Evolution Equations for Edge Waves and Shear Waves on Longshore Uniform Beaches

A general formalism for computing the nonlinear interactions between triads of coastally-trapped gravity and vorticity waves is developed. An analysis of the linearized problem reveals that gravity (or edge) waves and vorticity (or shear) waves exist as members of the same non-Sturm-Liouville eigenvalue problem, with unstable shear waves representing the complex eigenvalue portion of the resulting spectrum. Interaction equations derived here cover resonant interactions between three edge waves, three shear waves, or a shear wave and two edge waves. Numerical examples are shown for the case of three edge waves on a planar beach in the absence of a longshore current. It is found that edge waves can exchange signiicant amounts of energy over time scales on the order of ten wave periods, for realistic choices of edge wave parameters. Introduction The low frequency wave climate on an open coastal beach contains a complex mix of trapped gravity wave motions (edge waves) as well as vorticity (or shear) waves associated with the instability of the longshore current. To date, there has been a tendancy in the literature to treat both classes of motion as isolated systems in which the principle eeect of nonlinearity is through amplitude dispersion. Formulations of this type typically treat the wave eld in terms of a wave envelope modulated by cubic nonlinearity, leading to the cubic Schrr odinger equation for conservative edge wave systems (Yeh, 1985) or Ginzburg-Landau equation for marginally unstable shear waves (Feddersen, 1998). However, in eld conditions, all of these motions occur in a relatively dense spectral environment, and the existence of combinations of waves satisfying three-wave resonance conditions makes it likely that the dominant nonlinear mechanism aaecting edge or shear waves would be through resonant interactions at second order.