The Scalar Equations of Infinitesimal Elastic-Gravitational Motion for a Rotating, Slightly Elliptical Earth

Abstract

We derive the infinite set of coupled ordinary differential equations over radius that govern the infinitesimal free elastic-gravitational oscillations of a rotating, slightly elliptical Earth with an isotropic perfectly elastic constitutive relation and a hydrostatic prestress field. We show how the symmetries of such a body restrict the most general form of the displacement eigenfunctions. We discuss situations in which finite sets of coupled equations may yield good approximate eigenfunctions and describe the sense in which the solution to such a set approximates the solution of the infinite set. Finally we describe in some detail the equations governing a particular class of finite expansions and show how these may be put in a form convenient for numerical solution. The method used to convert tensor equations to scalar equations is an extension of the generalized surface spherical harmonic expansion of Phinney & Burridge and should be useful in other non spherically-symmetric applications.