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The superharmonic instability of finite-amplitude water waves

Published online by Cambridge University Press:  20 April 2006

P. G. Saffman
P. G. Saffman
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125
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Abstract

Zakharov's (1968) Hamiltonian formulation of water waves is used to prove analytically Tanaka's (1983) numerical result that superharmonic disturbances to periodic waves of permanent form exchange stability when the wave energy is an extremum as a function of wave height. Tanaka's (1985) explanation for the non-appearance of superharmonic bifurcation is also derived, and the non-existence of stability exchange when the wave speed is an extremum is explained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 159 , October 1985 , pp. 169 - 174
Copyright
© 1985 Cambridge University Press

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References

Chen, B. & Saffman, P. G. 1980 Numerical evidence for the existence of new types of gravity wave of permanent form on deep water. Stud. Appl. Maths. 62, 1–21.Google Scholar
Gantmacher, F. R. 1960 Matrix Theory. Chelsea.
Garabedian, P. R. 1965 Surface waves of finite depth. J. d'Anal. Math. 14, 161–169.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. I. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. 1984 On the stability of steep gravity waves. Proc. R. Soc. Lond. A 396, 269–280.Google Scholar
Mclean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three dimensional instability of finite amplitude water waves. Phys. Rev. Lett. 46, 817–820.Google Scholar
Saffman, P. G. 1980 Long wavelength bifurcation of gravity waves on deep water. J. Fluid Mech. 101, 567–581.Google Scholar
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 2047–3055.Google Scholar
Tanaka, M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 000, 000–000.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zufiria, J. & Saffman, P. G. 1985 An example of stability exchange in a Hamiltonian wave system. Stud. Appl. Maths (to appear).Google Scholar