Edge waves on a sloping beach

TLDR

The spectrum of a mechanical system, including the mixed spectrum of a semi‑infinite canal bounded by a sloping beach, governs its resonant behavior. The authors analyze systems with discrete, continuous, and mixed spectra and experimentally verify the theory for frequencies 100–17 c/min at beach angles 37.6° and 29.5°. Resonance occurs at discrete frequencies and at cut‑off frequencies; at discrete frequencies the inviscid edge wave is confined to the beach, while at cut‑off frequencies the resonance extends along the canal but is confined near the beach by viscosity, with especially large resonances predicted for critical angles up to 30°, all confirmed experimentally.

Abstract

The set of eigenfrequencies of a mechanical system forms its spectrum. A discussion is given of systems with discrete, continuous and mixed spectra. It is shown that resonance occurs at discrete points of the spectrum, and at cut-off frequencies (end-points of the continuous spectrum). The motion in a semi-infinite canal of finite width closed by a sloping beach has a mixed spectrum. The inviscid theory predicts that at a discrete frequency the resonance is confined to the neighbourhood of the beach (inviscid edge wave), while at a cutoff frequency the resonance extends a long way down the canal. The latter resonance is confined to the neighbourhood of the beach (viscous edge wave) by viscosity which is important near a cut-off frequency. Especially large resonances are predicted for a series of critical angles, of which the largest is 30°. The theory is verified experimentally in the frequency range 100 to 17c/min for the angles 37⋅6 and 29⋅5°.