Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-20T07:03:24.474Z Has data issue: false hasContentIssue false
  • English
  • Français

The period-doubling of gravity–capillary waves

Published online by Cambridge University Press:  21 April 2006

Peter A. E. M. Janssen
Peter A. E. M. Janssen
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Permanent address: KNMI, De Bilt, Holland.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper an attempt is made to explain the period-doubling of wind-generated gravity-capillary waves as observed in the experiment of Choi (1977). It is conjectured that period-doubling is closely related to the phenomenon of second-harmonic resonance. In order to obtain a simple dynamical model, results of McGoldrick (1970) and Simmons (1969) are extended to include the effect of wind input and shear in the current. For pure gravity–capillary waves (no wind, no current) the condition for energy transfer from the second harmonic to the fundamental wave of Chen & Saffman (1979) is recovered. We also discuss the effect of wind and we find that wind input gives rise to a very sudden period-doubling. Qualitative agreement with experiment is obtained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 172 , November 1986 , pp. 531 - 546
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

Bretherton F. P.1964 Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20, 457–479.Google Scholar
Chen, B. & Saffman P. G.1979 Steady gravity—capillary waves in deep water—I. Weakly nonlinear waves. Stud. Appl. Math. 60, 183–210.Google Scholar
Chen, B. & Saffman P. G.1980 Steady gravity—capillary waves on deep water — II. Numerical results for finite amplitude. Stud. Appl. Math. 62, 95–111.Google Scholar
Choi I.1977 Contributions a l’étude des mecanismes physiques de la génération des ondes de capillarité-gravité à une interface air—eau. Thesis, Université d'Aix Marseille.
Favre, A. & Coantic M.1974 Activities in the preliminary results of air—sea interaction at IMST. Adv. Geophys. 18A, 391–405.Google Scholar
Gradshteyn, I. S. & Ryzhik I. M.1965 Tables of Integrals Series, and Products. Academic.
Kawai S.1979 Generation of initial wavelets by instability of a coupled shear flow and their evolution to wind waves. J. Fluid Mech. 93, 661–703.Google Scholar
Mcgoldrick L. F.1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305–331.Google Scholar
Mcgoldrock L. F.1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271.Google Scholar
Mcgoldrick L. F.1972 On the rippling of small waves: a harmonic nonlinear resonant interaction. J. Fluid Mech. 52, 725–751.Google Scholar
Phillips O. M.1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Ramamonjiarisoa A., Baldy, S. & Choi I.1978 Laboratory studies on wind-wave generation, amplification and evolution. In Turbulent Fluxes through the Sea Surface, Wave Dynamics, and Prediction (Ed. A. Favre & K. Hasselmann), pp. 403–420. Plenum.
Simmons W. F.1969 A variational method for weak resonant wave interactions Proc. R. Soc. Land. A 309, 551–575.Google Scholar
van Gastel, K., Janssen, P. A. E. M. & Komen, G. J. 1985 On phase velocity and growth rate of wind-induced gravity—capillary waves. J. Fluid Mech. 161, 199–216.Google Scholar
Whitham G. B.1974 Linear and Nonlinear Waves. Wiley.