It would not be far-fetched to say that the work of Lord Rayleigh on surface guided waves has had fundamental and far-reaching effects upon modern life and many things we take for granted today, stretching from mobile phones through to the study of earthquakes. Many of these things take advantage of surface waves that are topographically guided thereby allowing energy to be carried in specific directions along some topography of the surface. Much of the emphasis has so far been placed on devices which are essentially thin rectangular ridge-like defects upon the surface. Indeed, experiments with these so-called SAW devices have demonstrated the presence of trapped waves. It remains to provide any convincing theory that captures this trapping effect for surface waves—this is our aim. We seek to resolve the issue of whether topography can support a trapped wave, whose energy is localized to within some vicinity of the topography, and to explain physically how trapping occurs. The trapping is first addressed by developing an asymptotic scheme that exploits a small parameter associated with the surface topography. We then provide numerical evidence to support results obtained from the asymptotic scheme; however, no rigorous proof of existence is presented.

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.

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