# Definition:Underdamped

## 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 that the solution of $(1)$ is in the form:

$x = e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t}$

where $\alpha = \sqrt {a^2 - b^2}$.

Then $S$ is described as being underdamped.