= Solution
When $k^{(p)}\gg k_J^{(p)}$, omit the gravitational term and define $\omega(t)=c_s(t)k/a(t)$. Then
$$
\boxed{\ddot\delta_{\mathbf k}+2H\dot\delta_{\mathbf k}+\omega^2\delta_{\mathbf k}=0.}
$$
This is an acoustic <damped harmonic oscillator> with time-dependent coefficients. Freezing the background locally at $t_0$, its amplitude damping rate is $H_0$, so
$$
\boxed{\tau_{\mathrm{amp}}=H_0^{-1},\qquad
T_{\mathrm{osc}}=\frac{2\pi}{\sqrt{\omega_0^2-H_0^2}}
\simeq\frac{2\pi a_0}{c_{s0}k}\quad(\omega_0\gg H_0).}
$$
The 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 is
$$
\delta_{\mathbf k}\simeq e^{-H_0(t-t_0)}
\left[C\cos\bigl(\omega_d(t-t_0)\bigr)+D\sin\bigl(\omega_d(t-t_0)\bigr)\right],
\quad\omega_d=\sqrt{\omega_0^2-H_0^2}.
$$
For evolution over many expansion times, retain the varying frequency. Put $\delta_{\mathbf k}=\chi/a$. Direct substitution gives
$$
\ddot\chi+\left(\omega^2-\frac{\ddot a}{a}\right)\chi=0.
$$
When $\omega\gg H$ and $|\dot\omega|\ll\omega^2$, the <WKB approximation> yields
$$
\boxed{\delta_{\mathbf k}(t)\simeq
\frac{1}{a(t)\sqrt{\omega(t)}}
\left[C\cos\!\left(\int^t\omega(s)\,ds\right)+
D\sin\!\left(\int^t\omega(s)\,ds\right)\right].}
$$
These 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 $\delta_{\mathbf k}=[C\sin(kc_s\eta)+D\cos(kc_s\eta)]/\eta$, 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 $2/H$.
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