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Finite amplitude surface waves in a liquid layer

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh
Ali Hasan Nayfeh
Affiliation:
Aerotherm Corporation, Mountain View, California
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Abstract

An analysis is presented for the interaction of capillary and gravity waves in a liquid layer of finite depth. The method of multiple scales is used to obtain a third-order expansion uniformly valid for all times. Although this expansion is valid for a wide range of wave-numbers, it breaks down at two critical wave-numbers if the liquid depth is larger than √3/kc, kc = (ρg/T)½, where g is the gravitational acceleration, and ρ and T are the liquid density and surface tension, respectively. For a deep liquid, the singularities are at kc/√2 and kc/√3 respectively, as found by Wilton (1915), and Pierson & Fife (1961).

A second-order expansion valid for wave-numbers near the first critical value (corresponding to a wavelength of 2·44 cm in deep water) is obtained. This expansion shows that two different wave profiles could exist at or near the first critical wave-number. One of these profiles is gravity-like while the other is capillary-like.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 40 , Issue 4 , 9 March 1970 , pp. 671 - 684
Copyright
© 1970 Cambridge University Press

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