= Velocity-impulse response of a damped oscillator
{title2=$x(0)=0,\quad\dot x(0)=v_0$}
For $\ddot x+2\gamma\dot x+\omega_0^2x=0$, an initial <velocity> $v_0$ at zero displacement gives
$$
x(t)=\begin{cases}
(v_0/\omega_d)e^{-\gamma t}\sin(\omega_dt),&\gamma<\omega_0,\quad\omega_d=\sqrt{\omega_0^2-\gamma^2},\\
v_0t e^{-\omega_0t},&\gamma=\omega_0,\\
(v_0/s)e^{-\gamma t}\sinh(st),&\gamma>\omega_0,\quad s=\sqrt{\gamma^2-\omega_0^2}.
\end{cases}
$$
All three cases begin with slope $v_0$. Only <underdamping> produces repeated crossings through equilibrium; <critical damping> and <overdamping> remain on the initial side of equilibrium for this particular initial condition. The overdamped long-time tail decays at rate $\gamma-s$, which becomes slow for strong <damping>.
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