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Nonlinear progressive free waves in a circular basin

Published online by Cambridge University Press:  26 April 2006

Peter J. Bryant
Peter J. Bryant
Affiliation:
Mathematics Department, University of Canterbury, Christchurch, New Zealand
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Abstract

A nonlinear analysis is presented of waves propagating around the free surface of water contained in a circular basin of finite uniform depth. The property that distinguishes these waves from unidirectional gravity waves is the occurrence of low-order resonant interactions between the components composing them. Steady waves in the neighbourhood of resonance are calculated in the fully nonlinear problem, when it is shown that multiple families of free-wave solutions, each family having a different set of resonating wave components, are associated with each of the water depths at which resonance occurs. Linear stability calculations indicate that most steady waves dominated by resonating wave components are linearly unstable. The nonlinear time evolution of perturbed waves is calculated as a check on the linear stability results, leading to doubts about the relevance of the linear prediction when marginal instability is said to occur.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 205 , August 1989 , pp. 453 - 467
Copyright
© 1989 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Bryant, P. J. 1985 Doubly periodic progressive permanent waves in deep water. J. Fluid Mech. 161, 27–42.Google Scholar
Bryant, P. J. 1988 Nonlinear waves in circular basins. Nonlinear Water Waves (ed. K. Horikawa & H. Maruo). Springer-Verlag.
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water. I. Weakly nonlinear waves. Stud. Appl. Maths 60, 183–210.Google Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity-capillary waves on deep water. II. Numerical results for finite amplitude. Stud. Appl. Maths 62, 95–111.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.
Miles, J. W. 1976 Nonlinear surfaces waves in closed basins. J. Fluid Mech. 75, 419–448.Google Scholar
Miles, J. W. 1984a Internally resonant surface waves in a circular cylinder. J. Fluid Mech. 149, 1–14.Google Scholar
Miles, J. W. 1984b Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.Google Scholar
Schwartz, L. W. & Vanden-Broeck, J. M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 119–139.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar