Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T04:43:19.147Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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