Coupled Rayleigh surface waves in a slowly varying elastic waveguide

A combination of the two lowest Rayleigh-Lamb modes, for a waveguide of uniform thickness with traction-free surfaces, can approximate, on one surface, a Rayleigh surface wave. If the guide is one or more shear wavelengths thick, the Rayleigh wave will be weakly coupled to the opposite surface and will transfer to that surface and then back over a distance of several wavelengths. This is caused by the two modes moving into and out of phase. We examine the same phenomenon in a waveguide whose thickness changes slowly with respect to wavelength and we develop a mathematical framework to describe it. Using the variation in thickness over a wavelength as a small parameter, an asymptotic expansion, which represents the lateral propagation with rays, but the transverse variation with local Rayleigh-Lamb modes, is used. In adapting this expansion to in-plane elastic wave propagation, we reformulate the problem for the Rayleigh-Lamb modes as an eigenvalue problem whose eigenvector is comprised of the two displacement components and the two components of the traction acting on a plane perpendicular to the axis of the guide. This enables us to exploit the orthogonality between the modes given by the reciprocity relation. The zero-order analysis indicates that, for waveguides with increasing thickness, the coupling phenomenon endures indefinitely (in the absence of damping) if the rate of growth in thickness is, at most, logarithmic. For a waveguide with a slow sinusoidal thickness variation, the coupling persists more or less as it would in a uniform guide.