Rayleigh waves guided by topography

Abstract

We consider wave propagation along the surface of an elastic half-space, whose surface is flat except for a straight, infinite length, ridge or trench that does not vary in its cross-section. We seek to resolve the issue of whether such a perturbed surface can support a trapped wave, whose energy is localized to within some vicinity of the defect, and explain physically how this trapping occurs. First, the trapping is addressed by developing an asymptotic scheme, which exploits a small parameter associated with the surface variation, to perturb about the base state of a flat half-space (which supports a surface wave, as demonstrated by Lord Rayleigh in 1885). We then provide convincing numerical evidence to support the results obtained from the asymptotic scheme; however, no rigorous proof of existence is presented.