Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-09T22:14:50.624Z Has data issue: false hasContentIssue false
  • English
  • Français

Square patterns and secondary instabilities in driven capillary waves

Published online by Cambridge University Press:  26 April 2006

S. T. Milner
S. T. Milner
Affiliation:
AT&T Bell Laboratories, Murray Hill, NJ 07974, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Amplitude equations (including nonlinear damping terms) are derived which describe the evolution of patterns in large-aspect-ratio driven capillary wave experiments. For drive strength just above threshold, a reduction of the number of marginal modes (from travelling capillary waves to standing waves) leads to simpler amplitude equations, which have a Lyapunov functional. This functional determines the wavenumber and symmetry (square) of the most stable uniform state. The original amplitude equations, however, have a secondary instability to transverse amplitude modulation (TAM), which is not present in the standing-wave equations. The TAM instability announces the restoration of the full set of marginal modes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 81 - 100
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

Benjamin, T. B. & Ursell, F., 1954 Proc. R. Soc. Lond. A 225, 505.
Bretherton, F. T.: 1961 J. Fluid Mech. 10, 166.
Ciliberto, S. & Gollub, J., 1985 J. Fluid Mech. 158, 381.
Cox, R. G.: 1986 J. Fluid Mech. 168, 169.
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D., 1983 J. Fluid Mech. 127, 155.
Douady, S. & Fauve, S., 1988 Europhys. Lett. 6, 221.
Eckhaus, W.: 1965 Studies in Non-linear Stability Theory. Springer.
Ezebskii, A. B., Korotin, P. I. & Rabinovitch, M. I., 1985 Pis'ma Zh. Eksp. Teor. Fiz. 41, 129.
Ezebskii, A. B., Rabinovitch, M. I., Reutov, V. P. & Starobinets, I. M., 1986 Sov. Phys. J. Exp. Theor. Phys. 64, 1228.
Fabaday, M.: 1831 Phil. Trans. R. Soc. Lond. 121, 319.
Hocking, L. M.: 1987 J. Fluid Mech. 179, 253.
Landau, L. D. & Lifshitz, E. M., 1959 Fluid Mechanics, p. 98. Pergamon.
Landau, L. D. & Lifshitz, E. M., 1976 Mechanics (3rd Edn), p. 80ff. Pergamon.
Levin, B. V. & Tbubnikov, B. A., 1986 Pis'ma Zh. Eksp. Teor. Fiz. 44, 311.
Miles, J. W.: 1967 Proc. R. Soc. Lond. A 297, 459.
Newell, A. C. & Whitehead, J. A., 1969 J. Fluid Mech. 38, 279.
Riecke, H.: 1990 Stable wave-number kinks in arametrically excited standing waves. ITP preprint, to be published.Google Scholar
Simonelli, F. & Gollub, J. P., 1989 J. Fluid Mech. 199, 471.
Tufillaro, N. B., Ramshankar, R. & Gollub, J. P., 1989 Phys. Rev. Lett. 62, 422.
Zakhabov, V. E., L'vov, V. S. & Starobinets, S. S. 1971 Sov. Phys., J. Exp. Theor. Phys. 32, 656.