Modelling and simulation of surface water waves

The evolution of waves on the surface of a layer of fluid is governed by non-linear effects from surface deformations and dispersive effects from the interaction with the interior fluid motion. Several simulation tools are described in this paper and compared with real life experiments in large tanks of a hydrodynamic laboratory. For the full surface wave equations a numerical FEM/FD program is described that solves both the interior flow and the surface evolution; apart from being very efficient, the program performs remarkably well. For theoretical analysis, simplified equations are desired. These can be obtained by modelling the interior flow to some degree of accuracy, leading to a single equation for the surface elevation. As representatives of this class, we discuss KdV-type of equations and, for wave packets, the NLS equation. We show that even this last equation describes quite well the large, non-symmetric deformations of the envelope of bi-harmonic waves when formulated as a signalling problem