Generalized debye series for acoustic scattering from objects of separable geometric shape

Summary In a classical paper of 1908, Debye has resolved the electromagnetic field scattered by a dielectric cylinder into a series of waves multiply internally reflected in the cylinder. For acoustic scattering by elastic cylinders, a corresponding series was derived from the conventional solution (obtained by satisfying the overall or global continuity conditions) by Brill and Überall, taking into account mode conversions of longitudinal (L) into transverse (T, shear) waves, or vice versa, upon internal scattering in a some-what involved fashion. In a series of papers, Gérard has shown that this approach could be greatly simplified by introducing local reflection and transmission coefficients at each interface, which is suitable for generalizing the Debye series to the case of elastic waves coupled by the continuity conditions at the external and each of any possible (multiple) internal interfaces of the scattering object. The approach is then applicable to all elastic objects for which surface and interfaces form coordinate surfaces of any separable geometry; the corresponding derivation is given here in the most general fashion, and is concretely illustrated by the examples of an elastic plate, infinite cylinder and sphere.