Advances in Decision Sciences

Advances in Decision Sciences / 2001 / Article

Open Access

Volume 5 |Article ID 869125 | 11 pages | https://doi.org/10.1155/S1173912601000074

Weak formulation of free-surface wave equations

Abstract

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, η=n=1αn(t)En where the αn(t) are time-dependent coefficients. For a tank with vertical sides the En are eigenfunctions of the eigenvalue problem, 2+λ2E=0,  En^=0 on the tank side walls. Evolution equations for the αn(t) can be obtained by taking inner products of the linearised equation of motion, ρvt=1ρP+F with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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