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Resonant oscillations in shallow water with small mean-square disturbances

Published online by Cambridge University Press:  21 April 2006

J. G. B. Byatt-Smith
J. G. B. Byatt-Smith
Affiliation:
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh EH9 3JZ, UK
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Abstract

The ordinary differential equation $\epsilon^2 \ddot{y} + \epsilon^2 \delta \dot{y} = y^2 - \cos t - 1 - c$ which represents forced water waves on shallow water near resonance is considered when the dispersion ε and the constant c are small. Asymptotic and numerical methods are used to show that solutions which are bounded for all finite t can exist only if $c > - 2^{-\frac{2}{3}}\epsilon^{\frac{4}{3}}\times 1.4664 \ldots $.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 193 , August 1988 , pp. 369 - 390
Copyright
© 1988 Cambridge University Press

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