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Uniformly travelling water waves from a dynamical systems viewpoint: some insights into bifurcations from Stokes’ family

Published online by Cambridge University Press:  26 April 2006

C. Baesens  and
R. S. Mackay
C. Baesens
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
R. S. Mackay
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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Abstract

Numerical work of many people on the bifurcations of uniformly travelling water waves (two-dimensional irrotational gravity waves on inviscid fluid of infinite depth) suggests that uniformly travelling water waves have a reversible Hamiltonian formulation, where the role of time is played by horizontal position in the wave frame. In this paper such a formulation is presented. Based on this viewpoint, some insights are given into bifurcations from Stokes’ family of periodic waves. It is demonstrated numerically that there is a ‘fold point’ at amplitude A0 ≈ 0.40222. Assuming non-degeneracy of the fold and existence of an associated centre manifold, this explains why a sequence of p/q-bifurcations occurs on one side of A0, with 0 < p/q [les ] ½, in the order of the rationals. Secondly, it explains why no symmetry-breaking bifurcation is observed at A0, contrary to the expectations of some. Thirdly, it explains why the bifurcation tree for periodic uniformly travelling waves looks so much like that for the area-preserving Hénon map. Fourthly, it leads to predictions of a rich variety of spatially quasi-periodic, heteroclinic and chaotic waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 333 - 347
Copyright
© 1992 Cambridge University Press

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References

Amick C. J., Fraenkel, L. E. & Toland J. F. 1982 On Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214.Google Scholar
Amick, C. J. & KirchgaUssner K. 1989 A theory of solitary water-waves in the presence of surface tension. Arch. Rat. Mech. Anal. 105, 1–49.Google Scholar
Amick, C. J. & Turner R. E. L. 1989 Small internal waves in two-fluid systems. Arch. Rat. Mech. Anal. 108, 111–139.Google Scholar
Aston P. J. 1991 Analysis and computation of symmetry-breaking bifurcation and scaling laws using group theoretic methods. SIAM J. Math. Anal. 22, 181–212.Google Scholar
Benjamin T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Benjamin T. B. 1984 Impulse, flow 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 on deep water. J. Fluid Mech. 27, 417–430.Google Scholar
Benjamin, T. B. & Olver P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185.Google Scholar
Bridges T. J. 1991 Spatial Hamiltonian structure, energy flux and the water-wave problem. Preprint, Utrecht.
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
Eckmann, J.-P. & Procaccia 1990 The generation of spatio-temporal chaos in large aspect ratio hydrodynamics. Preprint.
HeAnon M. 1969 Numerical study of quadratic area-preserving mappings. Q. J. Appl. Maths 27, 291–312.Google Scholar
KirchgaUssner K. 1982 Wave solutions of reversible systems and applications. J. Diffl Equat. 45, 113–127.Google Scholar
KirchgaUssner K. 1988 Nonlinearly resonant surface waves and homoclinic bifurcations. Adv. Appl. Mech. 26, 135–181.Google Scholar
Longuet-Higgins M. S. 1984a On the stability of steep gravity waves Proc. R. Soc. Lond. A 396, 269–280.Google Scholar
Longuet-Higgins M. S. 1984b New integral properties for gravity waves of finite amplitude. J. Fluid Mech. 149, 205–215.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
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. 1982a Islets of stability beyond period doubling Phys. Lett. A 87, 321–324.Google Scholar
MacKay R. S. 1982b Renormalization in area-preserving maps. PhD thesis, Princeton (Univ. Microfilms, Ann Arbor).
MacKay R. S. 1987 Introduction to the dynamics of area-preserving maps. In Physics of Particle Accelerators (ed. M. Month & M. Dienes) Am. Inst. Phys. Conf. Proc. 153, vol. 1, pp. 534–602.
Maddocks J. H. 1987 Stability and folds. Arch. Rat. Mech. Anal. 99, 301–328.Google Scholar
Meyer K. R. 1970 Generic bifurcation of periodic points. Trans. Am. Math. Soc. 149, 95–107.Google Scholar
Mielke A. 1990 Inertial manifolds for elliptic problems in infinite cylinders. Preprint.
Mielke A. 1991 Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems. Lecture Notes in Mathematics, vol. 1489. Springer.
Rimmer R. 1983 Generic bifurcations from fixed points of involutory area-preserving maps. Mem. Am. Math. Soc. 41.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
Tanaka M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 156, 281–289.Google Scholar
Whitham G. G. 1974 Linear and Nonlinear Waves. Wiley.
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. 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