Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1145.76325
Craig, Walter; Guyenne, Philippe; Nicholls, David P.; Sulem, Catherine
Hamiltonian long-wave expansions for water waves over a rough bottom. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2055, 839-873 (2005). ISSN 1364-5021; ISSN 1471-2946/e

Summary: This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long-wave asymptotic regime, extending the results of {\it R. Rosales} and {\it G. Papanicolaou} [Stud. Appl. Math. 68, 89--102 (1983; Zbl 0516.76018)] on periodic bottoms for two-dimensional flows. We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharov's Hamiltonian [{\it V. E. Zakharov}, J. Appl. Mech. Tech. Phys. 9, 1990--1994 (1968)] for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length-scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length-scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales and Papanicolaou (loc. cit.).\par In the two-dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg-de Vries (KdV) equation that is obtained in Rosales and Papanicolaou (loc. cit.). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long-scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three-dimensional long-wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev-Petviashvili (KP) system in the appropriate unidirectional limit.\par The computations for these results are performed in the framework of an asymptotic analysis of multiple-scale operators. In the present case this involves the Dirichlet-Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

MSC 2000:: *76B15 Wave motions (fluid mechanics)
35Q53 KdV-like equations
37K05 Hamiltonian structures, etc.
76M45 Asymptotic methods, singular perturbations

Keywords: Water waves; variable depth; long-wave asymptotics; Hamiltonian perturbation theory

Citations: Zbl 0516.76018

Cited in: Zbl 1207.76016

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]