In this paper, we consider the scattering of a time-dependent electromagnetic wave by an elastic body immersed in the lower half-space of a two-layered background medium which is separated by an unbounded rough surface. By proposing two exact transparent boundary conditions (TBCs) on the artificial planes, we reformulate the unbounded scattering problem into an equivalent initial-boundary value problem in a strip domain with the well-posedness and stability proved using the Laplace transform, variational method and energy method. A perfectly matched layer (PML) is then introduced to truncate the interaction problem with two finite layers containing the elastic body, leading to a PML problem in a finite strip domain. We further verify the existence, uniqueness and stability estimate of solution for the PML problem. Finally, we establish the exponential convergence in terms of the thickness and parameters of the PML layers via an error estimate on the electric-to-magnetic (EtM) capacity operators between the original problem and the PML problem.

This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem.