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Three-dimensional instabilities of nonlinear gravity-capillary waves

Published online by Cambridge University Press:  21 April 2006

Jun Zhang and
W. K. Melville
Jun Zhang
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 021939, USA
W. K. Melville
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 021939, USA
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Abstract

Linear three-dimensional instabilities of nonlinear two-dimensional uniform gravitycapillary waves are studied using numerical methods. The eigenvalue system for the stability problem is generated using a Galerkin method and differs in detail from techniques used to study the stability of pure gravity waves (McLean 1982) and pure capillary waves (Chen & Saffman 1985). It is found that instabilities develop in the neighbourhood of the linear (triad, quartet and quintet) resonance curves. Further, both sum and difference triad ressonances are unstable for sufficiently steep waves in consequence of which Hasselmann's (1967) theorem is restricted to weakly nonlinear waves. The appearance of a superharmonic two-dimensional instability and bifurcation to three-dimensional waves are noted.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 174 , January 1987, pp. 187 - 208
Copyright
© 1987 Cambridge University Press

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References

Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.Google Scholar
Benney, D. J. 1976 Significant interactions between small and large scale surface waves. Stud. Appl. Math. 55, 93–106.Google Scholar
Chen, B. & Saffman, P. G. 1985 Three-dimensional stability and bifurcation of capillary and gravity waves on deep water. Stud. Appl. Math. 72, 125–147.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79, 703–714.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30, 737–739.Google Scholar
Hogan, S. J. 1980 Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96, 417–445.Google Scholar
Hogan, S. J. 1981 Some effects of surface tension on steep water waves. Part 3. J. Fluid Mech. 110, 381–410.Google Scholar
Lighthill, M. J. 1965 Contributions to the theory of waves in nonlinear dispersive systems. J. Inst. Maths Appls 1, 296–306.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superhamonics. Proc. R. Soc. Lond. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
McLean, J. W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
McLean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional instability of finite amplitude water waves. Phys. Rev. Lett. 46, 817–820.Google Scholar
Oppenheim, A. V., Willsky, A. S. & Young, I. T. 1985 Signal and Systems. Prentice-Hall.
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Stokes, G. G. 1880 Supplement to a paper on the theory of oscillatory waves. Mathematical and Physical Papers, Vol. 1, pp. 314–326. Cambridge University Press.
Wilkinson, J. H. 1978 Kronecka's cononical form and the QZ algorithm. NPL Rep. DNACS 10/78.Google Scholar
Zhang, J. & Melville, W. K. 1986 On the stability of weakly-nonlinear gravity-capillary waves. Wave Motion (in press).