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

  • Roxana Tiron (a1) and Wooyoung Choi (a1) (a2)


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).


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1. Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417430.
2. Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48, 377385.
3. Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitudes. J. Fluid Mech. 2, 532540.
4. Chen, B. & Saffman, P. G. 1985 Three-dimensional stability and bifurcation of capillary and gravity waves on deep water. Stud. Appl. Maths 72, 125147.
5. Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. 125, 756760.
6. Davies, J. T. & Vose, R. W. 1965 On the damping of capillary waves by surface films. Proc. R. Soc. Lond. A 286, 218234.
7. Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141167.
8. Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31, 301346.
9. Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary–gravity waves. J. Fluid Mech. 79, 703714.
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, 7379.
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.
12. Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25, 5597.
13. Hogan, S. J. 1985 The fourth-order evolution equation for deep-water gravity–capillary waves. Proc. R. Soc. Lond. A 402, 359372.
14. Hogan, S. J. 1988 The superharmonic normal mode instabilities of nonlinear deep-water capillary waves. J. Fluid Mech. 190, 165177.
15. Kinnersley, W. 1976 Exact large amplitude waves on sheets of fluid. J. Fluid Mech. 77, 229241.
16. Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
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, 303324.
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, 471488.
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, 489505.
20. MacKay, R. S. 1986 Stability of equilibria of Hamiltonian systems. In Nonlinear Phenomena and Chaos (ed. Sarkar, S. ). Adam Hilger.
21. MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115125.
22. McLean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315330.
23. McLean, J. W. 1982b Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331341.
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, 817821.
25. Nicholls, D. P. 2009 Spectral data for travelling water waves: singularities and stability. J. Fluid Mech. 624, 339360.
26. Ovsjannikov, S. J. 1974 To the shallow water theory foundation. Arch. Mech. 26, 407422.
27. Swarztrauber, P. N. 1982 Vectorizing the FFTs. In Parallel Computations (ed. Rodrigue, G. ), pp. 5183. Academic.
28. Vanden-Broeck, J.-M. & Keller, J. B. 1980 A new family of capillary waves. J. Fluid Mech. 98, 161169.
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, 190194.
30. Zhang, J. & Melville, W. K. 1987 Three-dimensional instabilities of nonlinear gravity–capillary waves. J. Fluid Mech. 174, 187208.
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Linear stability of finite-amplitude capillary waves on water of infinite depth

  • Roxana Tiron (a1) and Wooyoung Choi (a1) (a2)


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