Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation

Abstract

A general method for the derivation of asymptotic nonlinear models in shallow and deep water is presented. Starting from a general dimensionless version of the water wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet–Neumann operator and we show that many asymptotic models can be recovered by a simple analysis of this operator. Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to an uneven bottom the approach developed by Matsuno [Phys. Rev. E 47, 4593 (1993)] and Choi [J. Fluid Mech. 295, 381 (1995)]. This model is still valid in shallow water but with less precision than what can be achieved with the Green–Naghdi model when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equations with the fully nonlinear shallow water Green–Naghdi equations represents a relevant model for describing ocean wave propagation from deep to shallow waters.