Velocity-impulse response of a damped oscillator ($x(0)=0,\dot{x}(0)=v_0$)

For $\ddot{x}+2\gamma \dot{x}+{\omega }_0^{2}x=0$, an initial velocity $v_0$ at zero displacement givesAll 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.