\begin{definition}[Definition:Period of Underdamped Oscillation]
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
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While the behaviour of $S$ is not strictly speaking periodic, its oscillations can be defined to have a "period" as follows:
Let $T$ be the smallest value of $t$ such that:
:$x = 0$
:$x' < 0$
Then $T$ is the '''period''' of the oscillation of $S$.
\end{definition}