Subharmonic generation of edge waves by nonuniform incident waves

Abstract

Subharmonic resonance of edge waves by incident and reflected waves on a plane beach has been studied thus far either without any long-shore modulation or with long-shore modulation on the scale O( kϵ) −1 where k is the wave number of the edge wave and ϵ the characteristic wave slope. In the former case, the evolution equation of the edge wave amplitude is an ordinary differential equation involving a cubic term; in the latter it becomes coupled partial differential equations of the cubic Schrödinger type. We study here yet another kind of long-shore modulation as a result of nonuniformity in the envelope of the incident/reflected waves. In addition to the initial growth, nonlinear forced evolution is also examined. It is shown that incident reflected wave packet of long-shore width on the scale O( kϵ 3) −1 or longer will essentially leak O(1) subharmonic progressive edge waves from the sides.