Intercomparison of Truncated Series Solutions for Shallow Water Waves
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 117, Issue 2
Abstract
The relationship between truncated Fourier series solutions of several different long‐wave evolution equations is explored. The models are all designed to represent the same physics in the asymptote of small nonlinearity and frequency dispersion, and yet give different numerical solutions for given values of dimensionless parameters. It is found that the problem of obtaining a steady solution in a coupled‐mode model of shallow‐water wave evolution is related more to the problem of properly choosing the corresponding time‐dependent evolution equation than to the problem of truncating the infinite series representation of the solution to that equation. In the process, a particular lowest‐order coupled‐mode model is related to a particular modified form of the Korteweg‐deVries equation. It is shown that the existing lowest‐order model may be inherently inaccurate in the case of relatively high waves, due to the nonphysical behavior of the form of the nonlinear term inherent in that model. Finally, solitary and cnoidal wave solutions are given for the entire family of modified Korteweg‐deVries equations, and their properties are compared.
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References
1.
Benjamin, T. B., Bona, J. L., and Mahony, J. J. (1972). “Model equations for long waves in nonlinear dispersive systems.” Phil. Trans. Roy. Soc., Royal Society of London, A272, 47–78.
2.
Cayley, A. (1895). An elementary treatise on elliptic functions. Dover, N.Y.
3.
Elgar, S., Freilich, M. H., and Guza, R. T. (1990). “Recurrence in truncated Boussinesq models for nonlinear waves in shallow water.” J. Geophys. Res., 95(c7), 11547–11556.
4.
Fenton, J. (1972). “A ninth‐order solution for the solitary wave.” J. Fluid Mech., 53, 257–271.
5.
Flick, R. E., Guza, R. T., and Inman, D. L. (1981). “Elevation and velocity measurements of laboratory shoaling waves.” J. Geophys. Res., 86(c5), 4149–4160.
6.
Freilich, M. H., and Guza, R. T. (1984). “Nonlinear effects on shoaling surface gravity waves.” Phil. Trans. Roy. Soc. London, Royal Society of London, A311, 1–41.
7.
Goring, D. (1978). “Tsunamis—The propagation of long waves onto a shelf.” Report KH‐R‐38, W. M. Keck Laboratory of Hydr. and Water Resour., California Inst. of Tech., Pasadena, Calif.
8.
Hammack, J., Scheffner, N., and Segur, H. (1989). “Two‐dimensional periodic waves in shallow water.” J. Fluid Mech., 209(Dec.), 567–589.
9.
Liu, P. L.‐F., Yoon, S. B., and Kirby, J. T. (1985). “Nonlinear refraction‐diffraction of waves in shallow water.” J. Fluid Mech., 153(Apr.), 184–201.
10.
Mei, C. C. (1983). The applied dynamics of ocean surface waves, John Wiley and Sons, New York, N.Y.
11.
Synolakis, C. E. (1990). “Generation of long waves in laboratory,” J. Wtrway., Port, Coast., and Oc. Engrg., ASCE, 116(2), 252–266.
12.
Yoon, S. B., and Liu, P. L.‐F. (1989). “Stem waves along breakwater.” J. Wtrway., Port, Coast., and Oc. Engrg., ASCE, 115(Sep.), 635–648.
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Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 117 • Issue 2 • March 1991
Pages: 143 - 155
Copyright
Copyright © 1991 ASCE.
History
Published online: Mar 1, 1991
Published in print: Mar 1991
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