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Oscillatory spatially periodic weakly nonlinear gravity waves on deep water

Published online by Cambridge University Press:  21 April 2006

Juana A. Zufiria
Juana A. Zufiria
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
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Abstract

A weakly nonlinear Hamiltonian model is derived from the exact water wave equations to study the time evolution of spatially periodic wavetrains. The model assumes that the spatial spectrum of the wavetrain is formed by only three free waves, i.e. a carrier and two side bands. The model has the same symmetries and invariances as the exact equations. As a result, it is found that not only the permanent form travelling waves and their stability are important in describing the time evolution of the waves, but also a new kind of family of solutions which has two basic frequencies plays a crucial role in the dynamics of the waves. It is also shown that three is the minimum number of free waves which is necessary to have chaotic behaviour of water waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 341 - 372
Copyright
© 1988 Cambridge University Press

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References

Abraham, R. & Marsden, J. E. 1978 Functions of Mechanics, 2nd edn. Benjamin/Cummings, Massachusetts.
Arnold, V. I. 1978 Mathematical Methods of Classical Mechanics. Springer.
Benjamin, T. B. 1984 Impulse, Force and Variational Principles. IMA J. Appl. Maths 32, 3–68.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains in deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430–446.Google Scholar
Brown, E. W. 1911 On the oscillating orbits about the triangular equilibrium points in the problem of three bodies. Mon. Not. R. Astr. Soc. 71, 492–502.Google Scholar
Buchanan, D. 1941 Trojan satellites (limiting case). Trans. R. Soc. Canada §3, 3, 9–25.Google Scholar
Caponi, E. A., Saffman, P. G. & Yuen, H. C. 1982 Instability and confined chaos in a nonlinear dispersive wave system. Phys. Fluids 25, 2159–2166.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
Devaney, R. L. 1976 Homoclinic orbits in Hamiltonian systems. J. Diffl Equat. 21, 431–438.Google Scholar
Doedel, E. J. & Kernevez, J. P. 1986 Software for continuation problems in ordinary differential equations with applications. Coltech Appl. Maths Rep.Google Scholar
Fermi, E., Pasta, J. & Ulam, S. 1955 Studies of nonlinear problems. In Collected Papers of Enrico Fermi, vol. 2, p. 978. University of Chicago Press, 1962.
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. Physicas 3 D, 468–486.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Hasimoto, H. & Ono, H. 1972 Nonlinear modulation of gravity waves. J. Phys. Soc. Japan 33, 805–811.Google Scholar
Jacobi, C. G. J. 1836 Sur le mouvement d'un point et sur un cas particulier du problème des trois corps. C. R. Acad. Sci. Paris 3, 59.Google Scholar
Lake, B., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiments. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Lerman, L. M. & Umanskii, I. L. 1984 On the existence of separatrix loops in four-dimensional systems similar to the integrable Hamiltonian systems. Physics Metals Metallogr. USSR 47, no. 3, 335–340.Google Scholar
Lighthill, M. J. 1965 Contributions to the theory of waves in nonlinear dispersive systems. J. Inst. Math. Appl. 1, 269.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplituade in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. 1985 Bifurcation in gravity waves. J. Fluid Mech. 151, 457–475.Google Scholar
Longuet-Higgins, M. S. 1986 Bifurcation and instability in gravity waves. Proc. R. Soc. Lond. A 403, 167–187.Google Scholar
MacKay, R. S. 1986a Stability of equilibria of Hamiltonian systems. In Nonlinear Phenomena and Chaos (ed. S. Sarkar), Adam Hilger.
MacKay, R. S. 1986b Introduction to the dynamics of area-preserving maps. Proceeding of the US Particle Accelerator School SLAC 1985, (ed. M. Month).
MacKay, R. S., Meiss, J. D. & Percival, I. C. 1984 Transport in Hamiltonian systems. Physica 13 D, 55–81.Google Scholar
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
Olver, P. J. 1986 Applications of Lie Groups to Differential equations. Springer.
Poincaré, H. 1985 Sur les courbes défines par les équations différentielles. J. Math. Pure Appl. ser. 4, 1, 167–244.Google Scholar
Poincaré, H. 1890 La problème des trois corps et les equations de la dynamique. Acta Math. 13, 1–270.Google Scholar
Routh, E. J. 1875 On Laplace's three particles, with a supplement on the stability of steady motion. Proc. Lond. Math. Soc. 6, 86–97.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
Shemer, L. & Stiassnie, M. 1985 Initial instabilities and long-time evolution of Stokes waves. The Ocean Surface: Wave Breaking, Turbulent Mixing and Radio Probing (ed. Y. Toba & H. Mitsuyasu), pp. 51–57. D. Reidel.
Stiassnie, M. & Kroszynski, U. I. 1982 Long-time evolution of an unstable water-wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Stiassnie, M. & Shemer, L. 1987 Energy computations for modulations of class I and II instabilities of Stokes wave. J. Fluid Mech. 174, 299–312.Google Scholar
Van der Meer, J. C. 1985 The Hamiltonian Hopf Bifurcation. Lectures Notes on Physics, 1160, Springer.
Van der Meer, J. C. 1986 Bifurcation at nonsemisimple 1:-1 resonance. Z. angew. Math. Phys. 37, 425–437.Google Scholar
Yuen, H. C. 1984 Order and chaos in the long-time evolution of a nonlinear wave train. Symposium on Wave Breaking, Turbulent Mixing, and Radio Probing of the Ocean Surface. Sendai, Japan (unpublished contribution).
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar
Zakharov, V. E. 1968 The instability 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. 1987a Weakly nonlinear nonsymmetric gravity waves on water of finite depth. J. Fluid Mech. 180, 371–385.Google Scholar
Zufiria, J. A. 1987b Nonsymmetric gravity waves on water of infinite depth. J. Fluid Mech. 181, 17–39.Google Scholar
Zufiria, J. A. 1987c Symmetry breaking in periodic and solitary gravity-waves on water of finite depth. J. Fluid Mech. 184, 183–206.Google Scholar
Zufiria, J. A. & Saffman, P. G. 1986 The superharmonic instability of finite amplitude surface waves on water of finite depth. Stud. Appl. Maths 74, 259–266.Google Scholar