The three-dimensional problem of wave trapping above a submerged round sill was first analysed by Longuet-Higgins on the basis of a linear shallow-water theory. The large responses predicted by the theory were, however, not well borne out by the experiments of Barnard, Pritchard & Provis, and this has motivated a more detailed study of the problem. A full linear theory for both inviscid and weakly viscous fluid, without any shallow-water assumptions, is presented here. It reveals important limitations on the use of shallow-water theory and the reasons for them. In particular, while the qualitative features of wave trapping are similar to those of shallow-water theory, the nearly resonant frequencies differ significantly, and, since the resonances are narrow, the observed amplitudes at a given frequency differ greatly. The geometry is strongly indicative of long waves, and the dispersion relation appears quite consistent with that, but the part of the motion at wavenumbers that are not small has, despite the small amplitude, a substantial effect on the response to excitation.
The Linear Shallow Water Theory: A Mathematical Justification
