Steep standing waves at a fluid interface

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Abstract
An algorithm is formulated for computing perturbation-series solutions for standing waves on the interface between two semi-infinite fluids of different but uniform densities. Using a comppter, the series solutions are computed to fifth order for a general value of r, the ratio of the density of the upper fluid to that of the lower fluid (0 ≤ r ≤ l), and to 21st order for five specific values of this ratio: r = 0, 10−3, 0·1, 5·0, 1·0. The series for the period, the energy, and the interface profile of the waves are summed using Padé approximants. The maximum wave height for each of the above five density ratios is estimated from the locations of the poles of the Padé approximants for the wave period and the wave energy. At maximum height the interface appears to be vertical at a point on the interface that is very near the crest for r = 10−3 and approaches the midpoint between the crest and the trough as r approaches 1·0.

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