Weak formulation of free-surface wave equations
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, where the are time-dependent coefficients. For a tank with vertical sides the are eigenfunctions of the eigenvalue problem, Evolution equations for the can be obtained by taking inner products of the linearised equation of motion, 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 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 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.