\begin{definition}[Definition:Amplitude 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 $(1)$ be subject to the initial conditions:
:$x = x_0$ at $t = 0$
:$x' = 0$ at $t = 0$
Let $b < a$, so as to make $S$ underdamped.
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The '''amplitude''' of the oscillation of $S$ is defined as:
:$A = \dfrac {x_0 a} {\sqrt {a^2 - b^2} } e^{-b t}$
\end{definition}