= Diffusion limit of the damped heat-transmission equation
{title2=$D=1/(2\alpha)$}
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.
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