A Fifth‐Order Stokes Theory for Steady Waves
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 111, Issue 2
Abstract
An alternative Stokes theory for steady waves in water of constant depth is presented where the expansion parameter is the wave steepness itself. The first step in application requires the solution of one nonlinear equation, rather than two or three simultaneously as has been previously necessary. In addition to the usually specified design parameters of wave height, period and water depth, it is also necessary to specify the current or mass flux to apply any steady wave theory. The reason being that the waves almost always travel on some finite current and the apparent wave period is actually a Dopplershifted period. Most previous theories have ignored this, and their application has been indefinite, if not wrong, at first order. A numerical method for testing theoretical results is proposed, which shows that two existing theories are wrong at fifth order, while the present theory and that of Chappelear are correct. Comparisons with experiments and accurate numerical results show that the present theory is accurate for wavelengths shorter than ten times the water depth.
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References
1.
Chappelear, J. E., “Direct Numerical Calculation of Wave Properties,” Journal of Geophysical Research, Vol. 66, 1961, pp. 501–508.
2.
Cokelet, E. D., “Steep Gravity Waves in Water of Arbitrary Uniform Depth,” Philosophical Transcripts Royal Society of London, Series A, Vol. 286, 1977, 183–230.
3.
Dailey, J. E., “Stokes V Wave Computations in Deep Water,” Journal of the Waterway, Port, Coastal, and Ocean Division, ASCE, Vol. 104, No. WW4, Nov., 1978, pp. 447–453.
4.
De, S. C., “Contributions to the Theory of Stokes Waves,” Proceedings, Cambridge Philosophical Society, Vol. 51, 1955, pp. 713–736.
5.
Fenton, J. D., “A High‐Order Cnoidal Wave Theory,” Journal of Fluid Mechanics, Vol. 94, 1979, pp. 129–161.
6.
Le Mehaute, B., Divoky, D., and Lin, A., “Shallow Water Waves: A Comparison of Theories and Experiments,” Proceedings, 11th Conference Coastal of Engineering, Vol. 1, 1968, pp. 86–107.
7.
Nishimura, H., Isobe, M., and Horikawa, K., “Higher Order Solutions of the Stokes and Cnoidal Waves,” Journal of the Faculty of Engineering, Univ. Tokyo, Series B, Vol. 34, 1977, pp. 267–293.
8.
Rienecker, M. M., and Fenton, J. D., “A Fourier Approximation Method for Steady Water Waves,” Journal of Fluid Mechanics, Vol. 104, 1981, pp. 119–137.
9.
Schwartz, L. W., “Computer Extension and Analytic Continuation of Stokes' Expansion for Gravity Waves,” Journal of Fluid Mechanics, Vol. 62, 1974, pp. 553–578.
10.
Schwartz, L. W., and Fenton, J. D., “Strongly‐nonlinear Waves,” Ann. Rev. Fluid Mech., Vol. 14, 1982, pp. 39–60.
11.
Skjelbreia, L., and Hendrickson, J., “Fifth Order Gravity Wave Theory,” Proceedings 7th Conference of Coastal Engineering, pp. 184–196.
12.
Tsuchiya, Y., and Yamaguchi, M., “Some Considerations on Water Particle Velocities of Finite Amplitude Wave Theories,” Coastal Engineering in Japan, Vol. 15, 1972, pp. 43–57.
13.
Ursell, F., “The Long‐Wave Paradox in the Theory of Gravity Waves,” Proceedings, Cambridge Philosophical Society, Vol. 49, 1953, pp. 685–694.
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Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 111 • Issue 2 • March 1985
Pages: 216 - 234
Copyright
Copyright © 1985 ASCE.
History
Published online: Mar 1, 1985
Published in print: Mar 1985
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