 Solution

The amplitude envelope falls by $e^{-1/2}$ when $e^{-\gamma t/2}=e^{-1/2}$, so $t=1/\gamma$. Dividing by the period $2\pi /{\omega }_d$ givesThis is a cycle-count estimate in the small-damping regime, not necessarily an integer. With zero damping there is no amplitude decay and no finite such time.