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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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References

Benjamin, T. B. & Ligeteill, M. J. 1954 On cnoidal waves and bores. Proc. Roy. Soc. A 224, 448–460Google Scholar
Boussinesq, J. 1871 Théorie de I'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C.R. Acad. Sci., Paris, 1871, p. 755.Google Scholar
Byatt-Smite, J. G. B. 1970 A n exact integral equation for steady surface waves. Proc. Roy. Soc. A 315, 405–418.Google Scholar
Friedrices, K. O. 1948 On the derivation of the shallow water theory. Appendix to The formation of breakers and bores, by J. J. Stoker. Commun. Pure Appl. Math. 1, 81–85.Google Scholar
Grimseaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611–622.Google Scholar
Hunt, J. N. 1955 On the solitary wave of finite amplitude. Houille Blanche, 10, 197–203.Google Scholar
Korteweg, D. J. & DE Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave. Phil. Mag., 39 (5), 422–443Google Scholar
Laitone, E. V. 1960 The second approximation to cnoidal and solitary waves. J. Fluid mech, 9, 430–444Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lenau, C. W. 1966 The solitary wave of maximum amplitude. J. Fluid Mech. 26, 309–320.Google Scholar
Ligethill, M. J. 1949 A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag., 40, 1179–1201.Google Scholar
Long, R. R. 1956 Solitary waves in the one- and two-fluid systems. Tullus, 8, 460–471.Google Scholar
Mccowan, J. 1891 On the solitary wave. Phil. Mag., 32 (5), 45–68.Google Scholar
Mccowan, J. 1894 On the highest wave of permanent type. Phil. Mag. 38 (5), 351–358.Google Scholar
Milne-Thomson, L. M. 1964 An exact integral equation for the solitary wave. Rev. Roum. Sci. Techn. Mec. Appl. 9, 1189–1194.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn, Macmillan.
Price, R. K. 1971 Bottom drift for the solitary wave, J. Geophyb. Res. 76, 1600–1602.Google Scholar
Rayleige, Lord 1876 On waves. Phys. Mag. 1 (6), 257–279. (See also Papers, vol. 1, p. 251. Cambridge University Press.)Google Scholar
Russell, J. S. 1844 Report on Wava. British Association Reports.
Seanks, D. 1955 Nonlinear transformations of divergent and slowly convergent sequences. J Math. Phys. 34, 1–42Google Scholar
Strelkoff, T. 1971 An exact numerical solution of the solitary wave. Proc. 2nd Int. Conf. Num Methods fluid dyn. Springer.
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. SOC. 49, 685–694.Google Scholar