Standing waves on water of uniform depth: on their resonances and matching with short-crested waves

Abstract

Numerical calculations of the resonant interactions of three-dimensional short-crested waves very near their two-dimensional standing wave limit are performed for water of uniform depth. A detailed study of the properties of the solutions indicates that both classes of waves admit multiple solutions that are connected to each other through turning points. It is also shown that the solutions match each other at the limit. Then a study on the superharmonic instabilities (resonant interactions) of short-crested waves was performed in the vicinity of the standing wave limit. The matching allowed extrapolation of the short-crested wave stability results to standing waves. The results are that for resonant waves, superharmonic instabilities associated with harmonic resonance are dominant. The possible jumps from one solution to another may lead to a drastic change of the wave itself. Since the superharmonic instability enhances this property one may conclude that this class of waves can be considered non-stationary. By contrast, non-resonant waves are weakly unstable or stable and are the only waves that are likely to exist. Thus, this class of waves can be considered as quasi-permanent.