Slowly varying shear-damped photon oscillator (${\delta }_{\gamma }=e^{-\frac{2}{15}{k}^2\int {\tau }_{c}d\tau }y,y^{\prime \prime }+[k^2/3-\frac{2}{15}{k}^2{\tau }_c^{\prime }+O({\tau }_c^2)]y=0$)

Removing the first derivative from the truncated photon diffusion equation gives the displayed transformed oscillator. Constant mean free time changes its oscillation frequency only at second order. For a slowly varying mean free time and subhorizon wavelengths, derivative corrections to the phase and prefactor are small; the leading solution is an acoustic sine/cosine multiplied by $\exp [-k^2/k_D^2]$, where $k_D^{-2}=2\int {\tau }_{c}d\tau /15$. Dropping the derivative correction requires slow variation as well as a small mean free time; it is not an exact solution for arbitrary time dependence.