= Surface gravity wave forced by a traveling bottom
{title2=$\frac ba=\frac{(kc)^2\operatorname{sech}(kd)}{(kc)^2-gk\tanh(kd)}$}
= Bottom-forced gravity wave
{synonym}
Consider fluid at rest with mean depth $d$ and a bottom displacement $a\sin(k(x-ct))$. The <linearized free-surface boundary conditions> at $y=0$ and moving-bottom <kinematic boundary condition> at $y=-d$ give a periodic particular response $\eta=b\sin(k(x-ct))$ and <velocity potential>
$$
\phi=-\frac b{kc}[g\cosh(ky)+kc^2\sinh(ky)]\cos(k(x-ct)).
$$
The bottom derivative fixes the signed response coefficient to the title formula. Here $k>0$, $c\ne0$, and the forcing must avoid $k^2c^2=gk\tanh(kd)$. This equality is <resonance> with the natural finite-depth <surface gravity wave>; nonzero undamped resonant forcing has no bounded periodic solution. The response reverses phase below resonance. General initial data can add free <surface gravity waves>, which do not decay in this inviscid model. Near resonance the resulting <wave steepness> and particle excursions, rather than the bottom displacement alone, determine whether the <linearization> remains valid.
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