Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations using weakly nonlinear theory, which leads to the nonlinear Schrödinger equation. The steeper events are shorter in both space and time than the lower events. Solutions of the nonlinear Schrödinger equation can be transformed from one steepness to another by suitable scaling of the length and time variables. We use this scaling on the modulations and find excellent agreement particularly for waves that do not grow too steep. Hence the number of waves in the initial modulation becomes an almost redundant parameter and allows wider use of each computation. A potentially useful property of the nonlinear Schrödinger equation is that there are explicit solutions which correspond to the growth and decay of an isolated steep wave event. We have also investigated how changing the phase of the initial modulation effects the first steep wave event that occurs.