= Fourier-mode Green function of the telegraph equation
{c}
{title2=$G_k(\tau)=\Theta(\tau)c^2e^{-\gamma\tau}\sinh(d_k\tau)/d_k$}
For operator $c^{-2}\partial_t^2+2\alpha\partial_t-\partial_x^2$, define $\gamma=\alpha c^2$ and $d_k^2=\gamma^2-c^2k^2$. Retarded support, continuity at zero time and the <derivative> jump $G_k'(0^+)=c^2$ 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.
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