Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T10:36:54.969Z Has data issue: false hasContentIssue false
  • English
  • Français

Faraday resonance in rectangular geometry

Published online by Cambridge University Press:  26 April 2006

Makoto Umeki
Makoto Umeki
Affiliation:
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan
Get accessRights & Permissions [Opens in a new window]

Abstract

The motion of subharmonic resonant modes of surface waves in a rectangular container subjected to vertical periodic oscillation is studied based on the weakly nonlinear model equations derived by both the average Lagrangian and the two-timescale method. Explicit estimates of the nonlinearity of some specific modes are given. The bifurcations of stationary states including a Hopf bifurcation are examined. Numerical calculations of the dissipative dynamical equations show periodic and chaotic attractors. Theoretical parameter-space diagrams and numerical results are compared in detail with Simonelli & Gollub's (1989) surface-wave modecompetition experiments. It is shown that the average Hamiltonian system for the present 2:1:1 external-internal resonance with suitable coefficients has homoclinic chaos, which was mathematically proven by Holmes (1986) for the specific case of 2:1:2 external-internal resonance.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991, pp. 161 - 192
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.)

Article purchase

Temporarily unavailable

References

Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion.. Proc. R. Soc. Lond. A 225, 505–517.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition is parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Crawford, J. D., Knobloch, E. & Riecke, H. 1990 Period-doubling mode interactions with circular symmetry.. Physica D 44, 340–396.Google Scholar
Feng, Z. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495–518.Google Scholar
Funakoshi, M. & Inoue, S. 1988 Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192, 219–247.Google Scholar
Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1989 Singularities and Groups in Bifurcation Theory, vol. 2. Springer.
Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd edn. Springer.
Holmes, P. 1986 Chaotic motions in a weakly nonlinear model for surface waves. J. Fluid Mech. 162, 365–388.Google Scholar
Kambe, T. & Umeki, M. 1990 Nonlinear dynamics of two-mode interactions in parametric excitation of surface waves. J. Fluid Mech. 212, 373–393.Google Scholar
Karatsu, M. 1988 Nonlinear resonance and chaos of surface waves of water. Master's thesis, University of Tokyo (in Japanese).
Mack, L. R. 1962 Periodic, finite amplitude, axisymmetric gravity waves. J. Geophys. Res. 67, 829–843.Google Scholar
Meron, E. & Procaccia, I. 1986 Low-dimensional chaos in surface waves: Theoretical analysis of an experiment.. Phys. Rev. A 34, 3221–3237.Google Scholar
Meron, E. & Procaccia, I. 1989 Reply to Miles. Phys. Rev. Lett. 63, 1437.Google Scholar
Miles, J. W. 1976 Nonlinear surface waves in a closed basin. J. Fluid Mech. 75, 419–448.Google Scholar
Miles, J. W. 1984a Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Miles, J. W. 1984b Internally resonant surface waves in a circular cylinder. J. Fluid Mech. 149, 1–14.Google Scholar
Miles, J. W. 1989 Symmetries of internally resonant, parametrically excited surface waves. Phys. Rev. Lett. 63, 1436.Google Scholar
Nagata, M. 1989 Nonlinear Faraday resonance in a box with a square base. J. Fluid Mech. 209, 265–284.Google Scholar
Nobili, M., Ciliberto, S., Loccaiaro, B., Faetti, S. & Fronzoni, L. 1988 Time-dependent surface waves in a horizontally oscillating container. Europhys. Lett. 7, 587–592.Google Scholar
Silber, M. & Knobloch, E. 1988 Parametrically excited surface waves in square geometry. Phys. Lett. 137, 349–354.Google Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effect of symmetry and degeneracy. J. Fluid Mech. 199, 471–494 (referred to herein as SG (1989)).Google Scholar
Tadjbakhsh, I. & Keller, J. B. 1960 Standing surface waves of finite amplitude. J. Fluid Mech. 8, 442–451.Google Scholar
Umeki, M. 1989 Nonlinear dynamics and chaos in Faraday resonance. J. Japan Soc. Fluid Mech. 8, 157–164 (in Japanese).Google Scholar
Umeki, M. & Kambe, T. 1989 Nonlinear dynamics and chaos in parametrically excited surface waves. J. Phys. Soc. Japan 48, 140–154.Google Scholar
Verma, G. H. & Keller, J. B. 1962 Three-dimensional standing surface waves of finite amplitude. Phys. Fluids 5, 52–56.Google Scholar