\begin{definition}[Definition:Logarithmic Decrement]
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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Let $T$ be the period of oscillation of $S$.
Let $x_1$ and $x_2$ be successive local maxima of $x$.
From Ratio of Successive Local Maxima for Underdamped Free Vibration:
:$\dfrac {x_1} {x_2} = e^{b T}$
The '''logarithmic decrement''' of $S$ is defined as:
:$\map \ln {\dfrac {x_1} {x_2} } = b T$
\end{definition}