Viscously damped monatomic-chain phonon (${\omega }_{\pm }=-i\Gamma /(2m)\pm \sqrt{{\omega }_0^2-{\Gamma }^2/(4m^2)}$)

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 $\omega \simeq -im{c}_s^2{k}^2/\Gamma$. Exactly uniform displacement remains a zero-restoring-force mode, so a constant displacement is not forced to vanish.