Trapped modes in elastic plates, ocean and quantum waveguides

Trapped modes and bound states occur in waveguides in a wide variety of physical situations, here we consider one class: the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. We consider the possibility of trapped modes in otherwise uniform elastic/ocean/quantum waveguides, that have a perturbation in thickness, or in curvature. These are all mathematically connected, so their study can be unified, and the same asymptotic procedure can be applied to them all. For bent waveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility, or otherwise, of trapping. An asymptotic scheme is developed to analyse whether trapped modes should be expected and to obtain the frequencies at which trapped modes are excited. A mathematical explanation and physical argument for the existence, or otherwise, of trapped modes, depending on different situations, is given. The asymptotic approach leads to an ordinary differential equation eigenvalue problem and its solutions are compared both with numerics, for the full governing equations, and with a further simplification that gives highly accurate estimates for the lowest eigenvalue.