Crest instabilities of gravity waves. Part 3. Nonlinear development and breaking

The ‘almost-highest wave’, which is the asymptotic form of the flow in a steep irrotational water wave of less than the limiting height, was recently shown to be unstable to infinitesimal disturbances (see Longuet-Higgins & Cleaver 1994). It was also shown numerically that the lowest eigenfrequency is asymptotic to that of the lower superharmonic instability of a progressive wave in deep water (Longuet-Higgins, Cleaver & Fox 1994). In the present paper these calculations are revised, indicating the presence of more than one such instability, in agreement with recent calculations on steep periodic and solitary waves (Longuet-Higgins & Tanaka 1997). The nonlinear development of the fastest-growing instability is also traced by a boundary-integral time-stepping method and the initial, linear growth rate is confirmed. The subsequent, nonlinear stages of growth depend as expected on the sign of the initial perturbation. Perturbations of one sign lead to the familiar overturning of the wave crest. Perturbations of the opposite sign lead to a smooth transition of the wave to a lower progressive wave having nearly the same total energy, followed by a return to a wave of almost the initial wave height. This appears to be the beginning of a nonlinear recurrence phenomenon.