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The stability of capillary waves on fluid sheets

  • M. G. Blyth (a1) and E. I. Părău (a1)


The linear stability of finite-amplitude capillary waves on inviscid sheets of fluid is investigated. A method similar to that recently used by Tiron & Choi (J. Fluid Mech., vol. 696, 2012, pp. 402–422) to determine the stability of Crapper waves on fluid of infinite depth is developed by extending the conformal mapping technique of Dyachenko et al. (Phys. Lett. A, vol. 221 (1), 1996a, pp. 73–79) to a form capable of capturing general periodic waves on both the upper and the lower surface of the sheet, including the symmetric and antisymmetric waves studied by Kinnersley (J. Fluid Mech., vol. 77 (02), 1976, pp. 229–241). The primary, surprising result is that both symmetric and antisymmetric Kinnersley waves are unstable to small superharmonic disturbances. The waves are also unstable to subharmonic perturbations. Growth rates are computed for a range of steady waves in the Kinnersley family, and also waves found along the bifurcation branches identified by Blyth & Vanden-Broeck (J. Fluid Mech., vol. 507, 2004, pp. 255–264). The instability results are corroborated by time integration of the fully nonlinear unsteady equations. Evidence is presented for superharmonic instability of nonlinear waves via a collision of eigenvalues on the imaginary axis which appear to have the same Krein signature.


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Akers, B. & Nicholls, D. P. 2012 Spectral stability of deep two-dimensional gravity water waves: repeated eigenvalues. SIAM J. Appl. Maths 72 (2), 689711.
Akers, B. & Nicholls, D. P. 2013 Spectral stability of deep two-dimensional gravity–capillary water waves. Stud. Appl. Maths 130 (2), 81107.
Akers, B. & Nicholls, D. P. 2014 The spectrum of finite depth water waves. Eur. J. Mech. (B/Fluids) 46, 181189.
Barlow, N. S., Helenbrook, B. T. & Lin, S. P. 2011 Transience to instability in a liquid sheet. J. Fluid Mech. 666, 358390.
Benjamin, T. B & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27 (03), 417430.
Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137185.
Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48 (377), 377385.
Billingham, J. 2006 Surface tension-driven flow in a slender wedge. SIAM J. Appl. Math. 66 (6), 19491977.
Blyth, M. G., Părău, E. I. & Vanden-Broeck, J.-M. 2011 Hydroelastic waves on fluid sheets. J. Fluid Mech. 689, 541551.
Blyth, M. G. & Vanden-Broeck, J.-M. 2004 New solutions for capillary waves on fluid sheets. J. Fluid Mech. 507, 255264.
Bridges, T. J. & Donaldson, N. M. 2011 Variational principles for water waves from the viewpoint of a time-dependent moving mesh. Mathematika 57 (01), 147173.
Byrd, P. F. & Friedman, M. D. 1971 Handbook of Elliptic Integrals for Engineers and Scientists. Springer.
Chen, B. & Saffman, P. G. 1980 Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Stud. Appl. Maths 62, 121.
Chen, B. & Saffman, P. G. 1985 Three-dimensional stability and bifurcation of capillary and gravity waves on deep water. Stud. Appl. Maths 77 (2), 125147.
Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. 125 (7), 756760.
Constantin, A. & Strauss, W. 2010 Pressure beneath a stokes wave. Commun. Pure Appl. Maths 63 (4), 533557.
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (06), 532540.
Crowdy, D. G. 1999 Exact solutions for steady capillary waves on a fluid annulus. J. Nonlinear Sci. 9 (6), 615640.
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141167.
Deconinck, B. & Trichtchenko, O. 2014 Stability of periodic gravity waves in the presence of surface tension. Eur. J. Mech. (B/Fluids) 46, 97108.
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31 (1), 301346.
Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary–gravity waves. J. Fluid Mech. 79 (04), 703714.
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996a Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1), 7379.
Dyachenko, A. I., Zakharov, V. E. & Kuznetsov, E. A. 1996b Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys. Rep. 22, 829840.
Groves, M. D. & Toland, J. F. 1997 On variational formulations for steady water waves. Arch. Rat. Mech. Anal. 137 (3), 203226.
Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25 (1), 5597.
Hogan, S. J. 1985 The fourth-order evolution equation for deep-water gravity–capillary waves. Proc. R. Soc. Lond. A 402 (1823), 359372.
Hogan, S. J. 1988 The superharmonic normal mode instabilities of nonlinear deep-water capillary waves. J. Fluid Mech. 190, 165177.
Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77 (02), 229241.
Krein, M. G. 1950 A generalization of some investigations of linear differential equations with periodic coefficients. Dokl. Akad. Nauk SSSR 73, 445448.
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. Part I. Superharmonics. Proc. R. Soc. Lond. A 360, 471488.
Mackay, R. S. 1987 Stability of equilibria of Hamiltonian systems. In Hamiltonian Dynamical Systems: A Reprint Selection, pp. 137153. IOP Publishing.
Mackay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406 (1830), 115125.
Peregrine, D. H., Shoker, G. & Symon, A. 1990 The bifurcation of liquid bridges. J. Fluid Mech. 212, 2539.
Rayleigh, J. W. S. 1896 The Theory of Sound, vol ii. Macmillan.
Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169174.
Sandstede, B. 2002 Stability of travelling waves. In Handbook of Dynamical Systems II (ed. Fiedler, B.), pp. 9831055. North-Holland.
Squire, H. B. 1953 Investigation of the instability of a moving liquid film. British J. Appl. Phys. 4 (6), 167169.
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. Part II. Waves on fluid sheets. Proc. R. Soc. Lond. A 253 (1274), 296312.
Tiron, R. & Choi, W. 2012 Linear stability of finite-amplitude capillary waves on water of infinite depth. J. Fluid Mech. 696, 402422.
Turner, M. R. & Bridges, T. J. 2016 Time-dependent conformal mapping of doubly-connected regions. Adv. Comput. Math. 42, 947972.
Vasan, V. & Deconinck, B. 2013 The bernoulli boundary condition for traveling water waves. Appl. Math. Lett. 26 (4), 515519.
Viotti, C., Dutykh, D. & Dias, F. 2014 The conformal-mapping method for surface gravity waves in the presence of variable bathymetry and mean current. Procedia IUTAM 11, 110118.
Wilkening, J. & Vasan, V. 2015 Comparison of five methods of computing the Dirichlet–Neumann operator for the water wave problem. Contemp. Maths 635, 175210.
Williamson, J. 1936 On the algebraic problem concerning the normal forms of linear dynamical systems. Amer. J. Math. 58, 141163.
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The stability of capillary waves on fluid sheets

  • M. G. Blyth (a1) and E. I. Părău (a1)


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