Resonance — Quick Summary

Mr Toogood's Physics · Periodic motion

AQA 3.6.1.4

f_driver = f₀

Resonance condition

φ = π/2

Phase, at resonance

φ → π

Phase, well above f₀

A → max

Amplitude at resonance

Free vs. forced oscillations

Free oscillation: no external driver — the system oscillates at its own natural frequency, f₀ (undamped: constant amplitude; damped: amplitude decays).
Forced oscillation: an external periodic driver forces the system to oscillate at the driver's frequency, not necessarily f₀.

What is resonance?

Resonance occurs when the driving frequency equals the system's natural frequency, f_driver = f₀. The oscillator's amplitude increases dramatically — the system is said to be resonating.

Graph of oscillation amplitude against driving frequency, peaking sharply at the natural frequency

Amplitude peaks sharply when the driving frequency matches f₀.

Without damping, energy is supplied continuously at exactly the right rate to keep increasing the amplitude, with no limit — in principle the system would grow until it destroyed itself (e.g. an opera singer shattering a glass, or the Tacoma Narrows Bridge collapse).

Phase relationships

The phase difference between the driving force and the resulting displacement changes as the driving frequency moves through resonance:

Driving frequency	Phase difference
f < f₀	less than π/2
f = f₀ (resonance)	exactly π/2
f > f₀	increases towards π

At resonance, the driving force is exactly in phase with the oscillator's velocity — this is what allows maximum energy transfer into the system on every cycle.

Barton's pendulums

A set of pendulums of different lengths hangs from a common horizontal bar, driven by one heavier "driver" pendulum. Only the pendulum whose length matches the driver's (same natural frequency) resonates with a large amplitude — the others respond weakly and with different phase lags, demonstrating resonance and phase change experimentally.

Exam essentials

Key definitions

Natural frequency, f₀: frequency of free oscillation.
Resonance: f_driver = f₀ → maximum amplitude response.

Phase quick reference

f < f₀: phase difference < π/2.
f = f₀: phase difference = π/2 (force ∥ velocity).
f > f₀: phase difference → π.

Common slips

Resonance is about frequency matching, not the driver's amplitude.
Real systems always have some damping — unlimited amplitude growth is an idealisation.
Phase difference is between the driving force and displacement — be precise about which quantities are being compared.