Point-Source Elastic Scattering by a Nested Piecewise Homogeneous Obstacle in an Elastic Environment

Abstract

A nested piecewise homogeneous elastic scatterer is embedded in a homogeneous elastic environment. The scatterer’s core may be rigid, cavity, Robin, or lossy penetrable. A 2D or 3D incident elastic field, generated by a point-source located in the homogeneous environment, impinges on the scatterer. The scattering problem is formulated in a dyadic form. The main purpose of this paper is to establish scattering relations for the elastic point-source excitation of a nested piecewise homogeneous scatterer. To this direction, we establish reciprocity principles and general scattering theorems relating the scattered fields with the corresponding far-field patterns. Furthermore, for a scatterer excited by a point-source and a plane wave, mixed scattering relations are derived. The optical theorem, relating the scattering cross-section with the field at the point-source’s location a is recovered as a corollary of the general scattering theorem. We present a detailed investigation for the 2D case and summarize the results for the 3D case, pointing out the main differences in the analysis.