The Physics of Resonance
Resonance is perhaps the most dramatic phenomenon in wave physics. When a driving force matches a system's natural frequency, even tiny periodic pushes can build up enormous oscillations. This principle explains how an opera singer can shatter a wine glass, how the Tacoma Narrows Bridge tore itself apart, and how every musical instrument produces sound. This simulation lets you tune the driving frequency and watch the amplitude response change in real time.
The Driven Harmonic Oscillator
The driven, damped harmonic oscillator is described by mx'' + bx' + kx = F₀cos(wt). Its steady-state solution has amplitude A = F₀/sqrt((w₀²-w²)² + (2bw/m)²), where w₀ = sqrt(k/m) is the natural angular frequency. At resonance (w = w₀), the amplitude becomes A = F₀/(2bw₀/m), limited only by damping. Without damping, the amplitude would grow without bound.
Quality Factor and Selectivity
The quality factor Q = w₀/(2b/m) quantifies the sharpness of the resonance peak. A high-Q system (low damping) responds strongly at one specific frequency — like a tuning fork (Q ~ 1000) or a laser cavity (Q ~ 10⁸). A low-Q system responds broadly — like a car suspension designed to absorb bumps across many frequencies. Q also determines how long oscillations persist after driving stops: the ring-down time is roughly Q/f₀ seconds.
Phase Relationship
At resonance, the oscillator's motion lags the driving force by exactly 90°. Below resonance, the system follows the driver nearly in phase. Above resonance, it opposes the driver, lagging by nearly 180°. This phase relationship is key to understanding energy transfer: maximum power transfer occurs at resonance because force and velocity are exactly in phase.