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 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 ${\omega }_{mn}^2=g{k}_{mn}\tanh (k_{mn}h)$. 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 $k_{mn}h\ll 1$, the shallow-water approximation gives ${\omega }_{mn}\sim \sqrt{gh}{k}_{mn}$; for $k_{mn}h\gg 1$, the deep-water gravity wave relation ${\omega }_{mn}^2\sim g{k}_{mn}$ follows. An oscillating-wall forcing of a rectangular tank couples to these same normal modes and becomes resonant at their natural frequencies.

For a tank $0<x<L$, $-h<z<0$, let the left wall move by $\mathrm{Re}(c{e}^{-i\omega t})$ with small amplitude, while the right wall and bottom remain fixed. A periodic velocity potential $\mathrm{Re}(f{e}^{-i\omega t})$ has $f_x(0,z)=-i\omega c$. Add the harmonic functionto obtain $G=f+Q$ with homogeneous wall and bottom Neumann boundary conditions. Thus $G=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (k_{n}x)\cosh (k_n(z+h))$, where $k_n=n\pi /L$. The linearized free-surface boundary conditions give $G_z-\lambda G=H$ at $z=0$, with $\lambda ={\omega }^2/g$ andIts Fourier cosine coefficients $H_n=(2/L){\int }_0^{L}H(x)\cos (k_{n}x)dx$ give the title expression for $n\ge 1$ and $a_0=-H_0/\lambda$. In particular $H_n=2i\omega c\lambda /(L{k}_n^2)$ for $n\ge 1$. The denominator vanishes at the corresponding normal modes of surface gravity waves in a rectangular tank, giving resonance. Exact undamped resonance produces secular growth instead of a bounded periodic response. Free transients do not generally decay in this inviscid model.