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Computer extension and analytic continuation of Stokes’ expansion for gravity waves

Published online by Cambridge University Press:  29 March 2006

Leonard W. Schwartz
Leonard W. Schwartz
Affiliation:
Department of Aeronautics and Astronautics, Stanford University, California
Present address: N.A.S.A. Ames Research Center, Moffett Field, California.
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Abstract

Stokes’ infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes’ expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Padé approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth. An alternative expansion procedure proposed by Havelock for the computation of waves short of the highest has been reconsidered and found to be defective.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 62 , Issue 3 , 11 February 1974 , pp. 553 - 578
Copyright
© 1974 Cambridge University Press

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References

Baker, G. A. 1965 The theory and application of the Padé approximant method. In Advances in Theoretical Physics (ed. K. Breuckner), vol. 1, p. 1. Academic.
Benjamin, T. B. & Feir, J. E. 1967 J. Fluid Mech. 27, 417.
Boussinesq, J. 1871 C. R. Acad. Sci., Paris, p. 755.
Chappelear, J. E. 1959 U.S. Army Corps Engrs, Beach Erosion Bd, Tech. Memo. no. 116.
De, S. C. 1955 Proc. Camb. Phil. Soc. 51, 713.
Domb, C. & Sykes, M. F. 1957 Proc. Roy. Soc. A, 240, 214.
Gerstner, F. J. V. 1804 Abh. bohm. Ges. Wiss. 1 (3) 1.
Grant, M. 1973 J. Fluid Mech. 59, 257.
Havelock, T. H. 1919 Proc. Roy. Soc. A, 95, 38.
Keller, J. B. 1948 Comm. Appl. Math. 1, 323.
Korteweg, D. J. & de Vries, G. 1895 Phil. Mag. 39 (5), 422.
Krasovskii, Yu. P. 1960 Dokl. Acad. Nauk SSSR, 130, 1237.
Michell, J. H. 1893 Phil. Mag. 36 (5), 430.
Nekrasov, A. I. 1921 Izv. Ivanovo-Voznesensk. Politekhn. Inst. 3, 52.
Schwartz, L. W. 1972 Ph.D. dissertation, Stanford University.
Shanks, D. 1955 J. Math. & Phys. 34, 1.
Stokes, G. G. 1849 Trans. Camb. Phil. Soc. 8, 441.
Stokes, G. G. 1880 Mathematical and Physical Papers, vol. 1, p. 314. Cambridge University Press.
Van Dyke, M. 1970 J. Fluid Mech. 44, 365.
Wehausen, J. V. 1965 Free surface flows. In Research Frontiers in Fluid Dynamics (ed. R. J. Seeger & G. Temple), p. 534. Interscience.
Whitham, G. B. 1967 J. Fluid Mech. 27, 399.
Wilton, J. R. 1914 Phil. Mag. 27 (6), 385.
Yamada, H. 1957a Rep. Res. Inst. Appl. Mech., Kyushu University, 5 (18), 37.
Yamada, H. 1957b Rep. Res. Inst. Appl. Mech., Kyushu University, 5 (18), 53.
Yamada, H. & Shiotani, T. 1968 Bull. Disas. Prev. Res. Inst., Kyoto University, 18 (135), 1.