Solution
= Solution
At the equilibrium thickness $H_0$,
$$
\Omega_z^2H_0=CH_0^{-\gamma}.
$$
Set $H=H_0+h$ and retain terms linear in $h$. The <linearization> is
$$
\ddot h+(\gamma+1)\Omega_z^2h=0.
$$
Thus the <vertical breathing mode of an astrophysical disk> has angular frequency
$$
\boxed{\omega_H=\sqrt{\gamma+1}\,\Omega_z}.
$$