= Oscillating-wall forcing of a rectangular tank
{title2=$a_n=\frac{H_n}{k_n\sinh(k_nh)-(\omega^2/g)\cosh(k_nh)}$}
For a tank $0<x<L$, $-h<z<0$, let the left wall move by $\operatorname{Re}(ce^{-i\omega t})$ with small <amplitude>, while the right wall and bottom remain fixed. A periodic <velocity potential> $\operatorname{Re}(f e^{-i\omega t})$ has $f_x(0,z)=-i\omega c$. Add the <harmonic function>
$$
Q(x,z)=\frac{i\omega c}{2L}[(z+h)^2-(L-x)^2]
$$
to obtain $G=f+Q$ with homogeneous wall and bottom <Neumann boundary conditions>. Thus $G=a_0/2+\sum_{n\geq1}a_n\cos(k_nx)\cosh(k_n(z+h))$, where $k_n=n\pi/L$. The <linearized free-surface boundary conditions> give $G_z-\lambda G=H$ at $z=0$, with $\lambda=\omega^2/g$ and
$$
H(x)=\frac{i\omega c}{L}\left[h-\frac\lambda2(h^2-(L-x)^2)\right].
$$
Its <Fourier cosine coefficients> $H_n=(2/L)\int_0^L H(x)\cos(k_nx)dx$ give the title expression for $n\geq1$ and $a_0=-H_0/\lambda$. In particular $H_n=2i\omega c\lambda/(Lk_n^2)$ for $n\geq1$. 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.
Back to article page
