Forced oscillations of a rectangular tank

The tank has width 2a and depth h and sways from side to side with a periodic motion of radian frequency ω. For a given frequency ω of oscillation the wavenumber k of waves in the tank is given by the real positive solution of

ω² = gk tanh kh .

The linearised solution is unbounded for ka = (n -1/2)π where n is a positive integer. These values of ka correspond to antisymmetric standing waves in a fixed tank.

Animations for various values of ka are given below.

For values of ka close to (n -1/2)π the amplitude of the fluid motion is large.

For values of ka not close to ka = (n -1/2)π, the amplitude of the fluid motion is small.

When the tank is oscillated with a higher frequency (equivalent to a larger value of ka) there are more spatial oscillations within the tank.