AN EIGENFUNCTION EXPANSION OF THE ELASTIC WAVE GREENS-FUNCTION FOR ANISOTROPIC MEDIA

We consider the propagation of waves in a three-dimensional, homogeneous, general anisotropic elastic medium of infinite extent. A special representation of the corresponding Green's function is derived. This representation expresses the Green's function as a bilinear expansion in the eigenfunctions of the linear elastic Hookean operator. The expansion follows from a completeness property which these eigenfunctions possess. Since the eigenfunctions of this operator correspond physically to freely-propagating plane waves, the eigenfunction-expansion representation affords a very natural physical interpretation of the Green's function as a modal and angular superposition of these freely-propagating modes. From this representation two additional results are derived: (1) A generalized completeness relation involving the freely-propagating modes which is shown to generate functional transform pairs which use the wave modes as the underlying basis set, and (2) A far-field asymptotic representation of the Green's function as a superposition of products of the freely-propagating modes with geometrically decaying spherical waves. The Green's-function representation and generalized completeness relation are valid for elastic media of general anisotropy. The asymptotic representation is here derived within the constraint that the elastic media be isotropic, for which a simple integration-by-parts procedure provides the full expansion.