Three-Dimensional Standing Surface Waves of Finite Amplitude

Abstract

Three‐dimensional gravity waves on the surface of an inviscid incompressible fluid of finite depth are considered. The waves are assumed to be periodic in time and in two perpendicular horizontal directions. The surface profile, potential function, pressure, and frequency of the motion are determined (to second order) as series in powers of the amplitude divided by the wavelength in one direction. As in the two‐dimensional case previously considered by Tadjbakhsh and Keller, it is found that the frequency increases with amplitude for depths less than a critical value and decreases with increasing amplitude for greater depths. The critical depth depends upon the wavelengths in the two horizontal directions. The three‐dimensional results do not reduce to the two‐dimensional ones when one of the wavelengths becomes infinite. This is because the motion remains three dimensional.