Amplitude response
The steady-state amplitude of a driven oscillator as a function of drive frequency — peaked near resonance.
Definition
The amplitude response of a driven damped oscillator is the function A(ω_d) giving the steady-state oscillation amplitude for a fixed driving force F₀ as the drive frequency ω_d is swept. For a linear second-order oscillator the formula is
A(ω_d) = F₀ / √[(ω₀² − ω_d²)² + γ²ω_d²]
where ω₀ is the natural frequency and γ is the damping rate. The function peaks near ω_d ≈ ω₀, with a peak height proportional to the quality factor Q = ω₀/γ and a peak width inversely proportional to Q. High-Q systems (a quartz crystal) have tall, narrow peaks; low-Q systems (a shock absorber) have broad, shallow ones.
Amplitude response is the everyday face of resonance. It explains why a violin string sings sweetly at some pitches and hardly at all at others, why a well-tuned AM radio rejects every station except one, and why the Tacoma Narrows bridge found a wind frequency that matched its natural mode and destroyed itself. Engineering a system to resonate strongly (MRI, radio, laser) means maximising Q at the desired frequency; engineering it to survive (buildings, bridges, electronics) means damping any resonance that could be excited by the environment.