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On the Korteweg—de Vries equation for a gradually varying channel

Published online by Cambridge University Press:  19 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla
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Abstract

Two integral invariants of Shuto's (1974) generalization of the Korteweg—de Vries equation for a unidirectional wave in a channel of gradually varying breadth b and depth d are derived. The second-order (in amplitude) invariant measures energy, as expected, but the first-order invariant measures mass divided by b½d¼; accordingly, mass is conserved only if either the mean free-surface displacement vanishes or bd½ is constant. This difficulty is associated with the reflected wave that is excited by the channel variation but neglected in the KdV approximation. The total mass flux is resolved into a primary (KdV) flux and a residual flux that is proportional to the mean displacement of the primary wave. The reflected wave associated with the residual flux is constructed by neglecting both nonlinearity and dispersion (even though both are significant for the primary wave). The results are applied to a slowly varying cnoidal wave, which is fully determined by conservation of mass and energy and the known results for a uniform channel, and to a slowly varying solitary wave, for which mass is not conserved and both trailing and reflected residuals are excited. The development of the Boussinesq equations for a gradually varying channel and their reduction to Shuto's equation are sketched in an appendix.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 91 , Issue 1 , 9 March 1979 , pp. 181 - 190
Copyright
© 1979 Cambridge University Press

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