Deep water gravity wave triad resonances on uniform flow

Abstract

Conditions for three-wave resonance between surface gravity waves on uniform current in deep water are found. The resonances occur because waves propagating downstream are elongated (relative to their still water form), whereas waves propagating upstream are contracted. Under certain conditions, the elongation and contraction is to such an extent that the difference between the wavenumbers closes resonance with a third wave. In contrast, the existing literature assumes there is no deep water gravity wave triad resonance on uniform current. Rather, the lowest order nonlinear resonance is assumed to be a four-wave interaction. In this sense, the results represent a new class of resonances. Moreover, since triads are of a lower order than four-wave interactions, the effects of the new triad resonances will be, by definition, an order of magnitude greater (in wave slope ka) than the strongest known resonances for the assumed conditions. Thus, the results represent a new and important mechanism of wave growth and energy exchange between surface gravity waves.