= Solution
For a <barotropic equation of state>, $\delta P=c_s^2\delta\rho=c_s^2\bar\rho\,\delta$. A spatial <Fourier mode> of comoving wavenumber $k$ therefore satisfies
$$
\ddot\delta_k+2H\dot\delta_k
+\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta_k=0.
$$
The <Jeans wavenumber> is defined by equality of the pressure and self-gravity terms:
$$
\boxed{k_J=\frac{a\sqrt{4\pi G\bar\rho}}{c_s}}.
$$
For $k\gg k_J$, pressure dominates and produces acoustic oscillations whose amplitude is affected by <Hubble friction>. For $k\ll k_J$, self-gravity dominates and a growing mode develops: this is <Jeans instability>.
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