Eigenvector expansions of Green's dyads with applications to geophysical theory

Abstract

The treatment of boundary value problems in various vector-separable regimes is unified and facilitated with the use of the eigenvector expansion of their corresponding Green's tensors. In particular, the method is useful for the vector Laplace, Poisson, Helmholtz and Navier equations in spherical and cylindrical polars. Several examples are given; among them, an evaluation of the dynamic and static response of an elastic sphere to shear dislocations requires particular mention. It is found that the static displacement field in a sphere which is associated with the Legendre polynomial of the first degree (l= 1) poses some problems. In such a case, one must incorporate additional conditions, namely, that the angular momentum of the sphere about its centre is zero and that the centre of mass of the sphere is not displaced. It is recommended that eigenvector expansions be adopted in geophysical theory. Its inherent elegance and compactness make it an excellent tool for the construction of theoretical Earth-models.