Driven Resonance
Drive spring-mass, sweep freq. Builds on Damped Oscillations.
A soprano singer holds a steady note near a crystal wine glass. The glass begins to vibrate. She adjusts her pitch slowly, and at one specific frequency, the glass vibrations grow dramatically — until the glass SHATTERS.
Why does the glass shatter only at that one specific frequency?
Resonance! When the singer’s frequency matches the glass’s natural vibration frequency, each sound wave arrives at exactly the right moment to push the glass further. Energy builds up cycle after cycle until the glass’s elastic limit is exceeded. A tiny force, perfectly timed, can destroy a wine glass — that’s the terrifying power of resonance.
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Sweep the driving frequency and watch the amplitude response. At resonance, the amplitude skyrockets. Notice how even a small driving force produces enormous oscillations at the right frequency.
A(ω) = F₀/m / √[(ω₀² − ω²)² + (2γω)²]
At ω = ω₀, the (ω₀² − ω²) term vanishes and only damping limits the amplitude. Less damping → taller resonance peak. Zero damping → infinite amplitude (the system would break).
In 1940, wind-driven oscillations matched the bridge’s natural frequency. The bridge oscillated with increasing amplitude until it tore itself apart, captured in famous footage.
A singer (or speaker) producing the exact natural frequency of a wine glass can shatter it. The glass’s Q factor determines how many cycles it takes to reach the breaking point.
Every radio station broadcasts at a specific frequency. Your radio’s tuner is an LRC circuit whose resonant frequency is adjusted to match. Only the matching station’s signal gets amplified.
A guitar body resonates at certain frequencies, amplifying those notes from the strings. The body shape and wood determine which frequencies are enhanced, giving each guitar its unique tone.
“Resonance is nature’s amplifier. A tiny periodic push, delivered at the right frequency, can topple bridges, shatter glass, tune radios, and make music. Timing is everything.”
Resonance Lab
Drive a spring-mass system with an adjustable periodic force. Sweep the driving frequency to find resonance. Adjust damping to see how it shapes the resonance curve. Watch the phase relationship between the driver and the oscillator change as you cross resonance.
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Start with low damping and slowly sweep the driving frequency from well below ω₀ to well above it. Watch the amplitude peak sharply at resonance. Now increase damping — the peak gets shorter and broader. At critical damping, there’s barely a peak at all! Watch the phase: below resonance the mass moves WITH the driver; at resonance it lags 90°; above resonance it moves AGAINST the driver. Try finding the frequency where the mass is exactly 90° behind the driving force.
Resonance is the most important single concept in oscillation physics. It explains why soldiers break step on bridges, why opera singers can shatter glass, why your radio can pick out one station from thousands, and why atoms absorb specific frequencies of light. The universe is full of oscillators waiting to resonate.