On the crest instabilities of steep surface waves

Abstract

The forms of the superharmonic instabilities of irrotational surface waves on deep water are calculated for wave steepnesses up to 99.9% of the limiting value. It is found that as the limiting wave steepness is approached the rates of growth of the lowest two unstable modes (n=1 and 2) increase according to the asymptotic law suggested by the theory of the almost-highest wave (Longuet-Higgins & Cleaver 1994; Longuet-Higgins, Cleaver & Fox 1994; Longuet-Higgins & Dommermuth 1997). Moreover, each eigenfunction becomes concentrated near the wave crest, with a horizontal scale proportional to the local radius of curvature at the crest. These are therefore ‘crest instabilities’ in the original sense.Similar calculations are carried out for the normal-mode instabilities of solitary waves in shallow water, at steepnesses up to 99.99% of the limiting steepness. Similar conclusions are found to apply, though with greater accuracy.