Diffusion limit of the damped heat-transmission equation ($D=1/(2\alpha )$)

At small wavenumber compared with $\alpha c$, the slow decay rate of the damped wave operator is $k^2/(2\alpha )$ and the other root decays on time $1/(2\alpha c^2)$. Long-time, long-wavelength motion therefore approaches heat diffusion with the displayed diffusivity. The original hyperbolic equation nevertheless has finite signal speed $c$, unlike the parabolic diffusion approximation.