Definition:Resonance

Definition
Consider a physical system $S$ whose behaviour is defined by the second order ODE:
 * $\dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$

where:
 * $K \in \R: k > 0$
 * $a, b \in \R_{>0}: b < a$

which has the general solution:
 * $(1): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$

where:
 * $\alpha = \sqrt {a^2 - b^2}$
 * $\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$

Let $\omega$ be such that the amplitude of the steady-state component of $(1)$ is at a maximum.

Then $S$ is said to be in resonance.

Also see

 * Condition for Resonance in Forced Vibration of Underdamped System