Directional Spreading and Non-Linear Wave Focussing on Deep Water
Observations of waves on the open sea are consistent with weakly non-linear modifications to a linear random process, at least most of the time. The occurrence of large waves can then be attributed to the random alignment of many small independent components. This random focussing by frequency and directional dispersion can be represented by numerical or physical experiments on focussed wave groups. Both fully non-linear and non-linear Schrodinger based simulations of focussed wave groups show the marked effect of directional spreading. A 1-D group contracts along the mean wave direction and becomes considerably taller than predicted by linear theory. The non linear dynamics are active for many wave periods and cumulative, leading to solitons and recurrence at least in the NLS equation. For groups with realistic directional spreading, there is still significant contraction along the mean wave direction but the group also becomes considerably more long-crested. Some extra crest elevation may result but this is much smaller than in 1-D. Although the non-linear dynamics occur over a much shorter period, the effects are still significant and the spectral content well after focus is markedly different to that before. Although not quantitatively accurate, the NLS equation and its solutions are useful in interpreting the effects of wave directionality and we suggest the use of an NLS conserved quantity [Ax2]−2[Ay2]−2ko4 [A4] as comparable to a Benjamin-Feir index for directionally spread groups.