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A high-order cnoidal wave theory

Published online by Cambridge University Press:  19 April 2006

J. D. Fenton
J. D. Fenton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, Australia 2033
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Abstract

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.

Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 94 , Issue 1 , 11 September 1979 , pp. 129 - 161
Copyright
© 1979 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. New York: Dover.
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. A 224, 448–460.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. A 286, 183–230.Google Scholar
De, S. C. 1955 Contributions to the theory of Stokes waves. Proc. Camb. Phil. Soc. 51, 713–736.Google Scholar
Dean, R. C. 1970 Recent results obtained from a numerical wave theory for highly nonlinear water waves. Proc. Symp. on Long Waves, pp. 129–152. University of Delaware, Newark, Delaware.
Fenton, J. D. 1972 A ninth-order solution for the solitary wave. J. Fluid Mech. 53, 257–271.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Table of Integrals, Series and Products. New York: Academic.
Iwagaki, Y. & Sakai, T. 1970 Horizontal water particle velocity of finite amplitude waves. Proc. 12th Conf. Coastal Engng. 1, 309–325.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. (5) 39, 422–443.Google Scholar
Laitone, E. V. 1960 The second approximation to cnoidal and solitary waves. J. Fluid Mech. 9, 430–444.Google Scholar
Laitone, E. V. 1965 Series solutions for shallow water waves. J. Geophys. Res. 70, 995–998.Google Scholar
Le Méhauté, B., Divoky, D. & Lin, A. 1968 Shallow water waves: a comparison of theories and experiments. Proc. 11th Conf. Coastal Engng. 1, 86–107.Google Scholar
Lighthill, M. J. 1949 A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag. 40, 1179–1201.Google Scholar
Littman, W. 1957 On the existence of periodic waves near critical speed. Commun. Pure Appl. Math. 10, 241–269.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. A 342, 157–174.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave. II. Proc. R. Soc. A 340, 471–493.Google Scholar
Monkmeyer, P. L. 1970 A higher order theory for symmetrical gravity waves. Proc. 12th Conf. Coastal Engng. 1, 543–561.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Skjelbreia, L. & Hendrickson, J. 1961 Fifth order gravity wave theory. Proc. 7th Conf. Coastal Engng, pp. 184–196.