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A Fourier approximation method for steady water waves

Published online by Cambridge University Press:  20 April 2006

M. M. Rienecker and
J. D. Fenton
M. M. Rienecker
Affiliation:
School of Mathematics, University of New South Wales, P.O. Box 1, Kensington, N.S.W., Australia, 2033
J. D. Fenton
Affiliation:
School of Mathematics, University of New South Wales, P.O. Box 1, Kensington, N.S.W., Australia, 2033
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Abstract

A method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed is presented. No analytical approximations are made. A finite Fourier series, similar to Dean's stream function series, is used to give a set of nonlinear equations which can be solved using Newton's method. Application to laboratory and field situations is emphasized throughout. When compared with known results for wave speed, results from the method agree closely. Results for fluid velocities are compared with experiment and agreement found to be good, unlike results from analytical theories for high waves.

The problem of shoaling waves can conveniently be studied using the present method because of its validity for all wavelengths except the solitary wave limit, using the conventional first-order approximation that on a sloping bottom the waves at any depth act as if the bed were horizontal. Wave period, energy flux and mass flux are conserved. Comparisons with experimental results show good agreement.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 104 , March 1981, pp. 119 - 137
Copyright
© 1981 Cambridge University Press

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References

Chaplin, J. R. 1980 Developments of stream-function wave theory. Coastal Engng. 3, 179–205.Google Scholar
Chappelear, J. E. 1961 Direct numerical calculation of wave properties. J. Geophys. Res. 66, 501–508.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. G. 1965 Stream function representation of nonlinear ocean waves. J. Geophys. Res. 70, 4561–4572.Google Scholar
Dean, R. G. 1970a Relative validities of water wave theories. J. Waterways & Harbors Div. A.S.C.E. 96, 105–119.Google Scholar
Dean, R. G. 1970b Recent results obtained from a numerical wave theory for highly non-linear water waves. Proc. Symp. on Long Waves, pp. 129–152. University of Delaware, Newark.
Dean, R. G. 1974 Evolution and development of water wave theories for engineering application. Vols I and II. Special Report no. 1, U.S. Army Coastal Engng Res. Center, Fort Belvoir, Virginia.
Eagleson, P. S. 1956 Properties of shoaling waves by theory and experiment. Trans. Am. Geophys. Union 37, 565–572.Google Scholar
Fenton, J. D. 1979 A high-order cnoidal wave theory. J. Fluid Mech. 94, 129–161.Google Scholar
Hansen, J. B. & Svendsen, I. A. 1979 Regular waves in shoaling water: experimental data, Inst. of Hydrodynamics and Hydraulic Engng. Tech. Univ. of Denmark, series paper no. 21.Google Scholar
Iwagaki, Y. & Sakai, T. 1970 Horizontal water particle velocity of finite amplitude waves. Proc. 12th Conf. Coastal Engng. 1, 309–325.Google Scholar
Koh, R. C. Y. & Le Méhauté, B. 1966 Wave shoaling. J. Geophys. Res. 71, 2005–2012.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
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. A 245, 535–581.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. Roy. Soc. A 340, 471–493.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. 184–196.Google Scholar
Stiassnie, M. & Peregrine, D. H. 1980 Shoaling of finite-amplitude surface waves on water of slowly-varying depth. J. Fluid Mech. 97, 783–805.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Svendsen, I. A. & Brink-Kjaer, O. 1972 Shoaling of cnoidal waves. Proc. 13th Conf. Coastal Engng. 1, 365–384.Google Scholar
Vanden-Broeck, J.-M. & Schwartz, L. W. 1979 Numerical computation of steep gravity waves in shallow water. Phys. Fluids 22, 1868–1871.Google Scholar