Abstract
The method of multiple scales is used to obtain a third-order expansion for the interaction of gravity and capillary waves. The expansion is found to break down for wavelengths of 2.44 and 2.99 cm in deep water, in agreement with Wilton. A second expansion valid for wavelengths near the latter critical wavelength is obtained. The latter expansion shows that three different wave profiles could exist at wavelengths near 2.99 cm. One is a gravity-like profile (wave velocity increases with increasing amplitude) having triple dimples. The other two are capillary-like profiles (wave velocity decreases with increasing amplitude) having quintuple dimples.

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