Nonlinear waves and solitons in water

Abstract

A new theoretical model is introduced for evaluating three-dimensional gravity-capillary waves in water of uniform depth to various degrees of validity for predicting nonlinear dispersive water wave phenomena. It is first based on two basic equations, one being the continuity equation averaged over the water depth, and the other the horizontal projection of the momentum equation at the free surface. These two partial differential equations are both exact (for flows assumed incompressible and inviscid), but involve three unknowns: the horizontal velocity at the free surface (in two horizontal dimensions), û; the depth-mean horizontal velocity, u; and the water surface elevation, ξ. Clsure of the system for modeling fully nonlinear and fully dispersive water waves is accomplished by finding for the velocity field a third exact equation relating these unknowns. Interesting phenomena in various cases are illustrated with review and discussion of literature.