A group-theoretical formulation of geophysical elastodynamics

Abstract

The methods described in an earlier paper, which are based on the theory of the irreducible representations of the three-dimensional rotation group, are applied in an analysis of rotationally invariant, self-adjoint systems of differential equations and boundary conditions. Particular emphasis is placed on the tensor field Helmholtz equation and the dynamical elastic equation for both homogeneous and inhomogeneous media, and with boundary conditions for both finite and infinite media. Consideration of the appropriate Green dyadic leads to a representation of the source field in terms of an infinite set of multipoles, each of which may in turn be related to a geometrical configuration of tensor point sources. A new formula is derived which describes the effect on an arbitrary tensor field of a translation of the origin, and thus ensures that the theory summarized above is applicable to a source region at a general location. Geophysical applications include the calculation of the excitation of the modes of free oscillation of the Earth by earthquake sources, and, more generally, the multipolar representation of seismic sources.