= Solution
The denominators in part (v) vanish when the forcing frequency equals the natural <surface gravity wave> frequency
$$
\boxed{\omega=\Omega
=\sqrt{gk}
=\left(\frac{\pi g\sqrt5}{L}\right)^{1/2}}.
$$
At this value, the initially quiescent resonant solution is
$$
H(t)=-\frac{kp_0}{2\rho\Omega},t\sin(\Omega t),
$$
$$
F(t)=-\frac{p_0}{2\rho\Omega}
\bigl(\sin(\Omega t)+\Omega t\cos(\Omega t)\bigr).
$$
Both contain a term growing linearly in time. This is <resonance>: the pressure forcing repeatedly supplies energy in phase with the box's $(1,2)$ standing-wave mode. In the ideal inviscid linear model there is no damping or nonlinear saturation, so its amplitude is unbounded. This is the <resonance of a pressure-forced rectangular surface-gravity mode>.
Back to article page
