Abstract
The formulation of Hamilton's principle for the Eulerian description of the motion of an ideal fluid leads to the identification of the pressure as a Lagrangian density. The resulting variational principle for irrotational gravity waves on the surface of a homogeneous fluid yields both Laplace's equation in the interior and the free-surface boundary conditions. The velocity potential at, and the displacement of, the free surface are canonical variables in Hamilton's sense. The corresponding canonical equations are of Boussinesq's type in the regime of weak dispersion and weak nonlinearity; they are valid only for relatively long waves but may be either stable or unstable with respect to short waves, depending on whether the corresponding Hamiltonian is or is not positive definite, as originally pointed out by Broer. The Korteweg-deVries and related equations are discussed.
Similar content being viewed by others
References
Ablowitz MJ, Kaup DJ, Newell AC and H. Segur (1974) Studies in Appl Math 53: 249.
Bateman H (1929) Proc Roy Soc Lond A 125: 598.
Benjamin TB (1974) Lectures in Appl Math 15: 3.
Benjamin TB, Bona JL and JJ Mahony (1972) Phil Trans Roy Soc Lond A 272: 47.
Boussinesq J (1872) J Math Pures Appl (2) 17: 55.
Broer LJF (1974a) Physica 76: 364.
Broer LJF (1974b) Appl Sci Res 29: 430.
Broer LJF (1975a) Appl Sci Res 31: 377.
Broer LJF (1975b) Physica 79A: 583.
Clebsch A (1859) J reine angew Math 56: 1.
Finlayson BA (1972) The Method of Weighted Residuals and Variational Principles, Ch. 8. Academic Press.
Fornberg B and GB Whitham (1978) Phil Trans Roy Soc Lond A 289: 373.
Gardner CS (1971) J Math Phys 12: 1548.
Gardner CS, Greene JM, Kruskal MD and RM Miura (1967) Phys Rev Lett 19: 1095.
Kelvin, Lord (1849) Mathematical and Physical Papers 1: 107.
Lamb H (1932) Hydrodynamics. Cambridge University Press.
Lin CC (1963) Liquid helium. Proc Int School of Physics, Course XXI (ed G Careri), p.93ff. Academic Press.
Luke JC (1967) J Fluid Mech 27: 395.
Milder DM (1977) J Fluid Mech 83: 159.
Miles JW (1976) J Fluid Mech 75: 419.
Miles JW (1977) J Fluid Mech 83: 153.
Miura RM, Gardner CS and MD Kruskal (1968) J Math Phys 9: 1204.
Peregrine DH (1966) J Fluid Mech 25: 321.
Seliger RE and GB Whitham (1968) Proc Roy Soc Lond A 305: 1.
Serrin J (1959) Handbuch der Physik 8/1, 144, 161, 203.
Truesdell C and RA Toupin (1960) Handbuch der Physik 3/1, 594.
Watson KM and BJ West (1975) J Fluid Mech 70: 815.
Whitham GB (1967) Proc Roy Soc Lond A 299: 6.
Whitham GB (1974) Linear and Nonlinear Waves. Wiley.
Zakharov VE (1968) J Appl Mech Tech 9: 190.
Zakharov VE and LD Faddeev (1972) Functional Anal Appl 5: 280.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Miles, J.W. Hamiltonian formulations for surface waves. Applied Scientific Research 37, 103–110 (1981). https://doi.org/10.1007/BF00382621
Issue Date:
DOI: https://doi.org/10.1007/BF00382621