the scatterer. There are three main types of boundary conditions in the theory of elasticity. In the first boundary value problem, known as the "rigid" problem, the displacement field is vanishing on the surface of the scatterer, i.e., ~(r)=0, *) rcS, The hat ..... on the top of a vector indicates that the vector has unit magnitude. (8) Vol. 38, 1987 On the scattering amplitudes for elastic waves 859 which describes the physical situation where the scatterer is a rigid body whose surface does not deform. In the second boundary value problem, or the "cavity" problem, the surface of the scatterer offers no resistance to exterior forces and deforms in such a way as to vanish the surface traction on it, i.e. r ~(r) = 0, r e S, (9) T = 2#A. V + 2 h d i v +/~h x rot (10) where is the surface traction operator and ~ is the outer unit normal on the surface of the scatterer *). When the scattering region is occupied by an elastic material characterized by physical parameters different from those of the surrounding medium, then besides the total ~ + on the exterior of the scatterer, there exists an interior wave field ~P- inside the scatterer. The two media are "glued" together on the boundary S, so that both the displacement as well as the traction field are continuous across S. These conditions form the third boundary value problem, or the "penetrable" problem of elastic scattering, and they are stated as S, T § (11) where T § and T - correspond to the exterior and the interior surface traction operators, respectively. In what follows, we will refer to the above boundary value problems as the "first", the "second" and the "third" scattering problem of the theory of elasticity. Using appropriate integral representations for the exterior field we were able to show, in [6], that the scattered field u exhibits the following asymptotic representation, as r -~ o% u(r) = gr(F, k) h(kpr) F + go(F, k') h(ksr ) t~ + g~o(~,Ir h(ksr) (/3 + 0 ~ , where ~, O, ~ are the unit vectors of the spherical coordinates, h(x) = eiX/ix and gr, go, ge are the normalized spherical scattering amplitudes, which are given by the following forms: IH go(F,k')=k 2 go(f,k")=k~ *) ( 2 /'| # In F| , (13) , : ( ] ' x / x ~+ 2F| D - ~- .#, (14) ~: ( [ x f x F + 2F | 4, (15) - ~o-2 By the smoothness of S, ~ exists and varies continuouslyon the surface of the scatterer. 860 G. Dassios, K. Kiriaki and D. Polyzos ZAMP where I~p = I~(k,) = ~1 ! ~(r') @ A' e-'k.~"ds(r ') (16) Op = O(k.) = ~1 ! T.. ~Y(r') ~ik,e./ ds(r') (17) os = O(ks) and (a | b):(c | d) = (a. d) (b. c). (18) We note that H" I expresses the scalar invariant while H" I x I expresses the vector invariant of the dyadic n. The asymptotic form (12) implies the decomposition of the scattered field u, far away from the scatterer, into the sum of three mutually orthogonal waves. The first term in the right hand side of (12) represents an expanding longitudinal spherical wave which propagates with phase velocity o9/k~, = %. The other two terms represent two expanding transverse spherical waves, one polarized along and the other polarized along ~, which propagate with phase velocity co/ks = % The physical meaning of the expansion (12) is even more clear if we incorporate the harmonic time dependence e-,or and write it in the form u(r) e -i~ = Gp. ~ | ~ eik'Cr-c'O + Gs" ( [ - ~ | f') eik~(r-c~t) 0 ( 1 ) + (19) where the vectorial scattering amplitudes Gp and Gs have the forms a p = - i k , II-t ,: ( Gs=-iks ~ s:(Ixlxf+2f| # [) - •ikpopl . (20) (21) The surface values of the displacement and the traction enter the amplitudes Gp and Gs via the dyadics /-/p, //s and the vectors Op, 0 s respectively. The directions of polarization, the spatial rate of decay and the outgoing character of the scattered field are explicitely indicated in the expressions (19). 3. Reciprocity in elastic scattering theory Consider two incident plane waves ~1, ~2 propagating in the directions kl, k2, respectively and let ul, u2 the corresponding scattered waves. The relative total fields are given by = ~i + ui, i = 1, 2. (22) Vol. 38, 1987 On the scattering amplitudes for elastic waves 861 Following Twersky's notation [15] we write {u, v}ae = f [u(r) 9To(r) - o(r). Tu(r)] ds(r), (23) af2 where the surface differential operator T is given by (10) and 8t2 is the bounding surface of a regular, bounded region in ~p~3. In what follows, we refer to a coordinate system whose origin coincides with the center of the smallest sphere SRo circumscribing the scatterer V-. Let SR be a sphere, centered at the origin, with radius R, large enough to include the scatterer in its interior, i.e. R > R o. Every direction passing through the origin is called "radial". Lemma 1. Suppose that ~ , i = 1, 2 are two solutions of the first, the second or the third scattering problem of the theory of elasticity, corresponding to the two radial directions of incidence ki, i = 1, 2. Then for every R > R o {7tl, ~2}sR = 0. (24) Proof. Betti's third formula [12] in the theory of elasticity reads {u, o}0o = ~ [u(r) 9A(*)o(r) - o(r). A (*) u(r)l dr(r) (25) where A(*) is given by (2). Applying formula (25) for the total fields 711, ~'2 in the region VR, exterior to the surface of the scatterer S and interior to the sphere SR, R > R o, we obtain {~gl, ~2}s, = {~1, ~2}s + j" [~1" A(*) ~g2 - ~2" A(*) ~1] dr(r) (26) VR which, in view of the fact that both ~ and ~2 satisfy the time independent Navier equation (1) in the region VR, implies that {~1, IIJz}sR = {~Ill, ~2}S" (27) Relation (27) holds true for every R > R 0. Therefore, it is enough to show that { ~ , ~2}s = 0 for each one of the three scattering problems. For the first, or the second problem, where the displacement or the traction is vanishing on the surface, the right hand side of (27) is obviously equal to zero since either ~1 = ~2 = 0, or T ~ 1 = T ~ 2 = 0 on S. For the third scattering problem, where both the displacement and the traction are assumed to be continuous across S, we conclude that {~1+, ~2+}s = ~ (It1+ 9T + ~2+ -- ~Y2+. r + ~1+) as S = S(~-. S T - 'F~- - ~ ' 2 - r-~i-)ds 862 G. Dassios, K. Kiriaki and D. Polyzos ZAMP = ~ (~Z" A(*)- ~ 2 - ~ f " A(*)- ~1-) ds V- (28) ~gf" ~ 1 ) ds = O. = -- 0)2 ~ (~1-" ~P2 - V- Hence, the proof of the Lemma is completed. Lemma 2. Under the hypotheses of Lemma 1 {r ~ } ~ = {r where r (29) ~1}~ u~ stand for the incident and the scattered wave, respectively. Proof. In view of (22) and the bilinearity of the form (23), Relation (24) reads {r r + {r u2)s + (ul, r + (ul, u~)~ = 0. (30) Since the only singularity of the incident plane wave is located at infinity it follows that {r r = I [r VRuV = __ (i)2 A(*) r ~ VR [r -- r r -- r A(*) r dv r dv = O. (31) k.) V - On the other hand, the scattered waves u 1 and u 2 expand as spherical outgoing waves in the radiation zone r >> 1, whose isolated singularity is localed at r = 0. As a consequence, Betti's formula (25) can be used to transform the integral { U l , U2}S over the surface of the scatterer S, to the integral {ul, U2}So ~ o v e r the surface of the sphere S~ centered at the origin with radius R >> 1. On the sphere So~ the displacement fields u 1, u 2 assume the asymptotic form (12) while the corresponding surface traction fields assume the form [6] Tu(r) = ikpc 2 g~(~, k) h(kpr) +ik~c~go(,,~)h(k,r)O+iksc2go(F,k)h(ksr)O+O( 1)~ . (32) Therefore, gr(F, fq) g,(F, k2) + i0)csh(k~r) go(~, kl) go(F, k2) + i0)c~h(k~r) go(~, ka) go(~, k2) U a 9 Tu 2 = i0)cph(kpr) (33) and the commutativity of (33) with respect to the directions of propagation kl,/~2 implies that {ul, U2}s = {ul, U2}s= = O. (34) Vol. 38, 1987 On the scattering amplitudes for elastic waves 863 Consequently, relation (30) becomes {~1, U2}s = - {ul, ~ 2 } s = { ~ 2 , ul}s, (35) which completes the proof of the Lemma. Our next objective is to find a relation between the above surface integrals { ", " }s and the scattering amplitudes g,, go, g~o. The following L e m m a provides the appropriate connection formulae. L e m m a 3. Under the hypotheses of L e m m a 1 the normalized spherical scattering amplitudes g,, go, g~o obtain the following representations over the surface S of the scatterer ik3 (u(r'), e - ' k , ~ / : } s , gr(f, k) = 4rcco2 (36) ik3 {u(r'), e-'k'r go(f, k) - 4rcco2 (37) go(f, k) - 4rcco2 {u(v'), e-iks~r'~} s. (38) Proof. The scattered field u has the integral representation [6] = 1 f[.(/) ,_ 9Tr r(r, v') _ . , , lo(r, r') Tr u(r )1 ds(r'), (39) where the fundamental tensor is written as the sum of a longitudinal and a transverse part if(r, r') = 10P(r, r') +/~(v, r'), (40) while the longitudinal part is given by eikpR loP(r, r') - c02R 3 [kER | R + (1 - ikpR) (I-- 3/~ | (4I) and the transverse part by lob(r, r') R = r - eiksR co2 R3 [ks2 R | R + (1 - i k s R) ( / ' - 3/~ | r', R = Iv - r'l, g = R/~. + k~a )d2k`R ~/', (42) (43) In the radiation zone we obtain lop(r,r,)=,| ik, h(kpr) e- ik,,./ + 0 ( i~) CT Cp , lob(v, r') = ( / ' - / : @ ~) 75- h(k~r) e -ik~" + 0 ~ Cs (44) . (45) 864 G. Dassios, K. Kiriaki and D. Polyzos ZAMP The uniformity of the expansions (44, 45) with respect to the variable ~, as well as the easily checked vector identity T~,[/(r') a | a] = [Tr, f(r') a] | a, (46) which holds for every f ~ C ~1) and any constant vector a, implies the asymptotic expansions T~,_PP(r, r') = ~ - h(kpr) [Tr,(e-i~,~ r'r3] | F + O ~ (47) Cp iks T~,/~S(r, r') = ZS- h(ksr) [T~,(e-ik"e'r'O~)] | Cs + ~ - h(ksr) [T~,(e-ik~e/~)] | ~ + 0 ~ Cs . (48) By means of (44), (45), (47) and (48) the representation (39) yields ikp u(r) - 4rcc 2 h(kpr) {u(r'), e-ik, e" r ~ + ~ik~ h(k~r) {u(r'), ik~ e_ik~e.,~} s (1) + 4rCC~ h(k~r) {u(r'), ~ + O ~5 9 e-ik~r O"}S (49) The representations (36)-(38) come out of direct comparison of (49) with (12) along the linearly independent directions described by ~, g, and ~b. This completes the proof of Lemma 3. Looking at the Formulae (36)-(38), more carefully, we see that the phase factors e ik,~.~'~., e-ik"r and e-ik"r'r'~ can be interpreted locally, as incident plane waves, propagating in the direction - ~ and polarized along ~ (longitudinal wave), ~ and ~ (transverse waves), respectively. Then (36)-(38) provide the spherical scattering amplitudes as a measure of the interaction between the scattered wave u and the incident plane wave, with the corresponding polarization on the surface of the scatterer. In other words, it is a longitudinal plane wave polarized along ~, that will reveal the radial scattering amplitude 9~, while the angular scattering amplitudes 9o, g~ will come out of a transverse plane wave polarized along the corresponding direction /7 or 4, as the case may be. Furthermore, in the expressions (36)-(38) for the scattering amplitudes, the direction of observation ~ appears in the phase of the exponential, while the direction of incidence k enters the corresponding formula through the incidence of the plane wave that is responsible for the excitation of the scattered wave u. We are now in a position to state the main "Reciprocity Theorem" for the normalized spherical scattering amplitudes in the theory of elastic scattering. Vol. 38, 1987 On the scattering amplitudes for elastic waves 865 Theorem 1. Under the hypotheses of Lemma I the following reciprocity relations hold true; (i) If both ~1 and ~2 are of the longitudinal type, then g,(~, k) = g , ( - k, - ~). (50) (ii) If both ~1 and t~ 2 are of the transverse type, then (i,1.03 go( , k) + (b:L.0) k) = (b2" 03 go(- k, - F) + (b2" O) g ~ ( - k, - F). (51) (iii) If ~1 is a longitudinal and ~2 a transverse wave, then -g~(?,k)=(kP']3[([~2.0")go(-k,-~)+(~z.O)g~o(-fr \k=) (52) Proof. (i) In view of Eqs. (6) and (36), Relation (29) implies g , ( - ~h, k2) = g , ( - k2, ~1) (53) and if we choose kl = - r and kz = k, Eq. (53) takes the form g,(s ]~) = g , ( - k, -- ~). (54) (ii) For the transverse wave (7), propagating in the direction F, the polarization vector b ties on a plane perpendicular to the direction ~ and therefore it assumes the decomposition = b . t = b. (~ | ~ + g | 0 +,~ | 0) = (b" 03 O+ (~. 0) <k. (55) Consequently, Relation (29) for transverse incidence is written as (~1" O') {Oeik'~zt'r,~12} 1 + (bl" 0) {Oe ik&', u2)s = (b2" 0") {Oeik&', ul} s + (b2" O) {Oe ikA~', ul}s (56) which in view of (50), (51) implies the relation ([Jl" 03 go(- kl, k2) + (bl" q3) ge( - kl, k2) = (b2" 0") go(-/%, kx) + (b2" ~) g o ( - k2, kl). (57) Formula (51) comes out of (57) via the substitution - k 1 = ~, k 2 = k. (iii) Substituting the incident waves (6), (7), into (29) we obtain {kl e~kd''', U2}s = (/~2" 03 {Oeik=r'=',ul} s + (~2" q3) {Oe ik&'', ul} s (58) which in view of (36)-(38) becomes -- g , ( - i l , k2) = (kP~ = [(b z 90") go(- k2, kl) + (~2" ~b) g o ( - k2, il)I. kks) (59) 866 G. Dassios, K. Kiriaki and D. Polyzos ZAMP The reciprocity relation (52) follows from (59) whenever - kl = r and k 2 = k, and this completes the proof of the theorem. Remark. The case where 9 1 is a transverse and 0 2 a longitudinal wave corresponds to a formula like (59) with the indices I and 2 interchanged. Introducing the tangential vector scattering amplitude o,(~, k) = g0(~, k) 0 + g~(~, k) 0 (60) as in [6] the reciprocity relations (51) and (52) can be written as b l u,(~, k) = ~2" u , ( - k, - ~) (61) - gr(?, k) = (kv~ a be" g , ( - (62) and \k=) ~, - r) respectively. By choosing the polarization vectors bt and b2 along ~ and q3, as the case may be, we obtain the more specific reciprocity relations given by the following. Corollary 1. (i) For O1, O2 polarized along go( ~, k) = go( - ~, - r). (63) (ii) For 9 1, O2 polarized along 0 g~o(~, k) = g g ( - k, - ~). (64) (iii) For 9 1 polarized along ~ and 9 2 along go( ~, k) = g g ( - k, - P). (65) (iv) For 9 1 polarized along 0 and 9 z along g~(~, k) = g o ( - k, - ~). (66) (v) For 9 1 longitudinal and 9 2 polarized along 0, or 0 \kU or kk=) The radial-radial, angular-angular and radial-angular connection of the normalized spherical scattering amplitudes g,, go, g~ is characterized by the corresponding polarization vectors of the incident waves. Vol. 38, 1987 867 On the scattering amplitudes for elastic waves The next theorem provides the radial scattering amplitude for two directions of incidence, in terms of an integral over directions of all three scattering amplitudes. We will refer to it as the "Radial Scattering Theorem". Theorem 2. Under the hypotheses of Lemma 1 and the assumption that both ~ and ~2 are longitudinal waves, the following formula g,(k~, k2) + g* (k2, kl) = - k3I(k~, k~; k2, k2)*), (68) where 1 I(kl, kl; k2, k2) = ~ lel~ [kp-3g*(F, k l ) gr(F, k2) # ^ + k 2 3 g*(F, k~) go(F, k2) + k~-3 ge(r, k~) g~o(F,k2)] dO(F) (69) holds true for all radial directions of incidence k~ and ka, while the asterisk indicates complex conjugation. Proof. For ~, i = 1, 2 as given by (22) and the bilinearity of the form { ", " }s, we derive the relation {~1", ~2}s = { ~ , ~2}s + { ~ , U2}s + {u~, ~2}s + {u~', u2} s. (70) Following a series of arguments similar to those that lead to {~ul, ~2}s = 0 in the proof of Lemma 1, as well as to (31) we verify that {~*, ~e}s -- { ~ , O2}s -- 0, (71) for every boundary condition that corresponds to the first, second and third scattering problem. The regularity of ul and u2 in the region exterior to S and interior to the sphere So of radius r >> 1 implies that {u*, u2} s = {u~, U2)s~ . (72) By means of the asymptotic forms (12) for the displacement and (32) for the surface traction we easily conclude, via (72), that {u'~, u2} s = 2i ~ [kvc~g*(F, kl) Or(F, k2) Ih(kvr)l z So~ + ksc~g*(F, kO go(F, kz) Ih(k~r)l 2 ^ + ksc~2 go:g (r, kl) go(F, k2)Ih(k~r)l 2] dO(F) = 2c~ J" [ k ; 3 g * ( F, k~) g~(r, k2) 1~1= 1 + go(F, = 4zoo2 i I ( ~ h , k t ; k2, k2) + dO(F) (73) *) In the notation I(ftl, bl ; ftz, b2), ai represents the propagation vector of ~i and bi its polarization. 868 G. Dassios, K. Kiriaki and D. Polyzos ZAMP because of (6). Furthermore, Formula (49) yields {#*, U2}s - 4roe)2 - - ik} g~(kl, k2) 4no 2 {u*, ~e}~ - -- 0r*(k~, ~1). (74) (75) Inserting (71), (73), (74) and (75) into (70), Formula (69) is derived and the Radial Scattering Theorem is proved. The corresponding "Angular Scattering Theorem" states as follows. Theorem 3. Under the hypothesis ofLemma I and the assumption that both ~ and ~2 are transverse waves the following formula (b, 90") go(kx, k2) -~ (~1" ~) gq~(kl, k2) ~t_(~2" 03 g~(k2, kl) "q'-(~2" ~) gff.( # ~2' kl) = -- k3 I(k~, b~; k2, b2) (76) holds true for all radial directions of incidence k, and k2 and all polarization vectors bl and be such that kl" ~1 = k2" b2 = 0. Proof. As in the proof of Theorem 2, Eq. (79) yields { ~ , U2}s + {u*, ~2}s + {u~, u2}s = 0, (77) where the last term in (77) is again expressed as in (73). The first two terms in (77) take the form 4nco 2 {~'I', U2}s = ik~ bl" [0g0(kl, k2) -t- q3 g~o(k1, k2)], 4redo2 {u*, ~ 2 } s - ik~ ~. [~a*(k~,~) + q,g~(k~, k0]. (78) (79) Formula (76) is confirmed through substitution of (73), (78) and (79) into (77). Finally, the "Radial-Angular Scattering Theorem" receives the subsequent form. Theorem 4. Under the hypotheses of Lemma 1 and the assumption that ~1 is a longitudinal wave propagating along the radial direction kl and ~2 is a transverse wave polarized along ~ and propagating along the radial direction k2, we obtain k;3or(kl, k2) + k;3([, 903 a~(k2, kO + k2a(~ 9O) ao* (k2, - kl) = - I (~, ~;~, T,). (80) Vol. 38, 1987 On the scattering amplitudesfor elastic waves 869 Proof. The proof follows immediately from (77) if we make use of (73), (74) and (79) for b2 = ~. Remark. The case where ~1 is a transverse wave polarized along b and II)2 is a longitudinal wave, gives the formula k23go(f'~, f'2) + k 2 3 0 ~ ( ~ , k2) + k ; 3 * - = - (81) The following Corollary is a simple consequence of Theorems 2, 3, and 4. Corollary 2. (i) If q~l and @2 are both polarized along 0 then g0(r,1, k2) + ~;'(k2, f,1) = - k~ I ( ~ , g; r,~, 03. (82) (ii) If @, is polarized along 0 and @2 is polarized along 4, or vice versa, then ^ k,) = - k,3 I(~1 ' #; k2, ~), o0(k,, k;) + O~, (k2, (83) gq>(kl, k2) -~- 9"(k2, ks) = - k~ I (kx, 4; k2, 0~), (84) or (iii) If t/i~ and {I}2 are both polarized along ~b then g~o(k~, k2) + g* (k2, kl) = -- ks3I(kl, r k2, (~), (85) (iv) If ~ , is a longitudinal wave and ~2 a transverse wave polarized along g, then kp3g,.(kl, k2) + k~-3g~(k2, kl) = - [(kl, kl; k2, 0"). (86) (v) If ~ , is a longitudinal wave and ~2 a transverse wave polarized along 4, then #.. - ka) = - I(k~, k~; k2, 4). k;3g,.(k,, k2) + k,-3 ge(k2, (87) Corollary 2 establishes the existence of particularly simple connection formulae between the normalized spherical scattering amplitudes whenever the polarization vectors are choosen appropriately. Remark. When the scatterer has inversion symmetry, i.e. when g~, go, and g~ are invariant under an interchange of the direction of observation ~' and the direction of incidence k, then the radial scattering theorem (68) reads Reg,(k,, k2) - k~ I(~1 k l ; k2, k2) 2 ' (88) Similarly, the special forms of the angular scattering theorem (82) and (85) assume the simpler forms Reg0(k~, k2) = k~ I(k~, O; k2, 0) 2 (89) 870 G. Dassios, K. Kiriaki and D. Polyzos ZAMP and Reg~o(kl, k2) - k~ 2 1(~1, ~; ~2, ~) (90) 4. The scattering cross-section The interaction of the incident wave with the discontinuity of the medium of propagation, or, in other words, the effect of the scatterer on the propagation of the incident wave, is measured by the amount of energy that the scatterer receives from the incident wave and reradiates in all directions. A normalized form of this energy is called scattering cross-section and it is defined to be the ratio of the t i m e average at which energy is scattered by the body, to the corresponding time average rate at which the energy of the incident wave crosses a unit area normal to the direction of incidence [3]. Because of the particular normalization of the scattering cross-section, as total energy m e a s u r e d ~ - u n i t s of energy per unit area, it has the dimensions of area and this fact justifies the characterization "cross-section". It actually indicates, the area of a plane surface perpendicular to the direction of incidence that "cuts" out of the incident wave an amount of energy equal to the energy "scattered" by the scatterer. In [6] it is proved that the scattering cross-sections o-p and as for longitudinal and transverse incidence are given by a p = 2 n k p I ( k , k; k, k) (91) a ~ = 2 n k ~ I ( k , [~; k, [~) (92) and respectively, where the integral expression I is given by (78). Formulae (91), (92) state that the scattering cross-sections are expressed via the integral (69) when the two incident waves ~1, ~2 are assumed to be identical. Using this observation we can easily prove the following. Corollary 3. (i) For longitudinal incidence the scattering cross-section is given by o"~ - 4n ~ Regr(k, k). (93) (ii) For transverse incidence, polarized along ~, the scattering cross-section is given by o-~ - 4n [(b. 0") Rego(k, k) + (b. ~) Reg,0(k, k)]. (94) Vol. 38, 1987 On the scattering amplitudes for elastic waves 871 (iii) In particular, for transverse incidence, polarized along ~, or 4, the scattering cross-section takes the simpler form o-s - 47c Reg0(k, k), (95) Reg~o(k,/~), (96) or o"s - 47~ respectively. The expressions (93) and (94) translated in their notation, have been proved independently by Barratt and Collins [3], by making use of Jones' Lemma for the asymptotic evaluation of double integrals. Formulae (93)-(96) form the elastic versions of the Optical Theorem for electromagnetic waves. They provide a quantitative link between the scattering amplitudes in the forward direction, as an interference pattern of the incident and scattered wave, and the overall effect of the disturbance caused by the scatterer. To a certain extent, they explain the mechanism of energy transferred during the scattering process. 5. Acoustic scattering In acoustic scattering [4, 5], the basic field is either the excess pressure field measuring the instantaneous deviation of the local pressure from its reference value, or the velocity potential. In both cases, the basic field is a scalar field satisfying the wave equation with a constant phase velocity c and a wave number k. The theory of scattering for acoustic waves, therefore, corresponds to the case of elastic scattering whenever the medium cannot substain transverse waves. The asymptotic form of the scattered acoustic wave takes the form [14, 15]: u(r) = g(f, k") h(kr) + O ~ (97) and it can be obtained from the corresponding elastic case (12) by restricting attention only to the longitudinal part along ~. The basic reciprocity theorem comes out of (50) in the known form [14, 15] g(~, f,) = g( - ~, - ~). (98) The radical scattering theorem (68) for the case of acoustics reduces to the classical scattering theorem [14, 15] g(k~, ~2) + g*(k2, kl) - 1 2• S g*(f, ks) g(f, k2) df2(~) Ir = 1 (99) 872 G. Dassios, K. Kiriaki and D. Polyzos ZAMP which is written as Reg(kl, k2) -- 1 le.~= 1 g*(g, k~) g(g, k2) d~2(~) 4n (loo) for scatterers with inversion symmetry. Finally the scattering cross-section is connected to the forward scattering amplitude via (93) which in our case is written as o- - 4~ k2 Reg(k, k). (101) Consequently, all the known relations [4, 5, 14, 15] referring to the normalized scattering amplitudes and scattering cross-section for acoustic waves are recovered from the above elastic counterparts. References [1] J.F. Ahner and G. C. Hsiao, A Neumann series representation for solutions to boundary-value problems in dynamic elasticity, Quart. Appl. Maths. 33, 73 (1975). [2] J.F. Ahner and G. C. Hsiao, On the two-dimensional exterior boundary-value problems of elasticity, SIAM J. App. Math. 31, 677 (1976). [3] P. Barratt and W. Collins, The scattering cross-section of an obstacle in an elastic solid for plane harmonic waves, Proc. Camb. Phil. Soc. 61, 969 (1965). [4] G. Dassios, Convergent low-frequency expansions for penetrable scatterers, J. Math. Phys. 18, 126 (1977). [5] G. Dassios, Low-frequency scattering theory for a penetrable body with an impenetrable core, SIAM J. Appl. Math. 42, 272 (1982). [6] G. Dassios and K. Kiriaki, The low-frequency theory of elastic wave scattering, Quart. Appl. Maths. 42, 225 (1984). [7] G. Dassios and K. Kiriaki, The rigid ellipsoid in the presence of a low-frequency elastic wave, Quart. Appl. Maths. 43, 435 (1986). [8] G. Dassios and K. Kiriaki, The ellipsoidal cavity in the presence of a low-frequency elastic wave, Quart. Appl. Maths. 44, 709 (1987). [9] N. Einspruch, E. Witterholt and R. Truell, Scattering of a plane transverse wave by a spher&al obstacle in an elastic medium, J. Appl. Phys. 31, 806 (1960). [10] D.S. Jones, Low-frequency scattering in elasticity, Q. J. Mech. Appl. Math. 34, 431, (1981). [11] K. Kiriaki, Low-frequency expansions for a penetrable ellipsoidal scatterer in an elastic medium (to appear). [12] V. Kupradze, Three Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North Holland, Amsterdam 1979. [13] R. Truell and C. Elbaum, Handbuch der Physik, Vol. 12 (2) Acoustics II, Springer Verlag, Berlin 1962. [14] V. Twersky, Certain transmission and reflection theorems, J. Appl. Physics 25, 859 (1954). [15] V. Twersky, Multiple scattering by arbitrary configuration in three dimensions, J. Math. Phys. 3, 83 (1962). [16] V. Twersky, Classroom notes in scattering theory, University of Illinois at Chicago (1973-74). [17] C. Ying and R. Truell, Scattering of a plane longitudinal wave by a spherical obstacle in an isotropically elastic solid, J. Appl. Phys. 27, 1086 (1956). [18] L. Wheeler and E. Sternberg, Some theorems in classical elastodynamics, Arch. Rat. Mech. Anal. 31, 51 (1968). Vol. 38, 1987 On the scattering amplitudes for elastic waves 873 Abstract Reciprocity and scattering theorems for the normalized spherical scattering amplitude for elastic waves are obtained for the case of a rigid scatterer, a cavity and a penetrable scattering region. Depending on the polarization of the two incident waves reciprocity relations of the radial-radial, radial-angular, and angular-angular type are established. Radial and angular scattering theorems, expressing the corresponding scattering amplitudes via integrals of the amplitudes over all directions of observation, as well as their special forms for scatterers with inversion symmetry are also provided. As a consequence of the stated scattering theorems the scattering cross-section for either a longitudinal, or a transverse incident wave is expressed through the forward value of the radial, or the angular amplitude, correspondingly. All the known relative theorems for acoustic scattering are trivially recovered from their elastic counterparts. (Received: July 14, 1986; revised: February 5, 1987)