Normal modes of surface gravity waves in a rectangular tank (${\omega }_{mn}^2=g{k}_{mn}\tanh (k_{mn}h)$)

For a tank with side lengths $a,b$, depth $h$, impermeable walls and no surface tension, the small-amplitude surface modes have horizontal factors $\cos (m\pi x/a)\cos (n\pi y/b)$ and $k_{mn}^2=(m\pi /a{)}^2+(n\pi /b{)}^2$. The velocity potential has vertical factor $\cosh (k_{mn}(z+h))$. Combining the linearized kinematic and dynamic conditions on the undisturbed surface gives ${\omega }_{mn}^2=g{k}_{mn}\tanh (k_{mn}h)$. The uniform $m=n=0$ elevation is excluded when the total water volume is fixed.