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The evolution of a weakly nonlinear, weakly damped, capillary-gravity wave packet

Published online by Cambridge University Press:  21 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

Longuet-Higgins's (1976) analysis of energy transfer within a narrow spectrum of gravity waves with approximately uncorrelated phases is generalized to accommodate capillarity and weak damping. The analysis is based on the corresponding generalization of Zakharov's (1968) evolution equation for weakly nonlinear, deep-water gravity-wave packets. The results for a symmetric normal spectrum are expressed in terms of elliptic integrals and depend, after appropriate scaling, on a single similarity parameter and on the sign of the curvature of the linear dispersion relation. Energy transfer is away from the peak of that spectrum if kl* < 0.393, where k is the wavenumber and l* is the capillary length (2.8 mm for water), but may be towards the peak if 0.343 < kl* < 0.707 (4.5 cm > 2π/k > 2.5 cm for water). The formulation is based on energy exchange through resonant quartets and is not valid in the neighbourhood of kl* = 0.707; at which the second harmonic of a capillary-gravity wave resonates with its fundamental (Wilton's ripples). The modulational instability of a weakly damped capillary-gravity wave is examined in an Appendix.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 187 , February 1988 , pp. 141 - 154
Copyright
© 1988 Cambridge University Press

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References

Ablowitz, M. J. & Segur, H. 1979 On the evolution of packets of surface waves. J. Fluid Mech. 92, 691–715.Google Scholar
Davey, A. 1972 The propagation of a weak nonlinear wave. J. Fluid Mech. 53, 769–781.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Djordjeviv, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79, 703–714.Google Scholar
Dungey, J. C. & Hui, W. H. 1979 Nonlinear energy transfer in a narrow gravity-wave spectrum. Proc. R. Soc. Lond. A 368, 239–265.Google Scholar
Fox, M. J. H. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum, II. Proc, R. Soc. Lond. A 348, 467–183.Google Scholar
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Math. Soc. (2) 7, 107–121.Google Scholar
Hasselmann, K. et al. 1973 Measurements of wind-wave growth and swell decay during the joint North Sea wave project. Ergänzungsheft zur Deuisch. Hydrogr. Z. A 12, 95 pp.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Longuet-Higgins, M. S. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum: a simplified model. Proc. R. Soc. Lond. A 347, 311–328.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.Google Scholar