Scattering Relations for Interior Acoustic Excitation of a Layered Obstacle with a Soft Or Hard Core

A layered scatterer with a soft or hard core is excited by a time harmonic spherical acoustic wave, generated by a point source located in the interior of the scatterer. Reciprocity and general scattering relations are established, relating the total, primary and secondary fields with the corresponding far field patterns. Moreover, we formulate the optical theorem, relating the scattering cross section with the field in the layer of the source