This paper extends recent theoretical work on sub-inertial trapped modes in bays to consider trapping of energy in the neighbourhood of estuary mouths on coastal shelves. The qualitative form of the theoretical predictions accords well with recent observations on the Scotian Shelf that show energy trapped near the Laurentian Channel at a frequency higher than that of the propagating waves on the shelf.
The trapping and scattering of shelf waves is modelled for a shelf–estuary or shelf–headland system by considering barotropic waves in a straight, infinite channel with an attached rectangular estuary or interrupted by a rectangular headland. Taking the depth to increase exponentially with distance from the coast and expanding in cross-shelf modes reduces the problem to a system of real linear algebraic equations.
Trapped modes with frequencies above the cutoff frequency of propagating waves are found near the mouth of the estuary. Waves propagating towards the estuary are strongly scattered and, for particular frequencies, incident energy can be either perfectly transmitted or totally reflected. An incident wave can be in resonance with the estuary causing energy to penetrate the estuary. Bounds on the frequencies of trapped and resonant solutions are given and allow an easy modal interpretation.
If the frequency of an incident wave is sufficiently high, waves cannot propagate past a headland. Energy at these frequencies can however tunnel through the region and appear as an attenuated wave on the far side. For particular frequencies all energy passes the headland and none is reflected. For headlands long compared with the incident wave, transmission coefficients for single-mode scattering follow from spatially one-dimensional wave mechanics.