Stable gravity wave of arbitrary amplitude in finite depth

Abstract

A combination and modification of two existing methods, which involves balancing static and dynamic pressure differences between points along the surface and conserving mass through cross sections below the surface in the reference frame moving with the phase velocity, is applied to surface gravity waves of arbitrary amplitude in water of finite depth. For a given still water depth and wave height the method determines in closed form the phase velocity, wavelength, and wave profile of the stable wave. The main assumption is that the horizontal component of the fluid velocity be independent of depth. The motion is not assumed to be irrotational. The wavelength of the stable wave is found to be about 3.6 times the still water depth for infinitesimal amplitude, and at finite amplitude the wavelength decreases as the amplitude increases. Therefore, shallow water waves are concluded to be unstable even at infinitesimal amplitude, for which the assumption is accurate. Previously it has been argued that only at finite amplitude will shallow water waves change form as they propagate. The wave profile is found to be sinusoidal for infinitesimal amplitude and to be asymmetric at finite amplitude, the crests being higher and narrower and the troughs shallower and broader. These results are consistent with well-known theoretical work and laboratory measurements.