The low-frequency scattering theory for a penetrable scatterer with an impenetrable core in an elastic medium

Abstract

We consider an incident time harmonic longitudinal or transverse plane wave which propagates in an infinite isotropic and homogeneous elastic medium and it is scattered by a penetrable body containing an impenetrable core. Integral representations for the total displacement field as well as the normalized scattering amplitudes are provided. A systematic procedure for finding the low-frequency expansions for the displacement field, the scattering amplitudes and the scattering cross section is developed. As a result the scattering problem is reduced to a sequence of elastostatic problems which can be solved using the Papkovich-Grodski-Neuber potentials. The leading low-frequency terms of the scattering amplitudes and the cross section are evaluated. The proposed procedure is applied to the scattering of a plane elastic wave by a penetrable spherical scatterer with a concentric spherical rigid inclusion. Particular values of the physical parameters correspond to special scattering problems and they are included as such.