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The first mechanism is Silk damping: finite photon mean free paths allow radiation to diffuse out of compressions, while velocity slip and radiation shear dissipate acoustic motion. A Fourier diffusion equation has $X^{\prime }=-D(\tau )k^{2}X$, hence $X\propto \exp [-k^2\int Dd\tau ]$. For the polarization-inclusive tightly coupled plasma, a useful expression isThe positive scattering rate is $\dot{\kappa }=a{n}_e{\sigma }_T$, and $R$ is the baryon loading parameter. The two terms represent heat conduction from velocity slip and photon shear viscosity. The mode amplitude has an approximately Gaussian cutoff in $k$; power is suppressed by $e^{-2k^2/k_D^2}$. The formula assumes tight coupling and slowly varying coefficients before decoupling.

The second mechanism is finite-width last-scattering damping. Photons emerge from a range of times and depths, so rapidly varying sources are averaged rather than observed at one sharply defined phase. Approximate the normalized cosmological visibility function by a Gaussian of width ${\sigma }_{\tau }$. For a radial plane-wave phase,The associated power factor is $e^{-(k\mu {\sigma }_{\tau }{)}^2}$. An acoustic factor $e^{\pm ik{c}_s\tau }$ is likewise averaged, replacing $k\mu$ by $k(\mu \pm c_s)$ in the constant-sound-speed approximation. Thus fine radial structure and rapid acoustic time oscillations are suppressed once their phase changes substantially across the visibility width. This is an averaging effect after the perturbations exist; Silk damping physically erases them by diffusion. Angular projection and the evolving source determine the final isotropic damping envelope, so the radial $\mu$ dependence must not be silently discarded.