= Normal modes of surface gravity waves in a rectangular tank
{title2=$\omega_{mn}^2=gk_{mn}\tanh(k_{mn}h)$}
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 $\omega_{mn}^2=gk_{mn}\tanh(k_{mn}h)$. The uniform $m=n=0$ elevation is excluded when the total water volume is fixed.
Back to article page
