= Solution
Use the forward spatial transform $\hat f(k)=\int e^{-ikx}f(x)\,dx$ and inverse $f(x)=\int e^{ikx}\hat f(k)\,dk/(2\pi)$. The retarded transformed <Green function> obeys
$$
\left[\frac1{c^2}\frac{d^2}{d\tau^2}+2\alpha\frac d{d\tau}+k^2\right]G_k(\tau)=\delta(\tau),\qquad \tau=t-t'.
$$
It vanishes for negative $\tau$. Continuity at zero excludes an unwanted $\delta'$ term; integrating across zero then fixes $G_k'(0^+)=c^2$. Define $\gamma=\alpha c^2$ and $d_k=\sqrt{\gamma^2-c^2k^2}$. The two homogeneous exponents are $-\gamma\pm d_k$. Imposing the two matching conditions gives the <Fourier-mode Green function of the telegraph equation>,
$$
\boxed{G_k(\tau)=\Theta(\tau)\frac{c^2}{d_k}e^{-\gamma\tau}\sinh(d_k\tau)
=\Theta(\tau)\frac{e^{-\alpha c^2\tau}}{\sqrt{\alpha^2-k^2/c^2}}
\sinh\!\left(\sqrt{\alpha^2c^4-k^2c^2}\,\tau\right).}
$$
For an explicit time-Fourier route, use $G_k(\tau)=\int e^{-i\omega\tau}\mathcal G_k(\omega)\,d\omega/(2\pi)$. Then
$$
\mathcal G_k(\omega)=\frac1{k^2-2i\alpha\omega-\omega^2/c^2},\qquad
\omega_\pm=-i\gamma\pm\sqrt{c^2k^2-\gamma^2}.
$$
For $k\ne0$, both <poles> lie below the real axis. Closing above for $\tau<0$ gives zero; closing below for $\tau>0$ is clockwise, giving $-i$ times the <residue> sum. Since $\mathcal G_k=-c^2/[(\omega-\omega_+)(\omega-\omega_-)]$, this is
$$
G_k(\tau)=ic^2\frac{e^{-i\omega_+\tau}-e^{-i\omega_-\tau}}{\omega_+-\omega_-}
\quad(\tau>0),
$$
which reduces to the same damped sine or hyperbolic sine above. The $k=0$ <pole> on the real axis is interpreted by the retarded limiting prescription, or directly by the initial-value matching above.
For $|k|<\alpha c$ the response is overdamped and nonoscillatory. For $|k|>\alpha c$, put $\Omega_k=\sqrt{c^2k^2-\gamma^2}$; analytic continuation gives $G_k=\Theta(\tau)c^2e^{-\gamma\tau}\sin(\Omega_k\tau)/\Omega_k$, a real damped <oscillation>. At equality the finite limit is $\Theta(\tau)c^2\tau e^{-\gamma\tau}$.
All nonzero <Fourier modes> decay. The uniform mode instead has $G_0=(1-e^{-2\gamma\tau})/(2\alpha)$ for positive time, leaving a constant <temperature> increment. At small wavenumber the slow decay exponent is $-k^2/(2\alpha)+O(k^4)$, so the <diffusion limit of the damped heat-transmission equation> has <diffusivity> $1/(2\alpha)$. The full hyperbolic equation has <finite propagation speed> $c$, unlike the limiting parabolic <heat equation>.
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