Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-20T00:57:24.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly nonlinear non-symmetric gravity waves on water of finite depth

Published online by Cambridge University Press:  21 April 2006

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

Abstract

A weakly nonlinear Hamiltonian model for two-dimensional irrotational waves on water of finite depth is developed. The truncated model is used to study families of periodic travelling waves of permanent form. It is shown that non-symmetric periodic waves exist, which appear via spontaneous symmetry-breaking bifurcations from symmetric waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 180 , July 1987 , pp. 371 - 385
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.)

References

Abraham, R. & Marsden, J. E. 1978 Fundations of Mechanics (2nd edn). Massachusetts: Benjamin/Cummings.
Arnol'D, V. I.1978 Mathematical Methods of Classical Mechanics. Springer.
Arnol'D, V. I. & Avez, A.1968 Ergodic Problems of Classical Mechanics. New York: Benjamin.
Benjamin, T. B. 1984 Impulse, force and variational principles. IMA J. of 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. 1980 Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Stud. Appl. Maths 62, 1–21.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
Doedel, E. J. & Kernevez, J. P. 1986 Software for continuation problems in ordinary differential equations with applications. Caltech. Applied Mathematics Rep.Google Scholar
Finn, J. M. 1974 Integral canonical transformations and normal forms for mirror machine Hamiltonians. Doctoral thesis, Appendix B, p 242. University of Maryland.
Goldstein, H. 1980 Classical Mechanics (2nd edn). Addison Wesley.
Grant, M. A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59, 257–262.Google Scholar
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
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
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
Longuet-Higgins, M. S. 1985 Bifurcation in gravity waves. J. Fluid Mech. 151, 457–475.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721–741.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.Google Scholar
Mackay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Norman, A. C. 1974 Expansions for the shape of maximum amplitude Stokes waves. J. Fluid Mech. 66, 261–265.Google Scholar
Olfe, D. B. & Rottman, J. W. 1980 Some new highest-wave solutions for deep-water waves of permanent form. J. Fluid Mech. 100, 801–810.Google Scholar
Radder, A. C. & Dingemans, M. W. 1985 Canonical equations for almost periodic, weakly nonlinear gravity waves. Wave Motion 7, 473–485.Google Scholar
Rimmer, R. 1978 Symmetry and bifurcation of fixed points of area preserving maps. J. Diff. Equns 29, 329–344.Google Scholar
Saffman, P. G. 1980 Long wavelength bifurcation of gravity waves on deep water. J. Fluid Mech. 101, 567–581.Google Scholar
Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.Google Scholar
Sattinger, D. H. 1980 Bifurcation and symmetry breaking in applied mathematics. Bull. Am. Maths Soc. 3, 779–819.Google Scholar
Sattinger, D. H. 1983 Branching in the presence of symmetry. CBMS-NSF Reg. Conf. in Appl. Maths SIAM, Philadelphia.
Stokes, G. G. 1849 Trans. Camb. Phil. Soc. 8, 441.
Tanaka, M. 1985a The stability of steep gravity waves. J. Fluid Mech. 156, 281–289.Google Scholar
Tanaka, M. 1985b The stability of solitary waves. Phys. Fluids (to appear).
Vanden-Broeck, J. M. 1983 Some new gravity waves in water of finite depth. Phys. Fluids 26, 2385–2387.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear waves. Wiley-Interscience.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zufiria, J. A. & Saffman, I. G. 1985 The superharmonic instability of finite amplitude surface waves on water of finite depth. Stud. Appl. Maths 74, 259–266.Google Scholar