Steady Three-dimensional Nite-depth Patterns on the Ocean Surface: a New Characterization of Short-crested Stokes Waves Interacting with a Mean Ow

ocean surface: a new characterization of short-crested Stokes waves interacting with a mean ow By T. J. Bridges1, F. Dias2 and D. Menasce2 1Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 5XH, UK 2 Institut Non-Lin eaire de Nice, Universit e de Nice-Sophia Antipolis 1361 route des Lucioles, 06560 Valbonne, France Abstract The formation of multi-periodic patterns on the surface of a uid layer with a uniform velocity eld and constant depth is considered. The problem of steady patterns is shown to have a novel variational formulation, which includes a new characterization of steady uniform mean ow. The formulation is applied to the problem of interaction of capillary-gravity short-crested waves with oblique travelling waves. By including oblique travelling waves and short-crested waves in the same analysis, new bifurcations of short-crested waves are found, which give rise to mixed waves which may have complicated spatial structure.