Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T07:53:39.632Z Has data issue: false hasContentIssue false
  • English
  • Français

Hysteresis in Faraday resonance

Published online by Cambridge University Press:  26 April 2006

S. P. Decent  and
A. D. D. Craik
S. P. Decent
Affiliation:
School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland, UK
A. D. D. Craik
Affiliation:
School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Faraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. Hysteresis occurs when both finite-amplitude solutions and the flat-surface solution are available. We derive a nonlinear model of Faraday resonance, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. The theoretical results are compared to previous experimental studies and to some new observations. It is found necessary to retain damping and forcing terms up to third-order in wave amplitude, and also the fifth-order conservative frequency shift, in order to achieve agreement with experiments. The latter fifth-order term was omitted from all previous studies of Faraday waves. The lower hysteresis boundary in forcing-frequency space is found in most cases to be defined by the lower boundary above which non-trivial stationary points exist. However, the stability of stationary points and the existence of limit cycles are also found to be factors in determining the lower hysteresis boundary. Our results also suggest an indirect method for estimating the coefficient of cubic damping, which is difficult to obtain either experimentally or theoretically.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 293 , 25 June 1995 , pp. 237 - 268
Copyright
© 1995 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 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–517.Google Scholar
Craik, A. D. D. 1994 The stability of some three-dimensional and time-dependent flows. IUTAM Symp. Potsdam, NY: Nonlinear Instability of Nonparallel Flows (ed. S. P. Lin, W. R. C. Phillips & D. T. Valentine), pp. 382–396. Springer.
Craik, A. D. D. & Armitage, J. 1995 Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank. Fluid Dyn. Res. 15, 129–143.Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.Google Scholar
Faraday, M. 1831 On the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 52, 319–340.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
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. W. 1993 On Faraday waves. J. Fluid Mech. 248, 671–683.Google Scholar
Miles, J. W. & Henderson, D. 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143–165.Google Scholar
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81–100.Google Scholar
Nagata, M. 1989 Nonlinear Faraday resonance in a box with a square base. J. Fluid Mech. 209, 265–284.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
Umeki, M. 1991 Faraday resonance in rectangular geometry. J. Fluid Mech. 227, 161–192.Google Scholar
Umeki, M. & Kambe, T. 1989 Nonlinear dynamics and chaos in parametrically excited surface waves. J. Phys. Soc. Japan 58 (1), 140–154.Google Scholar
Supplementary material: PDF

Decent and Craik supplementary material

Supplementary Material

Download Decent and Craik supplementary material(PDF)
PDF 177.8 KB