Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-31T21:06:50.964Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Faraday resonance in a box with a square base

Published online by Cambridge University Press:  26 April 2006

M. Nagata
M. Nagata
Affiliation:
Department of Mathematical Sciences, University of St Andrews, North Haugh, St Andrews KY16 9SS, Fife, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Surface wave motions in a container with a square base, which is subject to a vertical oscillation, are considered when the amplitude of the oscillation is small and the frequency of the oscillation is close to twice the natural frequency of the system. Subcritical wave motions are found for single modes as well as mixed modes. Here, single modes are described by either one of the two horizontal coordinates whereas mixed modes depend on both coordinates. It is found that in some subcritical region a stable single mode and a stable mixed mode coexist, producing complex basins of attraction.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 265 - 284
Copyright
© 1989 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

Battelino, P. M., Grebogi, C., Ott, E., Yorke, J. A. & Yorke, E. D. 1988 Multiple coexisting attractors, basin boundaries and basic sets. Physica D 32, 296–305.Google Scholar
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. 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
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge Univeristy Press.
Faraday, M. 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–346.Google Scholar
Feng, Z. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495–518.Google Scholar
Gollub, J. P. & Meyer, C. W. 1983 Symmetry-breaking instabilities on a fluid surface. Physica D 6, 337–346.Google Scholar
Gu, X. M., Sethna, P. R. & Narain, A. 1987 On three-dimensional nonlinear subharmonic resonant surface waves in a fluid. Part I. Theory. Trans. ASME E: J. Appl. Mech. 55, 213–219.Google Scholar
Holmes, P. J. 1986 Chaotic motions in a weakly nonlinear model for surface waves. J. Fluid Mech. 162, 365–388.Google Scholar
Jenkins, D. R. 1987 Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity. J. Fluid Mech. 178, 491–506.Google Scholar
Keolian, R., Turkevich, L. A., Putterman, S. J., Rudnick, I. & Rudnick, J. A. 1981 Subharmonic sequences in the Faraday experiment: departures from period doubling. Phys. Rev. Lett. 47, 1133–1136.Google Scholar
Larraza, A. & Putterman, S. 1984 Theory of non-propagating surface-wave solitons. J. Fluid Mech. 148, 443–449.Google Scholar
Meron, E. 1987 Parametric excitation of multimode dissipative systems. Phys. Rev. A 35, 4892–4895.Google Scholar
Miles, J. W. 1984a Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Miles, J. W. 1984b Parametrically excited solitary waves. J. Fluid Mech. 148, 451–460.Google Scholar
Nagata, M., Proctor, M. R. E. & Weiss, N. O. 1989 Transitions to asymmetry in magnetoconvection. Geophys. Astrophys. Fluid Dyn. (to appear).Google Scholar
Rayleigh, Lord 1883a On maintained vibrations. Phil. Mag. 15, 229–235 (Scientific Papers, Vol. 2, pp. 188–193).Google Scholar
Rayleigh, Lord 1883b On the crispations of fluid resting on a vibrating support. Phil. Mag. 16, 50–58 (Scientific Papers, Vol. 2, pp. 212–219).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
Swift, J. W. 1984 Bifurcation and symmetry in convection. Ph. D. disseration, University of California, Berkeley.
Wu, J., Keolian, R. & Rudnick, I. 1984 Observation of a non-propagating hydrodynamic soliton. Phys. Rev. Lett. 52, 1421–1424.Google Scholar