Boundary Integral Equation Method for Limit Surface Gravity Waves
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 113, Issue 4
Abstract
A numerical solution to periodic nonlinear irrotational surface gravity waves on a horizontal sea floor is developed using an iterative Boundary Integral Equation Method (BIEM). This solution technique is subsequently applied to determine the characteristics of limit waves for which the wave crest theoretically ceases to be rounded and become angled with an included angle of 120°. The analysis includes waves with and without mean currents and can also be used to estimate the wave set‐down. Solutions with zero mean current are parameterized as functions of relative water depth and presented in graphs. In the cases with current, the results are given in terms of the relative water depth and a dimensionless current speed.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Baker, G. R., Meiron, D. I., and Orzag, S. A. (1982). “Generalized Vortex Methods for Free‐Surface Flow Problems.” JFM 123, 477–501.
2.
Benjamin, T. B., and Feir, J. E. (1967). “The disintegration of wave trains on deep water, part I. theory.” JFM 27, 417–430.
3.
Boussinesq, J. (1877). “Essai sur la theorie des eaux courantes.” Memories Presents Par Divers Savants, Institu de France, Academie des Sciences, 23.
4.
Brebbia, C. A., and Wrobel, L. (1982). “Some applications of the boundary element method for potential problems.” Proc., 4th Inter. Conf., Hannover, Germany, Jun., 219–228.
5.
Cokelet, E. D. (1977). “Steep gravity waves in water of arbitrary uniform depth.” Phil. Trans. Roy. Soc., Ser. A 286, 769–774.
6.
Cruse, T. A., and Rizzo, F. J. (ed.) (1975). “Boundary‐integral equation method: computational applications in applied mechanics.” AMD, ASME.
7.
Dean, R. G. (1974). “Evaluation and development of water wave theories for engineering application.” Special Report No. 1, Vols. I and II. U.S. Army, Coastal Engrg. Research Center, Fort Belvoir, Va.
8.
Fenton, J. D., and Rienecker, M. M. (1980). “Accurate numerical solutions for nonlinear waves.” Proc. Coastal Engrg., Vol. I, 50–69.
9.
Jonsson, I. G., Skougaard, C., and Wang, J. D. (1970). “Interaction between waves and currents.” Proc., 12th Coastal Engrg. Conf., Washington, D.C., 489–507.
10.
Jonsson, I. G., and Wang, J. D. (1978). “Current‐depth refraction of water waves.” Inst., of Hydrodynamics and Hydr. Engrg., Technical Univ. of Denmark, Ser. 18.
11.
Korteweg, D. J., and DeVries, G. (1895). “On the change of form of long waves advancing in a rectangular canal on a new type of long stationary waves.” Philosophical Magazine, Ser. S. (London, Dublin and Edinburgh) 39, 442.
12.
Le Mehaute, B., Lu, C. C., and Ulmer, E. W. (1984). “A parameterized solution to the nonlinear wave problem.” J. Waterway Port, Coastal, and Ocean Engrg., ASCE, 110(3), Aug., 309–320.
13.
Liggett, J. A., and Liu, P. L.‐F. (1983). The boundary integral equation method for porous media flow. George Allen and Unwin, London, U.K.
14.
Lu, C. C. (1985). “A numerical solution to a nonlinear wave problem using boundary integral equation method with analysis of limit cases,” dissertation presented to Rosenstiel School of Marine and Atmospheric Science, University of Miami, at Miami, Fla., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
15.
Miche, R. (1944). “Mouvements ondulatoires des mers en profondeur constante on decroisante.” Annales des Ponts et Chausees, 25–78, 270–292, 369–406, 1311–1364.
16.
Michell, J. H. (1895). “On the highest waves in water.” Phil. Mag. 36, 430–435.
17.
Rienecker, M. M., and Fenton, J. D. (1981). “A fourier approximation method for steady water waves.” JFM. Vol. 104, 119–137.
18.
Schwartz, L. W. (1974). “Computer extension and analytic continuation of Stokes' expansion for gravity waves.” JFM. Vol. 62, 553–578.
19.
Shore Protection Manual, 1984. Vol. 1. CERC, U.S. Army Corps of Engrs., Fort Belvoir, Va.
20.
Stokes, G. G. (1847). “On the theory of oscillatory waves.” Trans. Camb. Phil. Soc. 8, 441–455.
21.
Stokes, G. G. (1880). “Supplement to a paper on the theory of oscillatory waves.” Mathematical and Physical Paper 1. Cambridge, England, 314–326.
22.
Toland, J. F. (1978). “On the existence of a wave of greatest height and Stokes' conjecture.” Proc. Royal Soc., London, Ser. A 363, 469–485.
23.
Williams, J. M. (1981). “Limiting gravity waves in water of finite depth.” Phys. Trans. Roy. Soc., Ser. A 302, 139–187.
Information & Authors
Information
Published In
Copyright
Copyright © 1987 ASCE.
History
Published online: Jul 1, 1987
Published in print: Jul 1987
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.