Limiting forms for capillary-gravity waves

The form of steep capillary waves is of interest as a possible initial condition for the formation of air bubbles at a free surface. In this paper the limiting forms of pure capillary waves and of quasi-capillary waves are studied analytically. Crapper’s finite-amplitude solution is expressed in a simple form, and is shown to be one of several exact elementary solutions to the pure-capillary free-surface condition. Among others are the solution z = w + sinh’w, where w is the velocity potential, and also z = w3. The latter solution, though it represents a self-intersecting flow, can be used as the first in a sequence of approximations to the form of the steepest wave. Hence it is shown that the influence of gravity on the shape of the limiting ‘bubble ’ is very small. The result is confirmed by an examination of Hogan’s numerical calculations of limiting capillary-gravity waves. In the crest of a limiting wave the particle velocity is almost constant and equal to the phase speed. This property makes i t possible to apply a quasi-static approximation so as to determine the form of the crest, and hence to find an expression for the complete profile of a capillary-gravity wave of limiting steepness. It appears that there exists a solitary wave of capillary-gravity type on deep water.