Finite Analytic Method for Mild-Slope Wave Equation
Publication: Journal of Engineering Mechanics
Volume 122, Issue 2
Abstract
A difference scheme for the mild-slope wave equation that governs the combined diffraction and refraction of nearshore waves is derived based on the finite analytic method. The nine-point scheme expresses the value of the dependent variable at the central point of a rectangular grid element as a linear combination of its values at the surrounding nodes. The coefficients of the linear combination depend on the local wave number as well as the width-to-length ratio of the grid element and are approximated, by Taylor expansion, as the polynomials of the relative mesh size with coefficients being functions of the width-to-length ratio of the grid element. Performance of the scheme is studied by comparing the numerical results with the exact solution for a Dirichlet problem and with the solution by separation of variables for the forced oscillation in a square basin. The scheme is also applied to the computation of wave diffraction by the spherical shoal in a wave channel, on which carefully measured data in laboratory are available for comparison.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Berkhoff, J. C. W. (1972). “Computation of combined refraction and diffraction.”Proc., 13th Coast. Engrg. Conf., ASCE, New York, N.Y., 471–490.
2.
Chen, C. J., and Chen, H. C.(1984). “Finite analytic method for unsteady two dimensional Navier-Stokes equations.”J. Comp. Physics, 53, 209–226.
3.
Chen, C. J., and Li, P. (1979). “Finite differential method in heat conduction—application of analytic solution technique.”Paper No. 79-WA/HT-50, ASME, New York, N.Y.
4.
Chwang, A. T., and Chen, H. C.(1987). “Optimal finite difference method for potential flows.”J. Engrg. Mech., ASCE, 113(11), 1759–1773.
5.
Dalrymple, R. A., Kirby, J. T., and Hwang, P. A.(1984). “Wave diffraction due to areas of energy dissipation.”J. Wtrwy., Port, Coast. and Oc. Engrg., ASCE, 110(1), 67–79.
6.
Hildebrand, F. B. (1987). Introduction to numerical analysis, Dover Publications, New York, N.Y.
7.
Ito, Y., and Tanimoto, K. (1972). “Wave diffraction in the region with caustics.”Proc., Coast. Engrg., 19, 325–329 (in Japanese).
8.
Madsen, P. A., Murray, R., and Sorensen, O. R.(1991). “A new form of the Boussinesq equations with improved linear dispersion characteristics.”Coast. Engrg., 15, 371–388.
9.
Mei, C. C. (1989). The applied dynamics of ocean surface waves . World Scientific, Singapore.
10.
Peregrine, D. H.(1967). “Long wave on beaches.”J. Fluid Mech., 27, 816–827.
11.
Radder, A. C.(1979). “On the parabolic equation method for water wave propagation.”J. Fluid Mech., 95, 159–176.
12.
Sleath, J. F. A. (1984). Sea bed mechanics, John Wiley & Sons, New York, N.Y.
13.
Yu, X., Isobe, M., and Watanabe, A.(1992a). “Numerical computation of wave transformation on beaches.”Coast. Engrg. in Japan, 35(1), 1–19.
14.
Yu, X., Isobe, M., and Watanabe, A.(1992b). “Finite element solution of wave field around structures in nearshore zone.”Coast. Engrg. in Japan, 35(1), 21–33.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 122 • Issue 2 • February 1996
Pages: 109 - 115
Copyright
Copyright © 1996 American Society of Civil Engineers.
History
Published online: Feb 1, 1996
Published in print: Feb 1996
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.