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Non-symmetric gravity waves on water of infinite depth

Published online by Cambridge University Press:  21 April 2006

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

Abstract

Two different numerical methods are used to demonstrate the existence of and calculate non-symmetric gravity waves on deep water. It is found that they appear via spontaneous symmetry-breaking bifurcations from symmetric waves. The structure of the bifurcation tree is the same as the one found by Zufiria (1987) for waves on water of finite depth using a weakly nonlinear Hamiltonian model. One of the methods is based on the quadratic relations between the Stokes coefficients discovered by Longuet-Higgins (1978a). The other method is a new one based on the Hamiltonian structure of the water-wave problem.

Type
Research Article
Copyright
© 1987 Cambridge University Press

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References

Amick, L. J., Fraenkel, L. E. & Toland, J. F. 1982 On the Stokes conjecture for the wave of extreme form. Acta Math., Stockh. 148, 193214.Google Scholar
Arnol'D, V. I.1978 Mathematical Methods of Classical Mechanics. Springer.
Arnol'D, V. I. & Avez, A.1968 Ergodic Problems of Classical Mechanics. Benjamin.
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430446.Google Scholar
Boussinesq, J. 1871 Théorie l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C.R. Acad. Sci. Paris 72, 755759.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. App. Maths 62, 121.Google Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison Wesley.
Green, J. M., Mackay, R. S., Vivaldi, F. & Feigenbaum, M. J. 1981 Universal behaviour in families of area-preserving maps. Physics 3 D, 468486.Google Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. W. Rabinowitz), pp. 359384. Academic.
Korteweg, P. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag. 39 (5), 422–443.Google Scholar
Longuet-Higgins, M. S. 1978a Some new relations between Stokes's waves coefficients in theory of gravity waves. J. Inst. Maths Applics 22, 261273.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360, 471488.Google Scholar
Longuet-Higgins, M. S. 1984 New integral relations for gravity waves of finite amplitude. J. Fluid Mech. 149, 205215.Google Scholar
Longuet-Higgins, M. S. 1985 Bifurcation in gravity waves. J. Fluid Mech. 151, 457475.Google Scholar
Mackay, R. S. 1986 Stability of equilibria of Hamiltonian systems. In Nonlinear Phenomena and Chaos (ed. S. Sarkar). Adam Hilger.
Mackay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. 406, 115125.Google Scholar
Rayleigh, Lord 1876 On waves. Phil. Mag. 1 (5), 257–279.Google Scholar
Saffman, P. G. 1980 Long wavelength bifurcation of gravity waves on deep water. J. Fluid Mech. 101, 567581.Google Scholar
Saffman, P. G. 1985 The superhamonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169174.Google Scholar
Sattinger, D. H. 1980 Bifurcation and symmetry breaking in applied mathematics. Bull. Am. Math. Soc. 3, 779819.Google Scholar
Sattinger, D. H. 1983 Branching in the presence of symmetry. CBMS-NSF Reg. Conf. in Appl. Math. SIAM, Philadelphia.
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 63, 553578.Google Scholar
Simmen, J. A. & Saffman, P. G. 1985 Steady deep-water waves on a linear shear current. Stud. Appl. Maths 73, 3557.Google Scholar
Stokes, G. G. 1849 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.Google Scholar
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 30473055.Google Scholar
Tanaka, M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 156, 281289.Google Scholar
Vanden-Broeck, J. M. 1983 Some new gravity waves in water of finite depth. Phys. Fluids 26, 23852387.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep fluid. J. Appl. Mech. Tech. Phys. 2, 190194.Google Scholar
Zufiria, J. A. 1987 Weakly nonlinear nonsymmetric gravity waves on water of finite depth. J. Fluid Mech.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, 259266.Google Scholar
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