Rule 27 of 36 · Chapter IV — Fields & Waves
Resonance rewards patience
Why this rule exists
A system driven at its natural frequency accumulates energy because each push arrives in phase with the motion, adding to it rather than fighting it. Off resonance, the drive and the response drift out of step, so energy fed in on one cycle is partly taken back on the next. Only sustained, correctly timed forcing lets a small periodic input build a large amplitude.
In practice
To excite a resonance, match the drive frequency to the system's natural frequency and wait — amplitude grows over many cycles, limited only by damping. To avoid one, detune the drive or add damping so energy leaks out as fast as it enters. The sharpness of the peak is set by the quality factor Q: high Q means a narrow, tall response.
When it doesn't apply
Real resonators have damping, which caps the amplitude and broadens the peak, so response is never truly infinite. Strong driving introduces nonlinearity, shifting the resonant frequency and distorting the simple picture.