resonance

Resonance occurs when a periodic driving force is applied to an oscillator at or near its natural frequency. The system absorbs energy efficiently, and the amplitude builds up to a peak limited only by damping. At exact resonance, the driving force is always in phase with the velocity, so every push adds energy.

The phenomenon is everywhere. A child on a swing learns resonance intuitively — push at the right moment and the arc grows. A wine glass shatters when a singer hits its natural frequency. The Tacoma Narrows Bridge collapsed in 1940 when wind vortices drove it at a resonant mode. MRI scanners use resonance to flip nuclear spins at precisely the right radio frequency.

Mathematically, the steady-state amplitude of a driven, damped oscillator is A(ω) = F₀/m / √((ω₀² − ω²)² + γ²ω²). The peak occurs near ω = ω₀ and its height is proportional to Q. Sharper peak, narrower bandwidth, more selective response — the same tradeoff that defines Q.