# Resonance: Conceptual misunderstanding

I am used to thinking of the resonance phenomenon in terms of a small driving force building up a large vibration amplitude. But when you solve the equation for a damped, driven oscillator, you get a transient part that decays with time, and a steady-state part that is, well, steady. How does this account for the slow buildup from small amplitude to large amplitude? I would have thought there would be a term with exponentially increasing amplitude.

• This is all correct, but you've missed something: a 'slow buildup' happens starting from a resting of nearly resting configuration. So take the general solution in the case of $x_0 = \dot{x}_0 = 0$. The system starts with zero energy and the driving force adds a little each cycle. The maximum size is still governed by the closeness of the driving and natural frequencies and the overall strength of the damping. – dmckee Jan 5 '18 at 7:51