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Linear stability of finite-amplitude capillary waves on water of infinite depth

Published online by Cambridge University Press:  06 March 2012

Roxana Tiron  and
Wooyoung Choi
Roxana Tiron
Affiliation:
Department of Ocean Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Wooyoung Choi*
Affiliation:
Department of Ocean Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
*
†Email address for correspondence: wychoi@njit.edu
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Abstract

We study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

JFM classification

Waves/Free-surface Flows: Capillary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 696 , 10 April 2012 , pp. 402 - 422
Copyright
Copyright © Cambridge University Press 2012

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References

1. Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
2. Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48, 377–385.CrossRefGoogle Scholar
3. Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitudes. J. Fluid Mech. 2, 532–540.CrossRefGoogle Scholar
4. Chen, B. & Saffman, P. G. 1985 Three-dimensional stability and bifurcation of capillary and gravity waves on deep water. Stud. Appl. Maths 72, 125–147.CrossRefGoogle Scholar
5. Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. 125, 756–760.Google Scholar
6. Davies, J. T. & Vose, R. W. 1965 On the damping of capillary waves by surface films. Proc. R. Soc. Lond. A 286, 218–234.Google Scholar
7. Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.CrossRefGoogle Scholar
8. Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31, 301–346.CrossRefGoogle Scholar
9. Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary–gravity waves. J. Fluid Mech. 79, 703–714.CrossRefGoogle Scholar
10. Dyachenko, A. L., Zakharov, V. E. & Kuznetsov, E. A. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 73–79.CrossRefGoogle Scholar
11. Garbow, B. S. 1978 Algorithm 535: the QZ algorithm to solve the generalized eigenvalue problem for complex matrices [F2]. ACM Trans. Math. Softw. 4 (4), 404.CrossRefGoogle Scholar
12. Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25, 55–97.CrossRefGoogle Scholar
13. Hogan, S. J. 1985 The fourth-order evolution equation for deep-water gravity–capillary waves. Proc. R. Soc. Lond. A 402, 359–372.Google Scholar
14. Hogan, S. J. 1988 The superharmonic normal mode instabilities of nonlinear deep-water capillary waves. J. Fluid Mech. 190, 165–177.CrossRefGoogle Scholar
15. Kinnersley, W. 1976 Exact large amplitude waves on sheets of fluid. J. Fluid Mech. 77, 229–241.CrossRefGoogle Scholar
16. Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
17. Li, Y. A., Hyman, J. M. & Choi, W. 2004 A numerical study of the exact evolution equations for surface waves in water of finite depth. Stud. Appl. Maths 113, 303–324.CrossRefGoogle Scholar
18. Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360, 471–488.Google Scholar
19. 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
20. MacKay, R. S. 1986 Stability of equilibria of Hamiltonian systems. In Nonlinear Phenomena and Chaos (ed. Sarkar, S. ). Adam Hilger.Google Scholar
21. MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
22. McLean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.CrossRefGoogle Scholar
23. McLean, J. W. 1982b Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.CrossRefGoogle Scholar
24. McLean, J. W., Ma, Y. C., Martin, D. U., Saffmann, P. G. & Yuen, H. C. 1981 Three dimensional instablity of finite-amplitude water waves. Phys. Rev. Lett. 46, 817–821.CrossRefGoogle Scholar
25. Nicholls, D. P. 2009 Spectral data for travelling water waves: singularities and stability. J. Fluid Mech. 624, 339–360.CrossRefGoogle Scholar
26. Ovsjannikov, S. J. 1974 To the shallow water theory foundation. Arch. Mech. 26, 407–422.Google Scholar
27. Swarztrauber, P. N. 1982 Vectorizing the FFTs. In Parallel Computations (ed. Rodrigue, G. ), pp. 51–83. Academic.CrossRefGoogle Scholar
28. Vanden-Broeck, J.-M. & Keller, J. B. 1980 A new family of capillary waves. J. Fluid Mech. 98, 161–169.CrossRefGoogle Scholar
29. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
30. Zhang, J. & Melville, W. K. 1987 Three-dimensional instabilities of nonlinear gravity–capillary waves. J. Fluid Mech. 174, 187–208.CrossRefGoogle Scholar