Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-06T07:22:37.476Z Has data issue: false hasContentIssue false
  • English
  • Français

The high wavenumber instabilities of a Stokes wave

Published online by Cambridge University Press:  18 April 2017

Dieter E. Hasselmann
Dieter E. Hasselmann*
Affiliation:
Meteorologisches Institut der Universität Hamburg, Germany
Get access Rights & Permissions [Opens in a new window]

Extract

A stability analysis for high wavenumber perturbations of a Stokes wave of wavenumber k1 and slope ϵ is presented. Except for a correction term the governing equation is shown to be of Hill's type. The analysis predicts instability at wavenumbers k2 = ¼(m + 1)2k1. The two lowest and strongest instabilities are the Benjamin- Feir instability at m = 1, and the quartet resonance at m = 2. Both are incorrectly treated by the present method. For m ≥ 3 the analysis should be asymptotically (ϵ → 0) correct, yielding instability O(ϵm) due to m-fold Bragg-scattering. The non-resonant perturbations behave as predicted by WKBJ theory. The instability is too weak for experimental detection; numerical tests should be possible, but are not available at present.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 93 , Issue 3 , August 1979 , pp. 491 - 499
Copyright
Copyright © Cambridge University Press 1979

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

Arscott, F. M. 1964 Periodic Differential Equations. Pergamon.Google Scholar
Bellman, R. 1964 Perturbation Techniques in Mathematics, Physics and Engineering. New York: Holt, Rinehart & Winston.Google Scholar
Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299, 59–67.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
Bretherton, F. P. 1964 Resonant interactions between waves. J. Fluid Mech. 20, 457–480.CrossRefGoogle Scholar
Garrett, C. & Smith, J. 1976 On the interaction between long and short surface waves. J. Phys. Oceanog. 6, 925.2.0.CO;2>CrossRefGoogle Scholar
Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity wave spectrum. Part 1. General Theory. J. Fluid Mech. 12, 481–500.CrossRefGoogle Scholar
Ince, E. L. 1926 Ordinary Differential Equations. Dover.Google Scholar
Lamb, H. 1879 Hydrodynamics (6th edition). Dover.Google Scholar
Longtjet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude. I. Superharmonics. Proc. Roy. Soc. A 360, 489–505.Google Scholar
Longtjet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude. II. Subharmonics. Proc. Roy. Soc. A 360, 507–528.Google Scholar
Longtjet-Higgins, M. S. & Phillips, O. M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12, 333–336.CrossRefGoogle Scholar
Meixner, J. & Schafke, F. W. 1954 Mathieusche Funktionen und Sphdroidfunktionen. Springer.CrossRefGoogle Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. J. Fluid Mech. 9, 193–217.CrossRefGoogle Scholar
Phillips, O. M. 1961 On the dynamics of unsteady gravity waves of finite amplitude. Part 2. J. Fluid Mech. 11, 143–155.CrossRefGoogle Scholar
Whittaker, E. T. & Watson, G. N. 1927 A course of modern analysis (4th edition). Dover.Google Scholar