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Weakly dispersive nonlinear gravity waves. (English) Zbl 0583.76013
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 a superficial pressure are derived under the assumption that the fluid moves in vertical columns. This assumption and the resulting equations of motion are equivalent to those of A. E. Green and P. M. Naghdi [ibid. 78, 237-246 (1976; Zbl 0351.76014)]. While these authors set out from conservation of energy and invariance under rigid translations, the Green-Naghdi (abbrev.: GN) equations and their invariants are, in the present paper, derived directly from Hamilton’s principle. A major advantage of this derivation is that consistent approximations to energy, impulse-momentum (for a level bottom) and potential vorticity are conserved if only approximations that preserve the original symmetries of the Lagrangian are introduced in the action integral. One of the aims of this paper is to establish the relation between the GN equations and those of Boussinesq. The fact that the potential vorticity vanishes in any flow originating from rest leads to a canonical formulation in which the evolution equations are equivalent - provided that the depth is uniform - to G. B. Whitham’s [e.g.: Proc. R. Soc. Lond., Ser. A 299, 6-25 (1967; Zbl 0163.211)] generalization of the Boussinesq equations for weak dispersion only. The further approximation on the assumption that both nonlinearity and dispersion are comparably weak leads to a canonical form of Boussinesq’s equations which exhibit the above-mentioned properties of conservation. The balance between nonlinearity and dispersion is intrinsic for the solitary waves, so that the GN and canonical Bouissinesq equations should be of comparable validity in the description of such waves and their interactions.
Reviewer: M.Biermann

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q99 Partial differential equations of mathematical physics and other areas of application
70H25 Hamilton’s principle
Full Text: DOI
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