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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
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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

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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.