Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-11T04:23:06.927Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth

Published online by Cambridge University Press:  21 April 2006

Juan A. Zufiria
Juan A. Zufiria
Affiliation:
Applied Mathematics Department, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A weakly nonlinear model is developed from the Hamiltonian formulation of water waves, to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, non-symmetric Wilton's ripples exist. They appear via a spontaneous symmetrybreaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and that further non-symmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low-gravity experiments is postulated.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 184 , November 1987 , pp. 183 - 206
Copyright
© 1987 Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

Abraham, R. & Marsden, J. E. 1978 Fundations of Mechanics, 2nd edn Benjamin/Cummings.
Amick, C. J. & Kirchgässner, K. 1987 On solitary water-waves in the presence of surface tension. (preprint).
Benjamin, T. B. 1982 The solitary wave with surface tension. Q. Appl. Maths 40, 231–234.Google Scholar
Benjamin, T. B. 1984 Impulse, force and variational principles. IMA J. Appl. Maths 32, 3–68.Google Scholar
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430–446.Google Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity–capillary waves on deep water. I. – Weakly nonlinear waves. Stud. Appl. Maths 60, 183–310.Google Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity–capillary waves on deep water. II. – Numerical results for finite amplitude. Stud. Appl. Maths 62, 95–111.Google Scholar
Doedel, E. J. & Kernevez, J. P. 1986 Software for continuation problems in ordinary differential equations with applications. Caltech. Applied Maths Rep.
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison Wesley.
Green, J. M., Mackay, R. S., Vivaldi, F. & Feigenbaum, M. J. 1981 Universal behavior in families of area-preserving maps. Physica 3D, 468–486.Google Scholar
Hartman, P. 1964 Ordinary Differential Equations. Wiley-Interscience.
Hunter, D. H. & Vanden-Broeck, J. M. 1983 Solitary and periodic gravity–capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219.Google Scholar
Jones, M. C. W. 1987 The bifurcation of Wilton ripples and other capillary–gravity waves in a channel of finite depth. (preprint).
Jones, M. C. W. & Toland, J. F. 1986 Symmetry and the bifurcation of capillary–gravity waves. Arch. Rat. Mech. Anal. 96, 29–53.Google Scholar
Keller, W. E. 1969 Helium-III and Helium-IV. Plenum.
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal and a new type of long stationary waves. Phil. Mag. 39(5), 422–443.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1978 Fluid Mechanics. Pergamon.
Miles, J. W. 1977 On Hamiltonian's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Nayfeh, A. H. 1970a Finite amplitude surface waves in a liquid layer. J. Fluid Mech. 40, 671–684.Google Scholar
Nayfeh, A. H. 1970b Triple and quintuple-dimpled wave profiles in deep water. Phys. Fluids 13, 545–550.Google Scholar
Pierson, W. J. & Fife, P. 1961 Some nonlinear properties of long-crested periodic waves with lengths near 2.44 centimetres. J. Geophys. Res. 66, 163–179.Google Scholar
Rimmer, R. 1978 Symmetry and bifurcation of fixed points in area preserving maps. J. Diffl Equat. 29, 329–344.Google Scholar
Sekerzh-Zenkovich, Y. I. 1956 On the theory of stationary capillary waves of finite amplitude on the surface of heavy fluid. Dokl. Akad. Nauk. SSSR 109, 913–918.Google Scholar
Schooley, A. H. 1960 Double, triple and higher-order dimples in the profiles of wind-generated water waves in the capillary–gravity transition region. J. Geophys. Res. 65, 4075–4079.Google Scholar
Toland, J. F. & Jones, M. C. W. 1985 The bifurcation and secondary bifurcation of gravity capillary waves. Proc. R. Soc. Lond. A 399, 391–417.Google Scholar
Vanden-Broeck, J. M. & Shen, M. C. 1983 A note on the solitary and cnoidal waves with surface tension. Z. angew. Math. Phys. 34, 112–117.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688–700.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep fluid. J. App. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zufiria, J. A. 1987a Weakly nonlinear non-symmetric gravity waves on water of finite depth. J. Fluid Mech. 180, 371–385.Google Scholar
Zufiria, J. A. 1987b Non-symmetric gravity waves on water of infinite depth. J. Fluid Mech. 181, 17–39.Google Scholar