New Form of the Hamiltonian Equations for the Nonlinear Water-Wave Problem, Based on a New Representation of the DTN Operator, and Some Applications

Abstract

We present a new Hamiltonian formulation for the non-linear evolution of surface gravity waves over a variable impermeable bottom. The derivation is based on Luke’s variational principle and the use of an exact (convergent up to the boundaries) infinite-series representation of the unknown wave potential, in terms of a system of prescribed vertical functions (explicitly dependent on the local depth and the local free-surface elevation) and unknown horizontal modal amplitudes. The key idea of this approach is the introduction of two unconventional modes ensuring a rapid convergence of the modal series. The fully nonlinear water-wave problem is reformulated as two evolution equations, essentially equivalent with the Zakharov-Craig-Sulem formulation. The Dirichlet-to-Neumann operator (DtN) over arbitrary bathymetry is determined by means of a few first modes, the two unconventional ones being most important. While this formulation is exact, its numerical implementation, even for general domains, is not much more involved than that of the various simplified models (Boussinesq, Green-Nagdhi) widely used in engineering applications. The efficiency of this formulation is demonstrated by the excellent agreement of the numerical and experimental results for the case of the classical Beji-Battjes experiment. A more complicated bathymetry is also studied.