Sloshing Modes in a Cuboidal Tank

Starting from Laplace's equation in Cartesian coordinates, assume that the boundary and free-surface conditions are all equal to zero and that the solution is separable. Let be the height of the liquid when it is still. The solution at time is:

,

where is the frequency and the gravitational acceleration.

References

[1] P. Hagedorn and A. DasGupta, Vibrations and Waves in Continuous Mechanical Systems, Chichester, UK: John Wiley & Sons, 2007.

[2] J. Kobine. Sloshing in a Square Tank [Video]. (Jan 31, 2013) www.youtube.com/watch?v=Yh3FWtecWIc.