Elastodynamic Completeness Relations for Scattered Wavefields

Abstract

Completeness relations that involve wavefields that have scattered from an inhomogeneity of compact spatial support are derived. The scatterer itself can be fully anisotropic, and is contained in an otherwise uniform three-dimensional isotropic elastic medium of infinite extent. Roughly, these relations are representations of the three-dimensional spatial Dirac delta function as a product of wavefields that are summed over all states. As such, the relations serve as the generators of transform pairs that use the wavefields as their underlying set of basis functions. Furthermore, by exploiting the causal nature of the scattering system, there are shown to be an infinite number of basis sets involving the wavefields corresponding to a given scatterer.