resonance

What happens when an external driving force pushes an oscillator at (or near) its _natural frequency_. Each push lands when the oscillator is already moving in the same direction — energy goes in coherently, amplitude grows. Off-resonance, pushes alternately add and subtract energy, and the response stays small. The steady-state amplitude as a function of driving frequency `ω` is `A(ω) = F / √((ω₀² − ω²)² + (2γω)²)`, which peaks near `ω = ω₀` for light damping. _Lighter damping → sharper, taller peak._ The same identity governs an opera singer breaking glass, the Tacoma Narrows bridge collapse, MRI nuclei flipping, and the radio dial picking out one station from a sky full. Resonance is _not_ a special force; it is energy injected on the beat the system already keeps.