 Solution

When $k^{(p)}\gg k_J^{(p)}$, omit the gravitational term and define $\omega (t)=c_s(t)k/a(t)$. ThenThis is an acoustic damped harmonic oscillator with time-dependent coefficients. Freezing the background locally at $t_0$, its amplitude damping rate is $H_0$, soThe oscillation-averaged quadratic oscillator energy decays twice as fast in this frozen-frequency model, with time $1/(2H_0)$. Stating which quantity is damped avoids a factor-of-two convention error. Locally the solution isFor evolution over many expansion times, retain the varying frequency. Put ${\delta }_{\mathbf{k}}=\chi /a$. Direct substitution givesWhen $\omega \gg H$ and $|\dot{\omega }|\ll {\omega }^2$, the WKB approximation yieldsThese are rapidly oscillating sound waves in comoving density, diluted by the expanding background. For constant $c_s$ their envelope scales as $a^{-1/2}$, not universally as $a^{-1}$, because $\omega \propto a^{-1}$. In the Einstein-de Sitter universe, introducing conformal time $d\eta =dt/a$ gives the exact constant-$c_s$ pressure-only solution ${\delta }_{\mathbf{k}}=[C\sin (k{c}_s\eta )+D\cos (k{c}_s\eta )]/\eta$, consistent with $a\propto {\eta }^2$. For that background the frozen-coefficient damping time is $3t/2$, while the slowly varying envelope has instantaneous decay time $2/H$.