 Fourier-mode Green function of the telegraph equation

For the operator $c^{-2}{\partial }_t^2+2\alpha {\partial }_t-{\partial }_x^2$ with $c>0$, the spatial Fourier mode $k$ of its causal Green function solvesSet $\gamma =\alpha c^2$ and $d_k^2={\gamma }^2-c^2{k}^2$. Causality, continuity at zero and the derivative jump $G_k^{\prime }(0^{+})=c^2$ giveImaginary $d_k$ gives a sine; when $d_k=0$, the quotient has limit $t$. These cases are one continuous formula, rather than separate arbitrary normalizations.

The spatially uniform mode isOnly for positive $\alpha$ does it tend to $1/(2\alpha )$. With zero damping it grows linearly, and negative $\alpha$ produces amplification. A stationary zero-mode limit therefore requires a sign hypothesis even though the causal formula itself covers all real $\alpha$.