Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T01:09:21.097Z Has data issue: false hasContentIssue false
  • English
  • Français

Faraday waves in 2:1 internal resonance

Published online by Cambridge University Press:  26 April 2006

Diane M. Henderson  and
John W. Miles
Diane M. Henderson
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla, CA 92093, USA
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Experiments are reported on Faraday waves in a circular cylinder, which are internally resonant with either the subharmonic mode (with frequency ¼ that of the forcing) or the superharmonic mode (with frequency equal to that of the forcing). For subharmonic resonance both modes achieved comparable amplitudes that were steady, or were modulated with one or two periods, or exhibited quasi-periodic or chaotic motions. A stability map of these responses is presented. Theoretical predictions of linear stability and growth rates are tested. Measurements of steady amplitudes, limit-cycle frequencies and wave slopes at breaking are presented. Some of the measured phase-plane trajectories are shown to have theoretical counterparts. For superharmonic resonance the amplitude of the superharmonic was never comparable with that of the Faraday wave. For low modes an energy exchange occurred during the initial period of growth, and a precession instability sometimes developed. For high modes for which both frequencies and wavenumbers are in a 2:1 ratio, superharmonic resonance occurred irreproducibly; it appeared to be overwhelmed by 1:1 interactions among the possible Faraday-wave modes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 449 - 470
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.)

References

Becker, J. & Miles, J., 1986 Parametric excitation of an internally resonant double pendulum, II. Z. angew. Math. Phys. 37, 641–650.Google Scholar
Becker, J. & Miles, J., 1991 Standing radial cross-waves. J. Fluid Mech. 222, 471–499.Google Scholar
Bloomfield, P.: 1976 Fourier Analysis of Time Series: An Introduction. Wiley.
Ciliberto, S. & Gollub, J. P., 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922–925.Google Scholar
Ciliberto, S. & Gollub, J. P., 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Concus, P.: 1962 Standing capillary-gravity waves of finite amplitude. J. Fluid Mech. 14, 568–576.Google Scholar
Gollub, J. P. & Meyer, C. W., 1983 Symmetry-breaking instabilities on a fluid surface. Physica 6(D), 337–346.Google Scholar
Gu, X. M. & Sethna, P. R., 1987 Resonant surface waves and chaotic phenomena. J. Fluid Mech. 183, 543–565.Google Scholar
Henderson, D. M. & Miles, J. W., 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95–109.Google Scholar
McGoldrick, L. F.: 1970 On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193–200.Google Scholar
Miles, J. W.: 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Miles, J. W.: 1984 Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Miles, J. & Henderson, D., 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143–165.Google Scholar
Phillips, O. M.: 1977 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Simmons, W. F.: 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309, 551–579.Google Scholar
Simonelli, F. & Gollub, J. P., 1989 Surface wave mode interactions: Effects of symmetry and degeneracy. J. Fluid Mech. 199, 471–494.Google Scholar
Tadjbakhsh, I. & Keller, J. B., 1960 Standing surface waves of finite amplitude. J. Fluid Mech. 8, 442–51.Google Scholar
Wilton, J. R.: 1915 On ripples. Phil. Mag. 29 (6), 688–700.Google Scholar
Wolfe, A., Swift, J. B., Swinney, H. L. & Vastano, J. A., 1985 Determining Lyapunov exponents from a time series. Physica 16(D), 285–317.Google Scholar