= 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 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> solves
$$
G_k''+2\alpha c^2G_k'+c^2k^2G_k=c^2\delta(t).
$$
Set $\gamma=\alpha c^2$ and $d_k^2=\gamma^2-c^2k^2$. Causality, continuity at zero and the <derivative> jump $G_k'(0^+)=c^2$ give
$$
G_k(t)=H(t)c^2e^{-\gamma t}\frac{\sinh(d_kt)}{d_k}.
$$
Imaginary $d_k$ 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 is
$$
G_0(t)=H(t)\begin{cases}(1-e^{-2\alpha c^2t})/(2\alpha),&\alpha\ne0,\\c^2t,&\alpha=0.\end{cases}
$$
Only 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 $\alpha$.
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