The resonant instabilities of long waves obliquely incident on a beach

Abstract

A monochromatic unidirectional wave train, incident on a plane beach and strongly reflected there, is shown to transfer energy to edge waves of lower frequency through a weak nonlinear interaction. For any angle of wave incidence the most readily excited edge wave perturbation consists of two low-mode progressive edge waves, generally having different frequencies and wave numbers, traveling in opposite directions along the beach. Standing edge waves, which might form stationary morphologic features with a regular longshore rhythm, are theoretically only excited when the primary surface waves are normally incident. However, edge waves generated by almost normally incident primary waves may be linked to features which slowly migrate along shore. On beaches bounded by headlands or jetties the progressive edge waves excited would be reflected at both ends, forming a complex pattern of standing waves. For beaches bounded at one end, only one of the edge waves would be standing. Regular beach cusps would therefore be expected in the vicinity of barriers. These cusps should decrease in relief with increasing distance from the obstacle as the reflected edge wave, which is not being actively forced, dies away due to viscous dissipation and further nonlinear interactions. Intriguingly, the cusps should have slightly different wavelengths on either side of the obstacle.