# Edge Waves on a Sloping Beach

F. Ursell

## 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 degrees. The theory is verified experimentally in the frequency range 100 to 17 c/min for the angles 37$\cdot$6 and 29$\cdot$5 degrees.