A new efficient Hamiltonian approach to the nonlinear water-wave problem over arbitrary bathymetry

Abstract: A new Hamiltonian formulation for the fully nonlinear water-wave problem over variable bathymetry is derived, using an exact, vertical series expansion of the velocity potential, in conjunction with Luke's variational principle. The obtained Euler-Lagrange equations contain infinite series and can rederive various existing model equations upon truncation. In this paper, the infinite series are summed up, resulting in two exact Hamiltonian equations for the free-surface elevation and the free-surface potential, coupled with a time-independent horizontal system of equations. The Dirichlet to Neumann operator is given by a simple and versatile representation, which is valid for any smooth fluid domain, not necessarily periodic in the horizontal direction(s), without limitations on the steepness and deformation of the seabed and the free surface. An efficient numerical scheme is presented and applied to the case of one horizontal dimension, establishing the ability of the new formulation to simulate strongly nonlinear and dispersive wave-bottom interactions by comparison with experimental measurements.