A small push, badly timed.

This plots the standard damped–driven–oscillator resonance response, Q(x) = 1/√((1−x²)² + (b̂x)²), where x = ω/ω₀ is the driving frequency as a fraction of the system's natural frequency and b̂ = b/ω₀ is the damping coefficient in the same units — equivalent to A(ω) = F₀/√((ω₀²−ω²)² + (bω)²) divided through by the static response F₀/ω₀². A simpler formula sometimes quoted, A ∝ 1/(ω₀²−ω²), is this equation's b→0 limit — it is undamped, and genuinely blows up to infinity exactly at resonance, which no real bridge, glass, or spring does. That undamped curve is drawn above as a dashed reference and clipped where it would run off the top of the chart, rather than actually plotted to infinity. Damping is doing real work in engineering: shock absorbers and tuned mass dampers (like the one in Taipei 101) exist specifically to hold b̂ high enough that a structure's resonance peak stays modest instead of catastrophic. The cleanest everyday example of the light-damping case is an opera singer shattering a wine glass by matching its natural frequency — a close-to-ideal single-resonance system. The 1940 Tacoma Narrows Bridge collapse is the anecdote most people reach for, but it's a shakier fit than the popular story suggests: the accepted modern explanation is aeroelastic flutter, a self-sustaining feedback between the bridge's motion and the wind flow around it, not simple forced resonance from a periodic gust matching the bridge's frequency. It's included here because it's the famous case, not because it's a textbook-clean example of this equation.