= Solution
{c}
The denominator of the travelling response vanishes at
$$
\boxed{c^*=\sqrt{g\tanh h}}.
$$
For the positive speeds in the paper, a finite travelling response exists whenever $c\ne c^*$. At $c=c^*$, the dynamic condition would reduce to $p_1/\rho=0$, which is impossible. The finite-depth <dispersion relation> of a <surface gravity wave> with <wavenumber> $k$ is $\omega^2=gk\tanh(kh)$. At $k=1$, its <phase speed> $\omega/k$ equals $c^*$: the moving <pressure> pattern drives a natural <surface gravity wave> at its own speed, producing <resonance>.
The restriction concerns a bounded travelling response. To distinguish this from existence of a solution of an <initial value problem>, set $T=\tanh h$, $P=p_1/\rho$, and $\omega_*^2=gT$. If
$$
\eta(x,t)=\operatorname{Re}(H(t)e^{ix}),\qquad
\phi(x,y,t)=\operatorname{Re}\!\left(
\frac{\dot H(t)}{\sinh h}\cosh(y+h)e^{ix}\right),
$$
the <linearized free-surface boundary conditions> reduce to the <forced harmonic oscillator>
$$
\boxed{\ddot H+\omega_*^2H=-PT\,e^{-ict}}.
$$
For example, if the forcing is switched on at $t=0$ from a flat surface with zero initial <velocity>, then, for $c\ne\omega_*$,
$$
H(t)=\frac{PT}{\omega_*^2-c^2}
\left[\cos(\omega_*t)-i\frac c{\omega_*}\sin(\omega_*t)-e^{-ict}\right].
$$
This satisfies $H(0)=\dot H(0)=0$ and contains both free oscillations and the forced component. At $c=\omega_*$ a solution of the <initial value problem> still exists, but its <particular solution> contains
$$
H_{\rm sec}(t)=-\frac{iPT}{2\omega_*}\,t e^{-i\omega_*t},
$$
which grows linearly with time. Thus the ideal <inviscid flow> model predicts resonant growth rather than a finite constant <amplitude>. Additional free oscillations can set the <initial conditions> but cannot remove that secular growth.
Back to article page
