Resonance is a phase relationship, not a peak
The resonance peak is not at the natural frequency, and with enough damping there is no peak at all. The 90 degree phase lag is always exactly there.
SHM Resonance Lab opens with a single readout: phase lag φ, spoken "phi", of 1.5708 rad, which it helpfully converts to 90.0°.
That number is π/2, and it is the most reliable thing about resonance.
The peak is not where you were told
A driven, damped oscillator responds most strongly not at its natural frequency ωn but at ωn√(1 − 2ζ²), where ζ, spoken "zeta", is the damping ratio. The peak sits below the natural frequency, and it moves as damping changes:
- ζ = 0.02 → peak at 0.9996 ωn
- ζ = 0.10 → peak at 0.9899 ωn
- ζ = 0.30 → peak at 0.9055 ωn
- ζ > 0.707 → no peak at all
Past ζ = 1/√2 the response curve simply decreases from the start. The system still has a natural frequency, still rings when struck, and has no resonant peak whatsoever.
So "resonance is where the amplitude is largest" fails as a definition. It gives the wrong frequency in every real case and no frequency in a large class of them.
What does not move
Now the phase. Drive the oscillator exactly at ωn and the response lags the drive by exactly 90°, for every value of ζ. Not approximately, and not only for light damping. The 0.02 case and the 0.7 case both give 90.000°.
The reason is visible in the equation. At ω = ωn the spring and inertia terms cancel exactly, because one goes as ω² and the other is constant and they are equal at that frequency. Nothing is left but the damping term, which is the one in quadrature with the displacement. A response governed purely by the damping term is exactly a quarter cycle behind.
This is why the definition worth carrying is the phase one. It is exact, it survives heavy damping, and it does not move.
Why it matters practically
At 90° the driving force is in phase with velocity rather than displacement, and force times velocity is power. So at resonance every push does positive work through the whole cycle, with none of it given back.
That is the mechanism, and it also explains the amplitude. Energy accumulates until the damping dissipates it as fast as the drive supplies it, so the steady amplitude is set by the damping alone: the gain is 1/(2ζ), or 25 times at ζ = 0.02.
Engineers call this the quality factor, Q, and finding resonance by watching for the 90° crossing rather than hunting a peak is standard practice, because near a sharp peak the amplitude is flat and hard to locate while the phase is sweeping fast.
RLC Resonance is the identical mathematics in a circuit: inductor and capacitor cancel, only the resistor remains, current and voltage come into phase, and Q is again the ratio of stored to dissipated energy.
The example everyone uses is wrong
Tacoma Narrows is the standard illustration and it was not resonance. There was no oscillating driver at the bridge's natural frequency. The wind was steady, and the deck's own twisting motion changed the airflow so as to feed more energy into that motion, a self-excited instability called aeroelastic flutter.
The distinction is not pedantry. Resonance is a response to an external frequency and is cured by moving the frequency or adding damping. Flutter needs no external frequency, and above the critical wind speed it grows regardless.
Soldiers breaking step on bridges is the honest example: a genuine periodic driver, deliberately detuned.
Measuring Q without finding the peak
Q also equals f₀ / Δf, where Δf is the width of the response between the points either side of resonance where the power has fallen by half.
This is the practical route, and it avoids the problem above. Locating a sharp peak means finding the flattest part of the curve, where the amplitude barely changes and noise dominates. The half-power points sit on the steep flanks where the signal is changing fastest, so they are exactly where a measurement is most precise.
At ζ = 0.02 the gain is 25 and the bandwidth is 4% of the centre frequency. Narrow and tall are the same statement.