Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-20T06:52:40.052Z Has data issue: false hasContentIssue false
  • English
  • Français

On Faraday waves

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093-0225, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The standing waves of frequency ω and wavenumber κ that are induced on the surface of a liquid of depth d that is subjected to the vertical displacement ao cos 2wt are determined on the assumptions that: the effects of lateral boundaries are negligible; ε = ka0 tanh kd [Lt ] 1 and 0 < ε−δ = O(δ3), where δ is the linear damping ratio of a free wave of frequency ω; the waves form a square pattern (which follows from observation). This problem, which goes back to Faraday (1831), has recently been treated by Ezerskii et al. (1986) and Milner (1991) in the limit of deep-water capillary waves (kd, kl* [Gt ] 1, where l* is the capillary length). Ezerskii et al. show that the square pattern is unstable for sufficiently large ε—δ, and Milner shows that nonlinear damping is necessary for equilibration of the square pattern. The present formulation extends those of Ezerskii et al. and Milner to capillary–gravity waves and finite depth and incorporates third-order parametric forcing, which is neglected in these earlier formulations but is comparable with third-order damping. There are quantitative differences in the resulting evolution equations (for kd, kl* [Gt ] 1), which appear to reflect errors in the earlier work.

These formulations determine a locus of admissible waves, but they do not select a particular wave. The hypothesis that the selection process maximizes the energy-transfer rate to the Faraday wave selects the maximum of the resonance curve in a frequency-amplitude plane.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 248 , March 1993 , pp. 671 - 683
Copyright
© 1993 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

Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Europhys. Lett. 6, 221– 226.Google Scholar
Ezerskii, A. B., Rabinovich, M. I., Reutov, V. P. & Starobinets, I. M. 1986 Spatiotemporal chaos in the parametric excitation of a capillary ripple. Sov. Phys. J. Exp. Theor. Phys. 64, 1228– 1236.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299– 340.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Manneville, P. 1990 Dissipative Structures and Weak Turbulence. Academic.
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419– 448.Google Scholar
Miles, J. W. 1984 Nonlinear Faraday resonance. J. Fluid Mech. 146, 285– 302.Google Scholar
Miles, J. 1992 On Rayleigh's investigation of crispations of fluid resting on a vibrating support. J. Fluid Mech. 244, 645– 648.Google Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143– 165 (referred to herein as MH).Google Scholar
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81– 100.Google Scholar
Rayleigh, Lord 1883 On the crispations of fluid resting on a vibrating support. Phil. Mag. 16, 50– 58 (Scientific Papers, vol. 2, pp. 212–219).Google Scholar
Tufillaro, N. B., Ramshankar, R. & Gollub, J. P. 1989 Order–disorder transition in capillary ripples. Phys. Rev. Lett. 62, 422– 425.Google Scholar