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When a damped oscillator is driven by an external periodic force $F = F_0\cos(\omega_d t)$, the steady-state response oscillates at the driving frequency $\omega_d$ (not $\omega_0$). The amplitude is $A = F_0/\sqrt{m^2(\omega_0^2-\omega_d^2)^2 + b^2\omega_d^2}$, which peaks near $\omega_d = \omega_0$ — this is resonance.

At exact resonance ($\omega_d = \omega_0$): $A_{\max} = F_0/(b\omega_0)$, limited only by damping. The displacement lags the driving force by exactly $\pi/2$, meaning the force is in phase with the velocity — maximum power transfer. Below resonance, the lag is less than $\pi/2$ (displacement nearly in phase with force). Above resonance, the lag approaches $\pi$ (nearly antiphase). The width of the resonance peak is characterized by the bandwidth $\Delta\omega = \omega_0/Q$ where $Q$ is the quality factor. Sharp resonance (high $Q$, low damping) means the system responds strongly only near $\omega_0$. Broad resonance (low $Q$, high damping) means response over a wide frequency range. Resonance is responsible for phenomena ranging from bridge oscillations to radio tuning to molecular absorption spectra.