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Weakly dispersive nonlinear gravity waves

Published online by Cambridge University Press:  20 April 2006

John Miles  and
Rick Salmon
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
Rick Salmon
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
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Abstract

The equations for gravity waves on the free surface of a laterally unbounded inviscid fluid of uniform density and variable depth under the action of an external pressure are derived through Hamilton's principle on the assumption that the fluid moves in vertical columns. The resulting equations are equivalent to those of Green & Naghdi (1976). The conservation laws for energy, momentum and potential vorticity are inferred directly from symmetries of the Lagrangian. The potential vorticity vanishes in any flow that originates from rest; this leads to a canonical formulation in which the evolution equations are equivalent, for uniform depth, to Whitham's (1967) generalization of the Boussinesq equations, in which dispersion, but not nonlinearity, is assumed to be weak. The further approximation that nonlinearity and dispersion are comparably weak leads to a canonical form of Boussinesq's equations that conserves consistent approximations to energy, momentum (for a level bottom) and potential vorticity.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 157 , August 1985 , pp. 519 - 531
Copyright
© 1985 Cambridge University Press

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References

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
Boussinesq M. J.1871 Théorie de l'intumescence liquide, appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire Acad. Sci. Paris, C. R. 72, 755–759.Google Scholar
Bretherton F. P.1970 A note on Hamilton's principle for perfect fluids. J. Fluid Mech. 44, 19–31.Google Scholar
Ertekin R. C.1984 Soliton generation by moving disturbances in shallow water: theory, computation and experiment. Ph.D. Dissertation, University of California, Berkeley.
Ertekin R. C., Webster, W. C. & Wehausen J. V.1984 Ship-generated solitons. Proc. 15th Symp. Naval Hydrodynamics. National Academy of Sciences, Washington, D.C. (In press.)
Ertel H.1942 Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z. 59, 277–281.Google Scholar
Goldstein H.1980 Classical Mechanics, 2nd edn. Addison-Wesley.
Green, A. E. & Naghdi P. M.1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.Google Scholar
Greenspan H. P.1969 The Theory of Rotating Fluids. Cambridge University Press.
Lin C. C.1963 Liquid helium. In Proc. Intl School of Physics, Course XXI (ed. G. Careri), pp. 93ff. Academic.
Lin, C. C. & Clark A.1959 On the theory of shallow water waves. Tsing Hua J. Chinese Studies 1, 55–61.Google Scholar
Long R. R.1964 The initial-value problem for long waves of finite amplitude. J. Fluid Mech. 20, 161–179.Google Scholar
Mei, C. C. & Le Méhauté B.1966 Note on the equations of long waves on an uneven bottom. J. Geophys. Res. 71, 393–400.Google Scholar
Miles J. W.1976 Solitary waves. Ann. Rev. Fluid Mech. 12, 11–43.Google Scholar
Miles J. W.1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Miles J. W.1985 Surface waves in basins of variable depth. J. Fluid Mech. 152, 379–389.Google Scholar
Noether E.1918 Invariante Variationsprobleme. Nachr. Ges. Gött. math.-phys. Kl pp. 235–257.Google Scholar
Peregrine D. H.1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Rayleigh Lord1876 On waves Phil. Mag. (5) 1, 257–279 (In Scientific Papers, vol. 1, pp. 251–271).Google Scholar
Salmon R.1983 Practical use of Hamilton's principle. J. Fluid Mech. 132, 431–444.Google Scholar
Seliger, R. L. & Whitham G. B.1968 Variational principles in continuum mechanics Proc. R. Soc. Lond. A 305, 1–25.Google Scholar
Van Saarloos W.1981 A canonical transformation relating the Lagrangian and Eulerian description of ideal hydrodynamics. Physica 108A, 557–566.Google Scholar
Wehausen, J. V. & Laitone E. V.1960 Surface waves. Encyc. Phys. 9, 667.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-Interscience.
Wu T. Y.1981 Long waves. J. Engng Mech. Div. ASCE 107, (EM3), 501–522.Google Scholar
Yamada H.1958 On approximate expressions of solitary wave. Rep. Res. Inst. Appl. Mech. Kyushu Univ. 6, 35–47.Google Scholar