Split‐Step Fourier Algorithm for Water Waves
Publication: Journal of Engineering Mechanics
Volume 116, Issue 2
Abstract
A parabolic model for propagation of water waves, implemented with an efficient split‐step Fourier transform algorithm, is presented. The split‐step Fourier transform algorithm, widely used in acoustics to solve the parabolic form of the Helmholtz equation and modified to approximately account for wave number dependency in the direction of propagation, has been successfully applied to model refraction and diffraction of water waves. The solution algorithm is exact for a constant depth ocean and gives a good approximation for a wave spectrum on a uniform slope. The results of the model for an elliptical shoal on a uniform slope are presented for both linear and nonlinear wave dispersion relationships and show good agreement with experimental results. The wide‐angle capabilities of the model are also examined for a circular shoal of elliptical cross section on a flat bottom. The algorithm is very efficient and stable, permitting large computational step sizes in the direction of propagation.
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References
1.
Bender, C. M., and Orszag, S. A. (1978). Advanced mathematical methods for scientists and engineers. McGraw‐Hill Book Co., New York, N.Y., 484–539.
2.
Berkhoff, J. C. W. (1972). “Computation of combined refraction‐diffraction.” Proc. 13th Int. Conf. Coastal Engrg., ASCE, 417–490.
3.
Berkhoff, J. C. W., Booij, N., and Radder, A. C. (1982). “Verification of numerical wave propagation models for simple harmonic linear waves.” Coastal Eng., 6, 255–279.
4.
Dalrymple, R. A., and Kirby, J. T. (1988). “Very wide angle water wave models and wave diffraction.” J. Fluid Mech., 192, 33–50.
5.
Dalrymple, R. A., et al. (1989). “Very wide angle water wave models and wave diffraction. Part 2. Irregular bathymetry.” J. Fluid Mech., 201, 299–322.
6.
Feit, M. D., and Fleck, J. A., Jr. (1978). “Light propagation in graded‐index optical fibers.” Appl. Opt., 17(24), 3990–3998.
7.
Fishman, L., and McCoy, J. J. (1985). “A new class of propagation models based on a factorization of the Helmholtz equation.” Geophys. J. Royal Astron. Soc., 80, 439–461.
8.
Fleck, J. A., Jr., Morris, J. R., and Feit, M. D. (1976). “Time‐dependent propagation of high energy laser beams through the atmosphere.” Appl. Phys., 10, 129–160.
9.
Greene, R. R. (1984). “The rational approximation to the acoustic wave equation with bottom interaction.” J. Acoust. Soc. Am., 76, 1764–1773.
10.
Hardin, R. H., and Tappert, F. D. (1973). “Applications of the split‐step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations.” SIAM Rev., 15, 423.
11.
Hedges, T. S. (1976). “An empirical modification to linear wave theory.” Proc., Institute Civil Engineers, 61, 575–579.
12.
Kirby, J. T. (1986). “Rational approximations in the parabolic equation method for water waves.” Coastal Eng., 10, 355–378.
13.
Kirby, J. T., and Dalrymple, R. A. (1983). “A parabolic equation for the combined refraction‐diffraction of Stokes waves by mildly varying topography.” J. Fluid Mech., 136, 453–466.
14.
Kirby, J. T., and Dalrymple, R. A. (1984). “Verification of a parabolic equation for propagation of weakly‐nonlinear waves.” Coastal Eng., 8, 219–232.
15.
Kirby, J. T., and Dalrymple, R. A. (1986). “An approximate model for nonlinear dispersion in monochromatic wave propagation models.” Coastal Eng., 9, 545–561.
16.
Knightly, G. H., Lee, D., and St. Mary, D. F. (1987). “A higher‐order parabolic wave equation.” J. Acoust. Soc. Am., 82, 580–587.
17.
Leontovich, M. A., and Fock, V. A. (1965). “Solution of the problem of propagation of electromagnetic waves along the earth's surface by the method of parabolic equations.” Electromagnetic Diffraction and Propagation Problems, Pergamon Press, Inc., Elmsford, N.Y., 213–234.
18.
Liu, P. L.‐F., and Tsay, T.‐K. (1984). “Refraction‐diffraction model for weakly nonlinear water waves.” J. Fluid Mech., 141, 265–274.
19.
McDaniel, S. T. (1975). “Parabolic approximations for underwater sound propagation.” J. Acoust. Soc. Am., 58(6), 1178–1185.
20.
Radder, A. C. (1979). “On the parabolic equation method for water‐wave propagation.” J. Fluid Mech., 95, 159–176.
21.
St. Mary, D. F., Lee, D., and Botseas, G. (1987). “A modified wide angle parabolic wave equation.” J. of Comput. Phys., 71, 304–315.
22.
Thomson, D. J., and Chapman, N. R. (1983). “A wide‐angle split‐step algorithm for the parabolic equation.” J. Acoust. Soc. Am., 74(6), 1848–1854.
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Copyright © 1990 ASCE.
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Published online: Feb 1, 1990
Published in print: Feb 1990
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