= Viscously damped monatomic-chain phonon
{title2=$\omega_\pm=-i\Gamma/(2m)\pm\sqrt{\omega_0^2-\Gamma^2/(4m^2)}$}
With real wave number and time convention $e^{-i\omega t}$, an on-site friction force $-\Gamma\dot u_j$ shifts the frequencies as displayed. Underdamped oscillations have amplitude envelope $e^{-\Gamma t/(2m)}$, while overdamped modes have two nonoscillatory rates. At small wave number a stable acoustic chain has slow decay $\omega\simeq-imc_s^2k^2/\Gamma$. Exactly uniform displacement remains a zero-restoring-force mode, so a constant displacement is not forced to vanish.
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