Embedded trapped modes in water waves and acoustics

Trapped modes, localized oscillations in unbounded media, are referred to in different contexts by various names; acoustic resonances, Rayleigh-Bloch waves, edge waves, array guided surface waves and bound states being examples. Most studies have concentrated on such phenomena in situations where they are associated with a cut-off frequency below which wave propagation is not possible. It is much more difficult to establish the existence of trapped modes in regions of parameter space which permit energy to travel to infinity. In this article we review recent results on these so-called embedded modes and discuss problems for future research. There are two distinct cases to consider. First, we consider trapped modes in waveguides governed by the Helmholtz equation. Such problems arise when considering obstacles in acoustic waveguides or bound states in quantum wires, for example. In two dimensions, the same equations govern water-wave channels after the depth dependence has been removed. Secondly, we examine situations from water-wave theory in which the potential satisfies Laplace’s equation with the frequency parameter now appearing in the free-surface boundary condition.