The fluid-solid interaction scattering problem with unknown buried objects

Consider the direct and inverse scattering problem of time harmonic acoustic waves by a two-dimensional elastic obstacle which contains an unknown impenetrable object inside. We apply the boundary integral equation method to solve the direct scattering problem. Since the obtained boundary integral system is a mixed form of scalar and vector equations, we consider the existence of the solution in the sense of a weak formulation. The factorization method is employed to solve the corresponding inverse scattering problem, that is, we try to recover the shape of the interaction surface from the knowledge of acoustic far-field patterns. With the help of a special data to pattern operator G, we can reconstruct the shape of the interaction surface by using the classical factorization method. Furthermore, a new mixed reciprocity relation is proposed to show the relation between the interior elastic field and unitary matrix S.