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Steep standing waves at a fluid interface

Published online by Cambridge University Press:  20 April 2006

James W. Rottman
James W. Rottman
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW
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Abstract

An algorithm is formulated for computing perturbation-series solutions for standing waves on the interface between two semi-infinite fluids of different but uniform densities. Using a comppter, the series solutions are computed to fifth order for a general value of r, the ratio of the density of the upper fluid to that of the lower fluid (0 ≤ r ≤ l), and to 21st order for five specific values of this ratio: r = 0, 10−3, 0·1, 5·0, 1·0. The series for the period, the energy, and the interface profile of the waves are summed using Padé approximants. The maximum wave height for each of the above five density ratios is estimated from the locations of the poles of the Padé approximants for the wave period and the wave energy. At maximum height the interface appears to be vertical at a point on the interface that is very near the crest for r = 10−3 and approaches the midpoint between the crest and the trough as r approaches 1·0.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 124 , November 1982, pp. 283 - 306
Copyright
© 1982 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. Brueckner), vol. 1, pp. 1–58. Acacemic.
Cabannes, H. (ed.) 1976 Padé Approximants Method and its Application to Mechanics. Lecture Notes in Physics, vol. 47. Springer.
Chabert-D'Hieres, G.1960 Etude du clapotis. Houille Blanche 15, 153–163.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
Edge, R. D. & Walters, G. 1964 The period of standing gravity waves of largest amplitude on water. J. Geophys. Res. 69, 1674–1675.Google Scholar
Fultz, D. 1962 An experimental note on finite-amplitude standing gravity waves. J. Fluid Mech. 13, 193–212.Google Scholar
Graves-Morris, P. R. 1973 Padé Approximants and their Applications. Academic.
Holyer, J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.Google Scholar
Hunt, J. N. 1961 Interfacial waves of finite amplitude. Houille Blanche 16, 515–531.Google Scholar
Longuet-Higgins, M. S. 1973 On the form of the highest progressive and standing waves in deep water. Proc. R. Soc. Lond. A 331, 445–456.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157–174.Google Scholar
Penney, W. G. & Price, A. T. 1952 Some gravity wave problems in the motion of perfect liquids. Part II. Finite periodic stationary gravity waves in a perfect liquid. Phil. Trans. R. Soc. Lond. A 244, 251–284.Google Scholar
Rayleigh, J. W. S. 1915 Deep water waves, progressive or stationary, to the third order of approximation. Proc. R. Soc. Lond. A 91, 345–353.Google Scholar
Saffman, P. G. & Yuen, H. C. 1979 A note on numerical computations of large amplitude standing waves. J. Fluid Mech. 95, 707–715.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Schwartz, L. W. & Whitney, A. K. 1977 A high order series solution for standing water waves. 6th Australasian Hydraul. and Fluid Mech. Conf., Preprints of Papers Part I, pp. 356–359.
Schwartz, L. W. & Whitney, A. K. 1981 A semianalytic solution for nonlinear standing waves in deep water. J. Fluid Mech. 107, 147–171.Google Scholar
Sekerzh-Zenkovich, Ya. I. 1947 On the theory of standing waves of finite amplitude on the surface of a heavy fluid (in Russian). Dokl. Akad. Nauk SSSR 58, 551–553.Google Scholar
Sekerzh-Zenkovich, Ya. I. 1951 On the theory of standing waves of finite amplitude on the surface of a heavy fluid of finite depth (in Russian). Izv. Akad. Nauk SSSR, Ser. Geogra. Geofiz. 15, 57–73.Google Scholar
Sekerzh-Zenkovich, Ya. I. 1961 Free finite oscillations of the surface of separation of two unbounded heavy fluids of different densities, (in Russian). Trudy Morsk. Gidrofiz. Inst. 23, 3–43.Google Scholar
Tadjbaksh, I. & Keller, J. B. 1960 Standing surface waves of finite amplitude. J. Fluid Mech. 8, 442–451.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218, 44–59.Google Scholar
Thorpe, S. A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32, 489–528.Google Scholar
Whitney, A. K. 1971 The numerical solution of unsteady free surface flows by conformal mapping. In Proc. 2nd Int. Conf. Numerical Methods in Fluid Dynamics (ed. M. Holt). Lecture Notes in Physics, vol. 8, pp. 458–462. Springer.