= 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 negligible <surface tension>, the small-amplitude <surface gravity wave> <normal 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 $\cosh(k_{mn}(z+h))$, satisfying the bottom <Neumann boundary condition>. Combining the <linearized free-surface boundary conditions> gives $\omega_{mn}^2=gk_{mn}\tanh(k_{mn}h)$. The uniform $m=n=0$ <surface elevation> is excluded when the total water volume is fixed. Taking one horizontal index zero gives the two-dimensional tank modes.
For $k_{mn}h\ll1$, the <shallow-water approximation> gives $\omega_{mn}\sim\sqrt{gh}\,k_{mn}$; for $k_{mn}h\gg1$, the <deep-water gravity wave> relation $\omega_{mn}^2\sim gk_{mn}$ follows. An <oscillating-wall forcing of a rectangular tank> couples to these same <normal modes> and becomes resonant at their natural frequencies.
Back to article page
