= Solution
In <tight coupling>, the scattering time $|\dot\tau|^{-1}$ is short compared with an acoustic period and an expansion time: $k/|\dot\tau|\ll1$ and $\mathcal H/|\dot\tau|\ll1$. Photons and baryons have nearly the same velocity; higher photon multipoles and the <photon-baryon velocity slip> are suppressed by powers of this small ratio. Here retain the collision operator as printed, which ignores <CMB polarization>. Its diffusion coefficient differs from the polarized result.
Neglect gravity and expansion, so $R$ is constant on the timescale under consideration. The quadrupole equation of the <photon Boltzmann hierarchy> is
$$
\dot\Theta_2+k\left(\frac37\Theta_3-\frac23\Theta_1\right)=\frac9{10}\dot\tau\Theta_2.
$$
To first order in $k/|\dot\tau|$, $\dot\Theta_2$ and $k\Theta_3$ are subleading relative to the dipole source. Hence
$$
\Theta_2\simeq-\frac{20}{27}\frac k{\dot\tau}\Theta_1,
\qquad \boxed{\Pi_\gamma\simeq-\frac49\frac k{\dot\tau}v_\gamma.}
$$
This is the <temperature-only tight-coupling quadrupole>. The negative collision rate is essential to its sign.
Put $\Delta=v_\gamma-v_b$. Subtracting the <baryon Euler equation with Thomson drag> from the <photon Euler equation> gives
$$
\dot\Delta=-\frac k4\delta_\gamma-\frac23k\Pi_\gamma
+\dot\tau\frac{1+R}{R}\Delta.
$$
The zeroth-order common velocity obeys $(1+R)\dot v=-k\delta_\gamma/4$. Solving the slip equation to its first nonzero order therefore gives
$$
\Delta\simeq\frac{R}{1+R}\frac{k\delta_\gamma}{4\dot\tau},\qquad
\dot\Delta\simeq\frac{R}{1+R}\frac{k\dot\delta_\gamma}{4\dot\tau}.
$$
The second expression assumes that $R$ and $\dot\tau$ vary only on the neglected background timescale. Terms from their variation would need retaining if cosmic expansion were restored. Multiplying the <baryon Euler equation with Thomson drag> by $R$ and adding it to the photon equation eliminates drag. Since $v_b=v_\gamma-\Delta$,
$$
(1+R)\dot v_\gamma=-\frac k4\delta_\gamma-\frac23k\Pi_\gamma+R\dot\Delta.
$$
Using $v_\gamma=3\dot\delta_\gamma/(4k)$ from the <photon continuity equation>, the quadrupole term contributes $(2/9)k\dot\delta_\gamma/\dot\tau$, and the slip term contributes $R^2k\dot\delta_\gamma/[4(1+R)\dot\tau]$. Hence
$$
\boxed{(1+R)\dot v_\gamma=-\frac k4\delta_\gamma
-\frac k4|\dot\tau|^{-1}\left(\frac{R^2}{1+R}+\frac89\right)\dot\delta_\gamma.}
$$
Differentiating the <photon continuity equation> gives the <photon-baryon diffusion damping equation>
$$
\boxed{\ddot\delta_\gamma+D\dot\delta_\gamma+c_s^2k^2\delta_\gamma=0,\quad
c_s^2=\frac1{3(1+R)},\quad
D=\frac{k^2|\dot\tau|^{-1}}{3(1+R)}\left(\frac{R^2}{1+R}+\frac89\right).}
$$
The sound speed reflects baryon inertia; the $R^2$ term is heat conduction through velocity slip, and the $8/9$ term is photon shear viscosity for the unpolarized hierarchy.
For constant coefficients the characteristic roots are $-D/2\pm\sqrt{D^2/4-c_s^2k^2}$. On the acoustic branch where $D<2c_sk$,
$$
\delta_\gamma(\eta)=e^{-D\eta/2}\left[A\cos(\omega\eta)+B\sin(\omega\eta)\right],\qquad
\omega=\sqrt{c_s^2k^2-D^2/4}.
$$
Thus the solutions are \b[damped acoustic oscillations], with positive diffusion damping growing as $k^2$. This is <Silk damping>. Slowly varying coefficients give an approximate envelope $\exp[-\frac12\int D\,d\eta]$ and acoustic phase $k\int c_s\,d\eta$. The mathematical critically damped and overdamped cases follow from the roots, but extrapolation beyond the tight-coupling range is not reliable. For very large $R$, the acoustic and damping scales must be compared explicitly rather than inferring underdamping from $k/|\dot\tau|\ll1$ alone.
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