On the nonlinear dynamics of wave groups produced by the focusing of surface-water waves

This paper concerns the focusing of surface-water waves whereby a large number of waves, of varying frequency and propagating in different directions, are superimposed at one point in space and time to produce a large transient wave group. A new numerical model is presented, the results of which are used to explain the nonlinear evolution of a number of steep wave groups observed by Johannessen & Swan in an earlier laboratory study. The numerical model is based on a unidirectional formulation, originally outlined by Fenton & Rienecker, which has been extended to allow accurate calculations of directionally spread wavefields. Both the instantaneous water-surface elevation and the associated velocity potential are described in space using series expansions, and the solution time-marched via the nonlinear free-surface boundary conditions. Calculations confirm that the model is in good agreement with existing laboratory data. Furthermore, the very high spatial resolution provided by the model allows insights into the evolution of a wavefield that cannot be obtained from laboratory data alone. In a number of steep-wave cases, it is shown that the large deviations from linear and second-order theory, based on the initial spectrum of free waves, are caused by a local widening of the free-wave regime. This nonlinear evolution, resulting in large changes in the wave-group envelope, can occur over the time taken for a single large wave to evolve. In comparison with other nonlinear evolution effects, such as the resonant interactions previously identified by Hasselmann, these changes are very rapid. In the present cases, it is shown that the shift of energy to higher-frequency wave components, within the extended free-wave regime, is more important than the bound-wave structure associated with even the steepest wave groups. As a result, formulations that estimate the bound-wave structure, relative to an assumed stationary regime of freely propagating wave components, may be highly inaccurate. Furthermore, the present study highlights the importance of the directionality of the wavefield. In particular, it is shown that while unidirectional waves appear to remain focused as they become highly nonlinear, short-crested wave groups do not. This is believed to be due to the directional variation in the wave-group envelope, and has important practical implications in that the apparent nonlinear defocusing leads to a reduction in the maximum crest elevation with increasing directional spread. The paper concludes by discussing the importance of the present effects to the ongoing debate concerning the occurrence of freak or rogue waves.