Parametrically excited standing edge waves

Abstract

The resonant excitation of weakly nonlinear, standing edge waves of frequency ω and longshore wavenumber k by a normally incident, non-breaking gravity wave of frequency 2ω and shoreline amplitude a to on a bottom that descends smoothly from a shoreline depth of zero and slope σ to an offshore depth h∞ is calculated for ka [Lt ] σ [Lt ] kh∞ [Lt ] 1. The analysis generalizes hose of Guza & Bowen (1976) and Rockliff (1978), which assume uniform slope and perfect reflection, and culminates in a pair of evolution equations for the slowly varying, quadrature amplitudes (or, equivalently, amplitude and phase) of the edge wave. Weak, linear damping (which implies imperfect reflection) is incorporated, and the resulting fixed points and bifurcation points of the evolution equations are determined. It is shown that the solution for prescribed initial conditions must tend to one of the stable fixed points, which correspond to an edge-wave amplitude of either zero or O((σa/k)½), depending on whether the damping exceeds or is inferior to a certain critical value. The restriction σ [Lt ] kh∞ is relaxed for the special depth profile h/h∞ = 1 −exp (−σx/h∞), for which the inviscid, shallow-water equations admit exact solutions. These solutions serve to validate the asymptotic (σ/kh↓O) approximations for arbitrary depth profiles.