Hostname: page-component-5c6d5d7d68-pkt8n Total loading time: 0 Render date: 2024-08-18T16:17:30.145Z Has data issue: false hasContentIssue false
  • English
  • Français

Third-harmonic resonance in the interaction of capillary and gravity waves

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh
Ali Hasan Nayfeh
Affiliation:
Aerotherm Corporation, Mountain View, California Present address: Engineering Mechanics Dept., Virginia Polytechnic Institute and State University, Blacksburg, Virginia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The method of multiple scales is used to determine the temporal and spatial variation of the amplitudes and phases of capillary-gravity waves in a deep liquid at or near the third-harmonic resonant wave-number. This case corresponds to a wavelength of 2·99 cm in deep water. The temporal variation shows that the motion is always bounded, and the general motion is an aperiodic travelling wave. The analysis shows that pure amplitude-modulated waves are not possible in this case contrary to the second-harmonic resonant case. Moreover, pure phase-modulated waves are periodic even near resonance because the non-linearity adjusts the phases to yield perfect resonance. These periodic waves are found to be unstable, in the sense that any disturbance would change them into aperiodic waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 48 , Issue 2 , 28 July 1971 , pp. 385 - 395
Copyright
© 1971 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

Barakat, R. & Houston, A. 1968 Non-linear periodic capillary—gravity waves on a fluid of finite depth J. Geophys. Res. 73, 6545.Google Scholar
Benney, D. J. 1962 Non-linear gravity wave interactions J. Fluid Mech. 14, 577.Google Scholar
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Math. Soc. A 7, 107.Google Scholar
Kamesvara Rav, J. C. 1920 On ripples of finite amplitude Proc. Indian Ass. Cultiv. Sci. 6, 175.Google Scholar
Mcgoldrick, L. F. 1965 Resonant interactions among capillary—gravity waves J. Fluid Mech. 21, 305.Google Scholar
Mcgoldrick, L. F. 1970a An experiment on capillary—gravity resonant wave interactions. J. Fluid Mech. 40, 251.Google Scholar
Mcgoldrick, L. F. 1970b On Wilton's ripples. J. Fluid Mech. 42, 193.Google Scholar
Nayfeh, A. H. 1965a Non-linear oscillations in a hot electron plasma Phys. Fluids, 8, 1896.Google Scholar
Nayfeh, A. H. 1965b A perturbation method for treating non-linear oscillation problems. J. Math. & Phys. 44, 368.Google Scholar
Nayfeh, A. H. 1968 Forced oscillations of the van der Pol oscillator with delayed amplitude limiting IEEE Trans. on Circuit Theory, 15, 192.Google Scholar
Nayfeh, A. H. 1970a Finite amplitude surface waves in a liquid layer. J. Fluid Mech. 40, 671.Google Scholar
Nayfeh, A. H. 1970b Triple- and quintuple-dimpled wave profiles in deep water Phys. Fluids, 13, 545.Google Scholar
Nayfeh, A. H. 1971a Two-to-one resonances near the equilateral libration points. AIAA J. 9, 23.Google Scholar
Nayfeh, A. H. 1971b Second-harmonic resonance in the interaction of capillary and gravity waves. J. Fluid Mech. (submitted for publication).Google Scholar
Nayfeh, A. H. & Kamel, A. A. 1970 Three-to-one resonances near the equilateral libration points AIAA J. 8, 2245.Google Scholar
Nayfeh, A. H. & Saric, W. S. 1971 Non-linear Kelvin-Helmholtz instability. J. Fluid Mech. (in the Press).Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. 1. The elementary interaction J. Fluid Mech. 9, 193.Google Scholar
Pierson, W. J. & Fife, P. 1961 Some nonlinear properties of long crested waves with lengths near 2·44 centimetres J. Geophys. Res. 66, 163.Google Scholar
Schooley, A. H. 1960 Double, triple, and higher-order dimples in the profiles of windgenerated water waves in the capillary-gravity transition region J. Geophys. Res. 65, 4075.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc. A 309, 551.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. New York: Academic.
Wilton, J. R. 1915 On ripples Phil. Mag. 29, 688.Google Scholar