= Solution
For a <viscously damped monatomic-chain phonon>, the extra force gives $m\ddot u_j+\Gamma\dot u_j$ on the left of the preceding equation. Keep a real <wave number> $k$ and permit complex frequency in the convention $u_j\propto e^{i(kja-\omega t)}$. The dispersion equation is
$$
\boxed{m\omega^2+i\Gamma\omega-m\omega_0^2(k)=0.}
$$
Writing $\gamma=\Gamma/(2m)$, its two roots are
$$
\boxed{\omega_\pm(k)=-i\gamma\pm\sqrt{\omega_0^2(k)-\gamma^2}.}
$$
The negative imaginary part corresponds to decay with the chosen time convention. Here $\Gamma$ has units of <mass> per time, while $\gamma$ is a decay rate. A driven experiment at specified real frequency would instead solve for a generally complex <wave number>; it is a different way of specifying the response, not a contradictory sign choice.
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