Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T01:27:23.924Z Has data issue: false hasContentIssue false
  • English
  • Français

An explicit Hamiltonian formulation of surface waves in water of finite depth

Published online by Cambridge University Press:  26 April 2006

A. C. Radder
A. C. Radder
Affiliation:
Rijkswaterstaat, Tidal Waters Division, P.O. Box 20907, 2500 EX The Hague, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

A variational formulation of water waves is developed, based on the Hamiltonian theory of surface waves. An exact and unified description of the two-dimensional problem in the vertical plane is obtained in the form of a Hamiltonian functional, expressed in terms of surface quantities as canonical variables. The stability of the corresponding canonical equations can be ensured by using positive definite approximate energy functionals. While preserving full linear dispersion, the method distinguishes between short-wave nonlinearity, allowing the description of Stokes waves in deep water, and long-wave nonlinearity, applying to long waves in shallow water. Both types of nonlinearity are found necessary to describe accurately large-amplitude solitary waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 237 , April 1992 , pp. 435 - 455
Copyright
© 1992 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

Abarbanel, H. D. I., Brown, R. & Yang, Y. M. 1988 Hamiltonian formulation of inviscid flows with free boundaries. Phys. Fluids 31, 2802–2809.Google Scholar
Abramowitz, M. & Stegun, I. A. 1968 Handbook of Mathematical Functions. Dover.
Anthony, K.-H. 1987 Entropy and dynamical stability - a method due to Lagrange-formalism as applied to thermodynamics of irreversible processes. In Trends in Applications of Mathematics to Mechanics, Proc. 7th. Symp. (ed. J. F. Besseling & W. Eckhaus), pp. 297–320. Wassenaar.
Baker, C. T. H. 1977 The Numerical Treatment of Integral Equations. Oxford.
Baker, C. T. H. & Miller, G. F. 1982 (eds) Treatment of Integral Equations by Numerical Methods. Proc. Symp. Durham, July 1982. Academic.
Benjamin, T. B. 1984 Impulse, flow force and variational principles. IMA J. Appl. Maths 32, 3–68.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
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430–446.Google Scholar
Broer, L. J. F. 1975 Approximate equations for long water waves. Appl. Sci. Res. 31, 377–395.Google Scholar
Broer, L. J. F., Groesen, E. W. C. van & Timmers, J. M. W. 1976 Stable model equations for long water waves. Appl. Sci. Res. 32, 619–636.Google Scholar
Broer, L. J. F. & Kobussen, J. A. 1972 Canonical transformations and generating functionals. Physica 61, 275–288.Google Scholar
Byatt-Smith, J. G. B. 1970 An exact integral equation for steady surface waves. Proc. R. Soc. Lond. A 315, 405–418.Google Scholar
Byatt-Smith, J. G. B. 1971 An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49, 625–633.Google Scholar
Creamer, D. B., Henyey, F., Schult, R. & Wright, J. 1989 Improved linear representation of ocean surface waves. J. Fluid Mech. 205, 135–161.Google Scholar
Davies, B. 1985 Integral transforms and their applications, 2nd edn. Springer.
Dingemans, M. W. 1992 Water Wave Propagation over Uneven Bottoms. World Scientific (to be published).
Fenton, J. D. 1985 A fifth-order Stokes theory for steady waves. J. Waterway, Port, Coastal Ocean Engng Div. ASCE 111, 216–234. (Also, Discussion and errata: 113 (1987), 437–438).Google Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison-Wesley.
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals, Series and Products. Academic.
Henyey, F. S. 1983 Hamiltonian description of stratified fluid dynamics. Phys. Fluids 26, 40–47.Google Scholar
Katopodes, N. D. & Dingemans, M. W. 1989 Hamiltonian approach to surface wave models. In Computational Modelling and Experimental Methods in Hydraulics (HYDROCOMP ′89) (ed. Č. Maksimović & M. Radojković), pp. 137–147. Elsevier.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lewis, D., Marsden, J., Montgomery, R. & Ratiu, T. 1986 The Hamiltonian structure for dynamic free boundary problems. Physica 18D, 391–404.Google Scholar
Lighthill, M. J. 1962 An Introduction to Fourier Analysia and Generalized Functions. Cambridge University Press.
Longuet-Higgins, M. S. 1974 On the mass, momentum, energy and circulation of a solitary wave. Proc. R. Soc. Lond. A 337, 1–13.Google Scholar
Longuet-Higgins, M. S. 1983 On integrals and invariants for inviscid, irrotational flow under gravity. J. Fluid Mech. 134, 155–159.Google Scholar
Madsen, P. A., Murray, R. & SoSrensen, O. R. 1991 A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Engng 15, 371–388.Google Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. Second printing with corrections. World Scientific.
Messiah, A. 1969 Quantum Mechanics. North-Holland.
Milder, D. M. 1977 A note regarding ‘On Hamilton's principle for surface waves'. J. Fluid Mech. 83, 159–161.Google Scholar
Milder, D. M. 1990 The effects of truncation on surface-wave Hamiltonians. J. Fluid Mech. 217, 249–262.Google Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Miles, J. W. 1980 Solitary waves. Ann. Rev. Fluid Mech. 12, 11–43.Google Scholar
Miles, J. W. 1981 Hamiltonian formulations for surface waves. Appl. Sci. Res. 37, 103–110.Google Scholar
Neyzi, F. & Kutku, Y. 1987 Canonical structures for dispersive waves in shallow water. J. Math. Phys. 28, 1499–1504.Google Scholar
Pennell, S. A. & Su, C. H. 1984 A seventeenth-order series expansion for the solitary wave. J. Fluid Mech. 149, 431–443.Google Scholar
Peregrine, D. H. 1985 Water waves and their development in space and time. Review lecture. Proc. R. Soc. Lond. A 400, 1–18.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
Saffman, P. G. 1988 Application of Hamiltonian methods to the structure and stability of water waves of permanent form. In Nonlinear Water Waves, IUTAM Symp., Tokyo 1987 (ed. K. Horikawa & H. Maruo), pp. 207–210. Springer.
Salmon, R. 1988 Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. 20, 225–256.Google Scholar
Schwartz, L. W. & Fenton, J. D. 1982 Strongly nonlinear waves. Ann. Rev. Fluid Mech. 14, 39–60.Google Scholar
Shields, J. J. & Webster, W. C. 1988 On direct methods in water wave theory. J. Fluid Mech. 197, 171–199.Google Scholar
Vujanovic, B. D. & Jones, S. E. 1989 Variational Methods in Nonconservative Phenomena. Academic.
West, B. J., Brueckner, K. A., Janda, R. S., Milder, D. M. & Milton, R. L. 1987 A new numerical method for surface hydrodynamics. J. Geophys. Res. 92 (C11), 11803–11824.Google Scholar
Whitham, G. B. 1967 Variational methods and applications to water waves. Proc. R. Soc. Lond. A 299, 6–25.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Witting, J. M. 1984 A unified model for the evolution of nonlinear water waves. J. Comput. Phys. 56, 203–236.Google Scholar
Woods, L. C. 1961 The Theory of Subsonic Plane Flow. Cambridge University Press.
Yasuda, T., Ukai, A. & Tsuchiya, Y. 1989 Group properties of swell propagating from deep to shallow water. Annuals of the Disaster Prevention Research Institute, Kyoto University, No. 32, B2, pp. 781–797 (in Japanese).
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. 1987 Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth. J. Fluid Mech. 184, 183–206.Google Scholar
Zwartkruis, T. J. G. 1991 Computation of solitary wave profiles described by a Hamiltonian model for surface waves. Part 1: Final report; Part 2: Appendices. ECMI-rep. Eindhoven University of Technology.