A class of elliptical free-surface flows

Exact solutions of the equations of motion for an inviscid fluid are rare. Using the formalism of John (1953), this paper presents a class of exact zero-gravity flows in which the free surface assumes the form of an ellipse having arbitrary but time-constant aspect ratio. The dynamically important region beneath the overturning crest of a breaking gravity wave is examined and the profile is found to be remarkably well approximated by a √3 aspect-ratio ellipse. The range of examples presented includes high-resolution computations in both deep and shallow water, and also the plunger-generated laboratory waves of Miller (1976). The ellipse solution is shown to model qualitatively certain essential features of the numerical waves. A recent self-similar solution due to Longuet-Higgins (1981, 1982), in which the free surface is a parametric cubic curve, is also discussed.