 Solution

Set ${\delta }_R=e^{-S}u$ and neglect $S^{\prime \prime }$, $(S^{\prime }{)}^2$, and slow variation relative to $k^2$. Cancelling the first derivative gives $S^{\prime }=\alpha k/(2\mathcal{H})$, hence $S={\int }^{\tau }\alpha k/(2\mathcal{H})d{\tau }^{\prime }$, and $u^{\prime \prime }+(c^2{k}^2/3)u\simeq 0$. Thus acoustic oscillations are exponentially damped, unlike pressureless matter perturbations, which have a growing mode.