The non-existence of standing modes in certain problems of linear elasticity

Abstract

AbstractSuppose that an elastic half-space, which contains certain surface defects, inclusions and cavities, is in free, two-dimensional, time-harmonic vibration, with the wave field at infinity ‘outgoing’ in character. It is shown that the elastic potentials representing such a ‘standing mode’ can be expressed in the form of contour integrals, for instanceU(t) being an analytic function of t. By considering the far field of these potentials, it is shown that U(t) is zero on a certain arc in the t-plane and is therefore identically zero. It follows that ø(r) is zero everywhere and this proves the non-existence of such standing modes in these configurations.This uniqueness theorem justifies the solution given by the author (Gregory (2)) for the problem in which time harmonic stresses act on the walls of a cylindrical cavity lying beneath the surface of an elastic half-space. It is also shown that if a Rayleigh surface wave is incident on any system of surface defects, inclusions and cavities, then energy must be transferred from the surface wave to scattered outgoing body waves of both P and S types.