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Hysteresis in Faraday resonance

Published online by Cambridge University Press:  26 April 2006

S. P. Decent  and
A. D. D. Craik
S. P. Decent
Affiliation:
School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland, UK
A. D. D. Craik
Affiliation:
School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland, UK
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Abstract

Faraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. Hysteresis occurs when both finite-amplitude solutions and the flat-surface solution are available. We derive a nonlinear model of Faraday resonance, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. The theoretical results are compared to previous experimental studies and to some new observations. It is found necessary to retain damping and forcing terms up to third-order in wave amplitude, and also the fifth-order conservative frequency shift, in order to achieve agreement with experiments. The latter fifth-order term was omitted from all previous studies of Faraday waves. The lower hysteresis boundary in forcing-frequency space is found in most cases to be defined by the lower boundary above which non-trivial stationary points exist. However, the stability of stationary points and the existence of limit cycles are also found to be factors in determining the lower hysteresis boundary. Our results also suggest an indirect method for estimating the coefficient of cubic damping, which is difficult to obtain either experimentally or theoretically.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 293 , 25 June 1995 , pp. 237 - 268
Copyright
© 1995 Cambridge University Press

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