Definition:Amplitude of Underdamped Oscillation

Definition
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
 * $(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$

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.


 * UnderdampedPeriodAmplitude.png

The amplitude of the oscillation of $S$ is defined as:
 * $A = \dfrac {x_0 a} {\sqrt {a^2 - b^2} } e^{-b t}$

Also see

 * Amplitude of Oscillation of Underdamped System is Exponential


 * Definition:Period of Underdamped Oscillation