= Solution
For the $n$th <normal mode>, the denominator just obtained is
$$
k_n\sinh(k_nh)-\frac{\omega^2}{g}\cosh(k_nh)=\frac{\cosh(k_nh)}g(\omega_n^2-\omega^2).
$$
Its forcing coefficient is nonzero for $c\ne0$ and $\omega>0$. Therefore the periodic response grows like $(\omega_n^2-\omega^2)^{-1}$ as $\omega$ approaches $\omega_n$: this is <resonance>. At exact resonance there is no bounded periodic particular solution in the linear inviscid model; the resonant amplitude instead contains a secular term proportional to $t\sin(\omega_nt)$ or $t\cos(\omega_nt)$. Physical damping and nonlinear effects eventually matter. Away from resonance, undamped free oscillations can persist alongside the periodic particular solution.
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