Stable deep-water waves propagating in one and two dimensions

Deep-water, narrow-banded wavetrains of uniform amplitude propagating in one horizontal dimension were shown to be unstable to small perturbations with nearly the same frequency and direction about 40 years ago. More recently, interactions of two narrow-banded wavetrains of uniform amplitude propagating in arbitrary directions were shown to be similarly unstable. In both cases, the instabilities have been described by either a single nonlinear Schrödinger (NLS) equation or by two coupled NLS equations. However, the inclusion in these equations of any amount of damping (of a particular type) stabilizes the instabilities. Experiments show that the evolution of perturbed wavetrains in both cases is more accurately described by the NLS models that include damping when the amplitudes are small or moderate. When the amplitude of either the underlying waves or the perturbations is large, neither the undamped NLS model nor the damped NLS model accurately predicts the observed behaviour, which includes frequency downshifting.