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∞αn(t)Enwhere theαn(t)are time-dependent coefficients. For a tank with vertical sides theEnare eigenfunctions of the eigenvalue problem,?2+λ2E=0,???E?n^=0?on the tank side walls.Evolution equations for theαn(t)can be obtained by taking inner products of the linearised equation of motion,ρ?v?t=?1ρ?P+Fwith 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 forceFis 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 assumeFirrotational. 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.
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