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The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

Published online by Cambridge University Press:  26 April 2006

T. R. Marchant  and
N. F. Smyth
T. R. Marchant
Affiliation:
Department of Mathematics, The University of Wollongong, PO Box 1144. Wollongong, NSW 2500, Australia
N. F. Smyth
Affiliation:
Department of Mathematics, The University of Wollongong, PO Box 1144. Wollongong, NSW 2500, Australia
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Abstract

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 221 , December 1990 , pp. 263 - 287
Copyright
© 1990 Cambridge University Press

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