Past exam of the condensed matter physics course of the University of Oxford / 2013 / 3 / d / Solution

For passive friction $\Gamma >0$, the time dependence of each spatial normal coordinate follows its damping regime:

For any fixed positive damping, sufficiently small nonzero $k$ is overdamped because ${\omega }_0\simeq c_s|k|$. The slow branch then has $\omega \simeq -i(m{c}_s^2/\Gamma )k^2$, a diffusive long-wavelength relaxation. At exactly $k=0$, uniform translation has no restoring force, so $u_0(t)=A+B{e}^{-\Gamma t/m}$ and the constant displacement need not decay. Thus saying every amplitude simply decays as $e^{-\Gamma t/(2m)}$ is valid only for the underdamped envelope, not for all wave numbers.