On the scattering of two-dimensional elastic point sources and related near-field inverse problems for small discs

Abstract

The problem of scattering of a point-generated elastic dyadic field by a bounded obstacle or a penetrable body in two dimensions is considered. The direct scattering problem for each case is formulated in a dyadic form. For two point sources, dyadic far-field pattern generators are defined and general scattering theorems and mixed scattering relations are presented. The direct scattering problem for a rigid circular disc is considered, and the exact Green function and the elastic far-field patterns of the radiating solution in the form of infinite series are obtained. Under the low-frequency assumption, approximations for the longitudinal and transverse far-field patterns of the scattered field are obtained, in addition to an asymptotic expansion for the corresponding scattering cross-section. A simple inversion scheme that locates the radius and the position of a rigid circular disc, which is based on a closed-form approximation of the scattered field at the location of the incident point source, is proposed.

Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

“…In this section we address the inverse scattering problem concerning partially coated obstacles which are bounded and simply connected domains in R 2 . Our aim is to recover the shape of the partially coated scatterer B from the knowledge of the near field data u sct (r), r ∈ C due to the point source u inc r 0 (r) ≡ Γ(•, r 0 ), r 0 ∈ C (see (1)). The latter is mathematically modelled by the following interior mixed boundary value problem:…”

“…Broadly speaking, classic scattering theory is concerned with the effect a bounded obstacle has on a time-harmonic incident wave, which is known as a scattering problem. A lot of scientific work has been done for direct scattering problems as well as for the inverse ones [1][2][3][4][5][6]. The first problems have the following interpretation: If the total field is the superposition of an incident field u inc and the scattered field u sct , then the direct scattering problem is to determine the scattered field from the knowledge of the incident wave field and the governing differential equation of the wave motion.…”