The nonlinear water wave equations with full dispersion

A 2D nonlinear water wave model with full dispersion isdeveloped. The model is based on the nonlinear kinematic and dynamic freesurface boundary conditions and is expressed in terms of free surfaceelevation \eta and the velocity potential \phi _\eta at the freesurface. The derivation of the equations is accurate to third order innonlinearity and keeps exact dispersion. The mild slope assumption isadopted and the derived equations can be seen as the extention of the mildslope equation of Berkhoff (1972) to the nonlinear and irregular wave case.The corresponding numerical scheme is presented, and the special attentionis paid on the treatment of the integration terms in the equations. Thevalidation of the model is made by simulating the first and second orderStokes waves and the nonlinear evolution of wave groups, the advantage ofthe model is shown by the good prediction of amplitude dispersion andfour-wave resonant interaction for the wave group evolution.