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Standing radial cross-waves

Published online by Cambridge University Press:  26 April 2006

Janet M. Becker  and
John W. Miles
Janet M. Becker
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA Present address: School of Mathematics, The University of New South Wales, Kensington, NSW 2033, Australia.
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

Standing radial cross-waves in an annular wave tank are investigated using Whitham's average-Lagrangian method. For the simplest case, in which a single radial cross-wave is excited, energy is transferred from the wavemaker to the cross-wave through the spatial mean motion of the free surface, as described by Garrett (1970) for a purely transverse cross-wave in a rectangular tank. In addition, energy is transferred through spatial coupling since, in contrast to the purely transverse cross-wave in the rectangular tank, the (non-axisymmetric) radial cross-wave is three-dimensional. It is shown in an Appendix that this spatial coupling does occur for a three-dimensional cross-wave in a rectangular tank. The equations that govern this single-mode resonance are isomorphic to those that govern the Faraday resonance of surface waves in a basin of fluid subjected to vertical excitation (Miles 1984a).

It is found that the second-order Stokes-wave expansion for deep-water, standing gravity waves, which is regular for rectangular containers, may become singular for circular containers (Mack (1962) noted these resonances for finite-depth, standing gravity waves in circular containers). The evolution equations that govern two distinct types of resonant behaviour are derived: (i) 2:1 resonance between a radial cross-wave and a resonantly forces axisymmetric wave, corresponding to approximate equality among the driving frequency, a natural frequency of the directly forced wave, and twice the natural frequency of a cross-wave; (ii) 2:1 internal resonance between a radial cross-wave and a non-axisymmetric second harmonic, corresponding to approximate equality among the driving frequency, the natural frequency of a non-axisymmetric wave of even azimuthal wavenumber, and twice the natural frc.quency of the cross-wave. The axisymmetric, directly forced wave in (i) is resonantly excited and exchanges energy with the subharmonic cross-wave through spatial coupling, whereas the cross-wave in (ii) is parametrically excited and exchanges energy with the non-axisymmetric second harmonic through spatial coupling. The equations governing case (i) are shown to exhibit chaotic motions; those governing (ii) are shown to be isomorphic to the equations governing 2:1 internal resonance in the Faraday problem (Miles 1984a, §6), which have been shown to exhibit chaotic motions (Gu & Sethna 1987).

Preliminary experiments on standing radial cross-waves are reported in an Appendix, and theoretical predictions of mode stability are in qualitative agreement with these experiments. For the single-mode theory, the interaction coefficient that is a measure of the energy exchange between the wavemaker and the cross-wave is evaluated numerically for a particular wavemaker. The maximum interaction coefficient for a fixed azimuthal wavenumber of the cross-wave typically occurs for that radial mode number for which the turning point of the cross-wave radial profile is nearest the wavemaker. The present experiments for standing radial cross-waves are compared with those of Tatsuno, Inoue & Okabe (1969) for progressive radial cross-waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 471 - 499
Copyright
© 1991 Cambridge University Press

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Footnotes

With an Appendix by Janet M. Becker and Diane M. Henderson.

References

Ciliberto, S. & Gollub, J. P., 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Garrett, C. J. R.: 1970 On cross-waves. J. Fluid Mech. 41, 837–849.Google Scholar
Gu, X. M. & Sethna, P. R., 1987 Resonant surface waves and chaotic phenomena. J. Fluid Mech. 183, 543–565.Google Scholar
Havelock, T. H.: 1929 Forced surface waves on water. Phil. Mag. 8 (7), 569–576.Google Scholar
Henderson, D. M. & Miles, J. W., 1989 Single-mode Faraday waves in small containers. J. Fluid Mech. 213, 95–109.Google Scholar
Hunt, J. N. & Baddour, R. E., 1980 Nonlinear standing waves bounded by cylinders. Q. J. Appl. Maths 33 (3), 357–371.Google Scholar
Jones, A. F.: 1984 The generation of cross-waves in a long deep tank by parametric resonance. J. Fluid Mech. 138, 53–74.Google Scholar
Lighter, S. & Chen, J., 1987 Subharmonic resonance of nonlinear cross-waves. J. Fluid Mech. 183, 451–465.Google Scholar
Lin, J. D. & Howard, L. N., 1960 Non-linear standing waves in a rectangular tank due to forced oscillation. MIT Hydrodynamics Laboratory Rep. 44.Google Scholar
Longuet-Hiooins, M. S.: 1967 On the trapping of wave energy round islands. J. Fluid Mech. 29, 781–821.Google Scholar
Luke, J. C.: 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.Google Scholar
Mack, L. R.: 1962 Periodic, finite-amplitude axisymmetric gravity waves. J. Geophys. Res. 67, 829–843.Google Scholar
Mahony, J. J.: 1972 Cross-waves. Part 1. Theory. J. Fluid Mech. 55, 229–244.Google Scholar
Martin, T. (ed.) 1932 Faraday's Diary, vol. 1, pp. 350–357. London: G. Bell.
Miles, J. W.: 1984a Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302; Corrigenda. J. Fluid Mech. 154 (1985), 535.Google Scholar
Miles, J. W.: 1984b Resonantly forced motion of two quadratically coupled oscillators. Physica 13D, 247–260.Google Scholar
Miles, J. W.: 1988 Parametrically excited, standing cross-waves. J. Fluid Mech. 186, 119–127.Google Scholar
Miles, J. W. & Becker, J. M., 1988 Parametrically excited, progressive cross-waves. J. Fluid Mech. 186, 129–146.Google Scholar
Miles, J. W. & Henderson, D. M., 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143–165.Google Scholar
Miloh, T.: 1984 Hamilton's principle, Lagrange's method and ship motion theory. J. Ship Res. 28 (4), 229–237.Google Scholar
Sethna, P. R.: 1965 Vibrations of dynamical systems with quadratic nonlinearities. J. Appl. Mech. 32, 576–582.Google Scholar
Sneddon, I. H.: 1972 The Use of Integral Transforms. McGraw-Hill.
Tatsuno, M., Inoue, S. & Okabe, J., 1969 Transfiguration of surface waves. Rep. Res. Inst. Appl. Mech. 17 (59), 195–215.Google Scholar
Tsai, W., Yue, D. K. P. & Yip, K. M. K. 1990 Resonantly excited regular and chaotic motions in a rectangular wavetank. J. Fluid Mech. 216, 343–380.Google Scholar