Resonance
Drive an oscillator near its natural frequency and the response explodes — the principle behind tuning radios, shattering glass, and toppling bridges.
Resonance
Resonance is what happens when you push an oscillator at just the right rhythm. As the driving frequency \omega of a Driven Oscillator sweeps toward its natural frequency \omega_0, the steady-state amplitude rises to a sharp peak. Each push arrives perfectly in phase to add a little more energy than damping removes, and small forces accumulate into large motions.
The response curve
Plotting amplitude against driving frequency (here in units of \omega_0, with light damping) gives the characteristic resonance peak — a Lorentzian lineshape:
Two features matter. The height of the peak is set by damping — less damping means a taller, more dangerous spike (in the frictionless limit it diverges). The width measures how selective the resonance is: a narrow peak responds only to a tiny band of frequencies, which is exactly what lets a radio pick one station out of many.
Resonance in the wild
- Tuning — an RLC circuit resonates at one frequency; turning the dial retunes \omega_0 to select a station.
- Music — the body of a violin and the column of air in an organ pipe resonate to amplify particular notes.
- Destruction — soldiers break step on bridges because a marching cadence near a structural \omega_0 can pump the span to failure. The same physics shatters a wine glass at its ringing pitch.