Weak formulation of water wave equations
by
Alfred Sneyd
University of Waikato

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement \eta is written as an eigenfunction expansion,

\eta = \infty å n=1  an (t) En ,

where the a_n(t) are time-dependent coefficients. For a tank with vertical sides the E_n are eigenfunctions of the eigenvalue problem,

Ñ2 E + \lambda2 E = 0,        ÑE · ^ n   = 0   on the tank side walls.

Evolution equations for the a_n(t) can be obtained by taking inner products of the linearised equation of motion,

\rho  \partialv \partialt = -  1 \rho ÑP + F

with the normal irrotational wave modes. For unforced waves each evolution equation is just a simple harmonic oscillator, but method is most useful when the body force F represents more than simple gravity. It is neatly manifested by a forcing term in the SHM evolution equation. It is not necessary to assume F irrotational.

Several applications will be considered: magnetic damping of surface waves, nonlinear oscillations in a tank, and the Faraday experiment.

Date received: September 8, 1999