Title:

Trapped modes of the Helmholtz equation

In the framework of the classical theory of linearised water waves in unbounded domains, trapped modes consist of nonpropagating, localised oscillation modes of finite energy occurring at some welldefined frequency and which, in the absence of dissipation, persist in time even in the absence of external forcing. Jones (1953) proved the existence of trapped modes for problems governed by the Helmholtz equation in semiinfinite domains. Trapped modes have been studied in quantum mechanics, elasticity and acoustics and are known, depending on the context, as bound states, acoustic resonances, RayleighBloch waves, sloshing modes and motion trapped modes. We consider trapped modes in two dimensional infinite waveguides with either Neumann or Dirichlet boundary conditions. Such problems arise when considering obstacles in acoustic waveguides or bound states in quantum wires for example. The mathematical model is a boundary value problem for the Helmholtz equation. Under the usual assumptions of potential theory, the solution is written in terms of a boundary integral equation. We develop a Boundary Element Method (BEM) program which we use to obtain approximate numerical solutions. We extend existing results by identifying additional trapped modes for geometries already studied and investigate new structures. We also carry out a detailed investigation of trapped modes, using the planewave spectrum representation developed for various characteristic problems from the classical theories of radiation, diffraction and propagation. We use simple planewaves travelling in diverse directions to build a more elaborate solution, which satisfies certain conditions required for a trapped mode. Our approach is fairly flexible so that the general procedure is independent of the shape of the trapping obstacle and could be adapted to other geometries. We apply this method to the case of a disc on the centreline of an infinite Dirichlet acoustic waveguide and obtain a simple mathematical approximation of a trapped mode, which satisfies a set of criteria characteristic of trapped modes. Asymptotically, the solution obtained is similar to a nearly trapped mode, which is a perturbation of a genuine trapped mode.
