# Definition:Period of Underdamped Oscillation

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## Definition

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.

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 $S$.

## Also see

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems