The effect of non-uniformity of modulated wavepackets on the mechanism of Benjamin–Feir instability

Abstract

The effect of non-uniformity on the development of a modulated, weakly nonlinear wavepacket is studied. The non-uniformity, characterized by slowly varying wavenumber and frequency of the primary wave, may lead to significant modification of the stability properties compared with the uniform case. As a specific example we consider a modulated Stokes wave on deep water. In the uniform case such a wave proves to be definitely unstable (Benjamin & Feir 1967). In the non-uniform case, on the other hand, the wave may become stable under certain conditions. One of these is an increase of the local group velocity in the direction of wave propagation. Then the Benjamin-Feir instability mechanism is quenched on a time scale determined by the degree of non-uniformity. In addition, a sufficient degree of non-uniformity leads to stability of the wave to linear perturbations. However, when the local group velocity decreases in the direction of wave propagation, non-uniformity has a destabilizing effect. A comparison is made with experiments. It is also shown that the analysis, based on this specific example, is readily applied to a greater variety of non-uniform, dispersive waves.