Nonlinear progressive edge waves: their instability and evolution

Abstract

The fundamental properties of nonlinear progressive edge waves are investigated experimentally in a physical model with a uniformly and mildly sloping beach with a straight shoreline. An evolution equation for the envelope of progressive edge waves is the nonlinear Schrödinger (NLS) equation. It is found that the timescale of viscous-dissipation effects in the experiments is comparable with the timescale of the theoretical evolution process for inviscid progressive edge waves. This lack of timescale separation indicates a major shortcoming of the NLS equation as a model of the laboratory experiments. Even with this limitation, a uniform train of edge waves is found to be unstable to a modulational perturbation as predicted by the NLS equation. However, behaviour of the evolution is both qualitatively and quantitatively different from the theoretical predictions. The evolution of the periodogram for the unstable wavetrain shows an asymmetric development of the sidebands about the fundamental frequency; instability growth is limited to the lower sideband. This behaviour leads to a sequential shift of wave energy to lower frequencies as the waves propagate. It is found that a locally soliton-shaped wave packet is unstable in the laboratory environment. It is estimated that a much-larger-scale experimental facility is required to achieve inviscid experiments for the NLS equation.