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A ninth-order solution for the solitary wave

Published online by Cambridge University Press:  29 March 2006

John Fenton
John Fenton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

Several solutions for the solitary wave have been attempted since the work of Boussinesq in 1871. Of the approximate solutions, most have obtained series expansions in terms of wave amplitude, these being taken as far as the third order by Grimshaw (1971). Exact integral equations for the surface profile have been obtained by Milne-Thomson (1964,1968) and Byatt-Smith (1970), and these have been solved numerically. In the present work an exact operator equation is developed for the surface profile of steady water waves. For the case of a solitary wave, a form of solution is assumed and coefficients are obtained numerically by computer to give a ninth-order solution. This gives results which agree closely with exact numerical results for the surface profile, where these are available. The ninth-order solution, together with convergence improvement techniques, is used to obtain an amplitude of 0.85for the solitary wave of greatest height and to obtain refined approximations to physical quantities associated with the solitary wave, including the surface profile, speed of the wave and the drift of fluid particles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 53 , Issue 2 , 23 May 1972 , pp. 257 - 271
Copyright
© 1972 Cambridge University Press

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