= Travelling-pressure response of finite-depth surface gravity waves
{title2=$A=-\frac{p_1}{\rho[g-c^2k\coth(kh)]}$}
For an <inviscid flow> of constant <mass density> $\rho$ and depth $h$, impose excess <pressure> $p_1\cos(k(x-ct))$ with $k>0$ on a <free surface>. Neglect <surface tension>. A travelling solution of the <linearized free-surface boundary conditions> is
$$
\eta=A\cos(k(x-ct)),\qquad
\phi=\frac{Ac}{\sinh(kh)}\cosh(k(y+h))\sin(k(x-ct)).
$$
The <velocity potential> satisfies the <Laplace equation> and the bed <no-penetration boundary condition>. Its surface <time derivative> is $-Ac^2k\coth(kh)\cos(k(x-ct))$, so the dynamic condition gives the displayed signed response. The nonnegative <amplitude> is $|A|$. It is finite except at
$$
c_*^2=\frac{g\tanh(kh)}k,
$$
the squared <phase speed> of a free <surface gravity wave> at that <wavenumber>. Equality produces <resonance>. This describes a travelling forced component; free <surface gravity waves> must be added to meet general <initial conditions>. At exact <resonance>, a solution of the <initial value problem> generally grows with time rather than having a bounded periodic <amplitude>.
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