Elastic waves in discontinuous media: Three-dimensional scattering

Abstract

This report contains an exact study of elastic wave propagation and its scattering in discontinuous media where hard reflectors are onionlike sets of surfaces. In order to reformulate the problem as a finite set of boundary integral equations, the wave motion between reflectors is represented by means of elastic potentials which involve vectorial densities on the surfaces. In the external medium, an outgoing asymptotic condition generalizes the Silver–Müller (and the Sommerfeld) condition to the case of coupled waves (S and P waves) moving with different velocities. The uniqueness of the Green’s function, which guarantees the uniqueness of the direct problem solution, is proven. For any incident wave and arbitrary number of surfaces, the transmission and scattering problems are studied, with and without the simplification obtained by assuming constant Poisson ratios. According to the parameter ranges, the equations which are obtained are well posed, either as second kind Fredholm equations, or because they reduce to the inverse of the sum of the identity operator and a ‘‘small norm’’ bounded operator. The results can be used to describe rigorously the three-dimensional scattering of elastic waves in the frequency domain for any kind of incident wave function (P,S,...) as well as the response to a localized source.