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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.

Put $\lambda ={\omega }^2/g$ andThis polynomial is a harmonic function: $Q_{xx}+Q_{zz}=0$. Also $Q_x(0,z)=i\omega c$, $Q_x(L,z)=0$ and $Q_z(x,-h)=0$. Thus $G$ satisfies Laplace's equation with homogeneous Neumann boundary conditions on both side walls and the bottom. Its Fourier cosine series consequently has the formThe surface condition becomes $G_z-\lambda G=H(x)$, whereFor a nonzero forcing frequency away from the natural frequencies, the coefficients of the oscillating-wall forcing of a rectangular tank areHere $k_0=0$ in the definition of $H_0$. In fact $H_n=2i\omega c\lambda /(L{k}_n^2)$ for $n\ge 1$, since ${\int }_0^L(L-x{)}^2\cos (k_{n}x)dx=2L/k_n^2$. The zero-mode coefficient ensures that the mean surface elevation is $hc/L$ in complex amplitude, exactly the first-order volume conservation adjustment to the moving wall.