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Two-timing, variational principles and waves

  • G. B. Whitham (a1)

In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.

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Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Waltham, Mass.: Blaisdell.
Kuzmak, G. E. 1959 Prikl. Mat. Mekh. Akad. Nauk S.S.S.R. 23, 515. (Translated in Appl. Math. Mech. 23, 73.)
Landau, L. D. & Lifshitz, E. M. 1960 Mechanics. Oxford: Pergamon.
Luke, J. C. 1966 Proc. Roy. Soc. A 292, 403.
Whitham, G. B. 1965a Proc. Roy. Soc. A 283, 238.
Whitham, G. B. 1965b J. Fluid Mech. 22, 27.
Whitham, G. B. 1967a J. Fluid Mech. 27, 39.
Whitham, G. B. 1967b Proc. Roy. Soc. A 299, 6.
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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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