Weak formulation of free-surface wave equations. Issue 2 (2001) - Sneyd, A. D.
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 = 1 ∞ α n ( t ) E n where theα n ( t ) are time-dependent coefficients. For a tank with vertical sides theE n are eigenfunctions of the eigenvalue problem, ∇ 2 + λ 2 E = 0,    ∇ E ⋅ n ^ = 0  on the tank side walls . Evolution equations for theα n ( t ) can be obtained by taking inner products of the linearised equation of motion, ρ ∂ v ∂ t = − 1 ρ ∇ P + 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 forceF 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 assumeF 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.