An approximate method for scattering in elastodynamics- the Born approximation II: SH-waves in infinite and half-space L. L. CHU,* A. ASKARt and A. S. CAKMAK Prmceto n Unwerstty, Department of Civtl Engmeermg, Princeton, NJ 08544 The general formulation presented earher is apphed to the scattering of SH-waves by cawnes and nDd mclusmns As an assessment of the range of vah&ty of the Born approxnnanon, the cases with a circular geometry are studmd for infinite and half spaces Both near and far field results are obtained The results m&cate that the approxtmatmn is good for up to ka -~ 1 (k = wave number, a = radms of the circle) for the details of the scattering. Excellent agreement Is observed however for almost the enttre range of ka for the global morion of a rigid inclusion 1 INTRODUCTION In this expression for g-~ g(x, x') and In a previous paper, 1 a general scheme for the scattering of k = coX/p/l~ elastm waves by soft (Dnchlet condmon) and hard (Neu- (5) mann condmon) inclusions ts presented. As an appllcanon r = ((X1--X'I) 2-t- (X2- X;)2) 1/2 of this general formulanon and an assessment of the range of vah&ty of the Born approxmaatxon, m this paper the where p is the densaty of the embedding material and k is scattering of anti-plane SH-waves xs stu&ed Antlplane the wave number of the incident waves, Ho O) is the zeroeth- waves are defined as waves where dxsplacements ua = u2 = 0 order Hankel function of the first kind as a result of taking and u3 = u(xl, x2) Of the six distract components of the the harmonic time dependence of u as exp (- toot). 2-3 stress tensor t, only txa and t23 remain. Strnllarly, only the The plan of this paper is as follows in Secnon 2, the G33 = G component of the mamx Green's function exists integral equations and the Born approxtmatlon for ann- and the nonzero components of H, the Green's stress plane waves are obtained from the general formulation, lm tensor, are. Secnon 3, the far field solunon and the scattering cross sectmn are obtained, m Sections 4 and 5, exphclt results are aG obtained for the case of a mrcular mhomogenelty embedded H331 = Hm = P -- 0x'1 respectwely in mfimte and sem~-mfimte spaces Th~se latter (1) problems can be solved exactly by separauon of vanables 0G Finally, a &scusslon of the companson of the results of the H332 = H23a = ~ - - exact and approximate solunons is presented All the above 0x~ calculanons are carried for the case of a cavity, fixed and where p is the shear modulus of the embedding materml movable ngld inclusions Slmdarly, for n being the normal to the boundary Here the term cavny denotes tunnels and plpehnes while ngM inclusion stands fm ngtd foundahons and OG bmled slluctures P tip H33p = Id an' ( 2 ) Furthermore, tot u and tH being independent of x3, lhe 2 NEAR FIELD SOLUTION mtegraUon over this coordinate can be perfomred In wew of the above dlscussmn equaUons (9) and (1 1) of ~ Reference 1 become 3g u, ds'-- g(kr-') t,s7 ' ds' GydA'= ( Gdx~ ]ds' (3) G O ~ O O Above ds' is the hne element along the boundary and z-3 (61 t u(x)=ut(x)4- p-- u ds'-- gt nds' g f Gdx3 -~pH(o')(kr On' = = -) {4} G 0" -~ Above, uS'=-=uS(x'), the superscripts s and l represent the scattered and mcMent field respectwely, while the total * Woodward-Clyde Consultants, 3 Embarcadero Center, San I'ran- field has no superscript osco, Cahforma, U3A ~" On leave from Bogazlcl Unwers~tesL Mathemahcs Departmenl, For points on the boundary or, singularities appear as Bebek. lstanbul, Turkey in the general case Hence, equation (13) of Reference 1 0261-7277/82]03102-15 $2 00 102 Sod Dynamws and Earthquake Engmeenng, 1982, Iiol 1, No 3 o 1982 CMI. Pubhcatlons