The limiting configuration of interfacial gravity waves

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Abstract
Progressive gravity waves at the interface between two unbounded fluids are considered. The flow in each fluid is taken to be potential flow. The problem is converted into a set of integrodifferential equations, reduced to a set of algebraic equations by discretization, and solved by Newton’s method together with parameter variation. Meiron and Saffman’s [J. Fluid Mech. 1 2 9, 213 (1983)] calculations showing the existence of overhanging waves are confirmed. However, the present calculations do not support Saffman and Yuen’s [J. Fluid Mech. 1 2 3, 459 (1982)] conjecture that the waves are geometrically limited (i.e., that solutions exist until the interface intersects itself). It is proposed that along a solution branch starting with sinusoidal waves of small amplitude, one reaches solutions with vertical streamlines and then overhanging waves. Continuing on this branch, one returns to nonoverhanging waves and then back toward a wave with vertical streamlines. It is suggested that this succession of patterns and accompanying oscillation in wave characteristics is repeated indefinitely. Graphs of the results are included.