Fourier-mode Green function of the telegraph equation ($G_k(\tau )=\Theta (\tau )c^2{e}^{-\gamma \tau }\sinh (d_k\tau )/d_k$)

For operator $c^{-2}{\partial }_t^2+2\alpha {\partial }_t-{\partial }_x^2$, define $\gamma =\alpha c^2$ and $d_k^2={\gamma }^2-c^2{k}^2$. Retarded support, continuity at zero time and the derivative jump $G_k^{\prime }(0^{+})=c^2$ fix the displayed normalization. Imaginary $d_k$ produces a damped sine; zero $d_k$ gives $c^2\tau e^{-\gamma \tau }$. The spatially uniform mode tends to $1/(2\alpha )$ rather than decaying to zero.

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.