In the framework of the classical theory of linearised water waves in unbounded domains,
trapped modes consist of non-propagating, localised oscillation modes of finite energy occurring
at some well-defined 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 semi-infinite domains. Trapped modes have been studied in quantum
mechanics, elasticity and acoustics and are known, depending on the context, as bound
states, acoustic resonances, Rayleigh-Bloch 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.