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Uniqueness in the inverse scattering problem in a piecewise homogeneous medium

The scattering of time-harmonic acoustic plane waves by an impenetrable obstacle in a piecewise homogeneous medium is considered. Having established the well posedness of the direct problem by the variational method, we prove a uniqueness result for the inverse problem, that is, the unique determination of the obstacle and its boundary condition from a knowledge of the far-field pattern for incident plane waves. The proof is based on a generalization of the mixed reciprocity relation.