The Far-Field Equations in Linear Elasticity — an Inversion Scheme

TLDR: In this paper, a pair of integral equations of the first kind which hold independently of the boundary conditions are constructed in the far-field region and the support of the body is found by noting that the solutions of the integral equations are not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interioir points.

Abstract: In this paper the far-field equations in linear elasticity for the rigid body and the cavity are considered. The direct scattering problem is formulated as a dyadic one. This imbedding of the vector problem for the displacement field into a dyadic field is enforced by the dyadic nature of the free space Green's function. Assuming that the incident field is produced by a superposition of plane dyadic incident waves it is proved that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed in the far-field region. The properties of the Herglotz functions are used to derive solvability conditions and to build approximate far-field equations. Having this theoretical framework, approximate far-field equations are derived for a specific incidence which generates as far-field patterns simple known functions. An inversion scheme is proposed based on the unboundedness for the solutions of these approximate “far-field equations” and the support of the body is found by noting that the solutions of the integral equations are not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interioir points. It is also pointed that it is sufficient to recover the support of the body if only one approximate “far-field equation” is used. The case of the rigid sphere is considered to illuminate the unboundedness property on the boundary.