= Logarithmic decrement
{title2=$\delta=\ln\frac{|x_n|}{|x_{n+1}|}$}
For <underdamping>, let $x_n,x_{n+1}$ be successive same-sign extrema of <displacement> relative to <equilibrium>, one full <oscillation period> apart. The <logarithmic decrement> is
$$
\delta=\ln\frac{|x_n|}{|x_{n+1}|}=\alpha T_{\mathrm d}
=\frac{2\pi\zeta}{\sqrt{1-\zeta^2}},\qquad
\zeta=\frac{\delta}{\sqrt{4\pi^2+\delta^2}}.
$$
Comparing extrema $N$ full cycles apart instead gives $\delta=N^{-1}\ln(|x_n|/|x_{n+N}|)$. Adjacent extrema of opposite sign are only half a cycle apart and give half this decrement. This distinction matters when inferring a <damping ratio>. Near <critical damping>, a full second same-sign extremum may be too small to resolve above <measurement noise>; a full-trace fit with uncertainty is then more informative than a peak ratio.
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