8.4 Resonance reborn — Q as bandwidth

The damped-driven oscillator from chapter 2.4 is, viewed as a system that turns force into displacement, a filter. Its transfer function is

H(\omega) \;=\; \frac{1/m}{(\omega_0^2 - \omega^2) + 2 i \gamma \omega},

with $\omega_0$ the natural frequency and $\gamma$ the damping rate. We derived in chapter 2 that the amplitude response peaks at $\omega \approx \omega_0$ with peak height $\propto 1/\gamma$ and full width at half maximum (FWHM) of the squared response $\Delta\omega = 2\gamma$ .

Reinterpreted as a filter, this is a band-pass filter centred at $\omega_0$ with bandwidth $\Delta\omega = \omega_0/Q$ , where the quality factor

Q \;\equiv\; \frac{\omega_0}{2\gamma}

now has a clean frequency-domain interpretation: it is the centre frequency divided by the bandwidth.

What Q means in three ways

We’re now in a position to see the three meanings of Q from chapter 2.5 as three facets of one number.

1. Energy storage. Time-domain interpretation: $Q$ is $2\pi$ times the energy stored in a cycle divided by the energy dissipated in a cycle. The free oscillation decays over $\sim Q$ radians ( $\sim Q / 2\pi$ cycles).

2. Decay time. $Q$ is the number of radians of free oscillation in which the energy decays by $1/e$ . For a resonator with $Q = 100$ at $f = 1$ kHz, the ring-down time is $Q / \omega_0 = Q / (2\pi f) \approx 16$ ms.

3. Bandwidth (frequency-domain). $Q = f_0 / \Delta f$ , where $\Delta f$ is the full width at half maximum of the energy response. High- $Q$ resonators are narrow-band; low- $Q$ are broad-band.

These three are the same fact viewed from three angles. They are related by the uncertainty principle: a narrow filter in frequency takes a long time to “ring up” — the impulse response of a high- $Q$ filter is a slowly-decaying sinusoid at the centre frequency. You cannot have a filter that is both narrow in frequency and short in time.

A Lorentzian

The squared magnitude of the resonant response is

|H(\omega)|^2 \;\propto\; \frac{1}{(\omega^2 - \omega_0^2)^2 + 4\gamma^2 \omega^2}.

Near the peak ( $\omega \approx \omega_0$ ), expand $\omega^2 - \omega_0^2 \approx 2\omega_0(\omega - \omega_0)$ and the response simplifies to the Lorentzian:

|H(\omega)|^2 \;\approx\; \frac{|H|_\text{peak}^2}{1 + 4(\omega - \omega_0)^2 / (2\gamma)^2}.

The Lorentzian shape appears wherever a damped oscillator is driven by a sinusoid: in atomic spectroscopy (natural linewidths), in NMR, in optical cavities, in laser line shapes, in the cochlea’s frequency-response functions. The shape is universal across the natural and engineering sciences.

Acoustic resonators in this language

Returning to acoustic resonators:

A typical room mode has $Q \sim 10$ – $100$ , depending on absorption.
An organ pipe at 200 Hz has $Q \sim 40$ , narrowly tuned.
A wine glass has $Q \sim 1000$ , very narrowly tuned.
A Helmholtz resonator (bottle) has $Q \sim 5$ – $20$ .
A wooden xylophone bar has $Q \sim 200$ .
A tuning fork has $Q \sim 1000$ – $10{,}000$ .
The cochlea’s filter at a given place has $Q \sim 1$ – $10$ — broad — but the cochlear amplifier sharpens these dramatically (hearing book, chapter 5).

Each value reflects a design tradeoff. Narrow-band ( $\text{high-}Q$ ) filters take a long time to ring up but resolve frequencies precisely. Broad-band ( $\text{low-}Q$ ) filters respond quickly but with less frequency selectivity. The cochlea splits the difference — and chapter 4 of the hearing book is the story of how.

Looking back at the chapter

Chapter 8 took the time-domain framework of the earlier chapters and gave it a Fourier dual. The wave equation becomes algebraic. Convolution becomes multiplication. Linear systems become transfer functions. Resonance becomes filter bandwidth. Most of the conceptual machinery of acoustics is in this chapter and the next two; what changes from chapter to chapter is the medium and its motion.

Next chapter: the medium itself moves. Doppler, Mach cones, sound in flow.