Nonlinear Asymmetric Excitation of Edge Waves

Abstract

The resonant excitation of two weakly nonlinear oppositely travelling edge waves on a gently sloping beach by a perfectly reflected obliquely incident gravity wave is calculated. The four first-order nonlinear differential equations that govern the slowly varying amplitudes and phases of the edge waves are associated with a Hamiltonian supplemented by a dissipation function that provides for both viscous and radiation damping. This system is reduced, through successive canonical transformations in action-angle variables, to two evolution equations that are isomorphic to those for the symmetric problem of normal incidence. Asymmetric edge waves of finite amplitude, like their symmetric counterparts, occur in a narrow band of frequency if and only if the dissipation is inferior to a certain critical value, in which case they appear as a slowly modulated (in both space and time) standing wave and induce a second-order mean flow along the shore. The effects of imperfect reflexion are described by reference to the symmetric problem