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Modulation of three resonating gravity–capillary waves by a long gravity wave

Published online by Cambridge University Press:  26 April 2006

Karsten Trulsen  and
Chiang C. Mei
Karsten Trulsen
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Chiang C. Mei
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

We consider a resonant triad of gravity–capillary waves, riding on top of a much longer gravity wave. The long-wave phase is assumed to vary on the same scale as the slow modulation of the short waves. Envelope equations are first deduced in the Lagrangian description. By perturbation analysis for a weak long wave, we then find that the long wave can resonate the natural modulation oscillations of the triad envelope, giving rise to various bifurcations in the Poincaré map. Numerical integration for a stronger long wave reveals that chaos can emerge from these bifurcations. The bifurcation criterion of Chen & Saffman (1979) for collinear Wilton's ripples is generalized to arbitrary non-collinear triads, and is found to play an important role as a criterion for the onset of chaotic behaviour.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 290 , 10 May 1995 , pp. 345 - 376
Copyright
© 1995 Cambridge University Press

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