Trapped modes in topographically varying elastic waveguides

Trapped modes within elastic waveguides have been shown to exist for bent guides, rods and thickened plates (in plane stress). Here we concentrate upon the complementary problem of thickened or thinned elastic guides in plane strain. We develop an asymptotic procedure that encapsulates the essential physics within a single ordinary differential equation. It is shown that this reduced model is functionally the same as that for a bent plate and hence the two problems are mathematically identical. Two-humped, or doubly bent, waveguides are also considered and it is shown that trapping can occur localised at the geometric variation whether it be at the humps or bends or in-between them. Physical arguments for the trapping are advanced and numerical simulations of the full elasticity equations are performed to both demonstrate the accuracy of the asymptotic model and to validate it.