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A general approach to linear and non-linear dispersive waves using a Lagrangian

  • G. B. Whitham (a1)
Abstract

The basic property of equations describing dispersive waves is the existence of solutions representing uniform wave trains. In this paper a general theory is given for non-uniform wave trains whose amplitude, wave-number, etc., vary slowly in space and time, the length and time scales of the variation in amplitude, wave-number, etc., being large compared to the wavelength and period. Dispersive equations may be derived from a variational principle with appropriate Lagrangian, and the whole theory is developed in terms of the Lagrangian. Boussinesq's equations for long water waves are used as a typical example in presenting the theory.

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Boussinesq, J. 1877 Essai sur la théorie des eaux courantes. Mém. Prés. Acad. Sci., Paris.
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, vol. i. New York: Interscience.
Korteweg, D. J. & de Vries, G. 1895 Phil. Mag. 39, 422.
Landau, L. D. & Lifshitz, E. M. 1960 Mechanics. London: Pergamon Press.
Noether, E. 1918 Invariant Variationsprobleme. Nachr. Ges. Göttingen, 235.
Rüssman, H. 1961 Arch. Rat. Mech. Anal. 8, 353.
Stokes, G. G. 1847 Camb. Trans. 8, (also Collected Papers, 1, 197).
Whitham, G. B. 1965 Proc. Roy. Soc. A, 283, 238.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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