Crest instabilities of gravity waves. Part 1. The almost-highest wave

Abstract

It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R ) ½ , where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45 R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.