Resonance

Summary

Resonance is the strong response of a linear system when it is forced at one of its own natural frequencies. For a linear equation with constant coefficients and forcing $e^{\lambda t}$, it happens when $\lambda$ is a root of the characteristic polynomial. Then the response is not a multiple of $e^{\lambda t}$ but picks up a power of $t$, and an undamped oscillator has unbounded solutions. It explains the collapse of structures under periodic loads and the tuning of circuits and instruments. See the definition and the worked forced oscillator.