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A general method for the derivation of asymptotic nonlinear shallow water anddeep water models is presented. Starting from a general dimensionless versionof the water-wave equations, we reduce the problem to a system of two equationson the surface elevation and the velocity potential at the free surface. Theseequations involve a Dirichlet-Neumann operator and we show that all theasymptotic models can be recovered by a simple asymptotic expansion of thisoperator, in function of the shallowness parameter (shallow water limit) or thesteepness parameter (deep water limit). Based on this method, a newtwo-dimensional fully dispersive model for small wave steepness is alsoderived, which extends to uneven bottom the approach developed by Matsuno\cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallowwater but with less precision than what can be achieved with Green-Naghdimodel, when fully nonlinear waves are considered. The combination, or thecoupling, of the new fully dispersive equations with the fully nonlinearshallow water Green-Naghdi equations represents a relevant model for describingocean wave propagation from deep to shallow waters.

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