 Fourier-mode Green function of the telegraph equation

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.