2.5 Power, Q, and bandwidth

The driven oscillator absorbs energy from the drive and dissipates it through damping. At resonance, both flows balance and the oscillator settles into a steady state. The dimensionless quality factor $Q$ captures the relationship between how much energy is stored, how much is dissipated per cycle, and how narrow the resonance peak is.

Power absorbed

The instantaneous power delivered by the drive to the oscillator is $P(t) = F(t) \dot x(t)$. For a steady sinusoidal drive $F = F_0 \cos(\omega t)$ and the response $x(t) = X(\omega) \cos(\omega t + \varphi)$, the time-averaged power is

which, using $\sin|\varphi| = 2\gamma\omega / \sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}$ and substituting, simplifies to

This is the power delivered by the drive — equivalently, dissipated by the damping — when the system is in steady state. It is a Lorentzian-shaped curve in $\omega$, peaked at resonance.

Quality factor

Define

For an underdamped oscillator, $Q$ has three equivalent interpretations, each useful in different contexts:

1. Energy storage: $Q = 2\pi \times$ (energy stored / energy dissipated per cycle). High $Q$ means the oscillator hangs on to its energy for many cycles.
2. Decay time: the free-oscillation energy decays as $e^{-\omega_0 t / Q}$. The number of radians of free oscillation in which the energy drops by $1/e$ is exactly $Q$.
3. Bandwidth: the full width of the power-absorption curve at half maximum (FWHM) is

a Lorentzian. Equivalently $Q = \omega_0 / \Delta\omega$.

The third interpretation is the most useful for acoustics. High-$Q$ resonators are narrow-band; low-$Q$ resonators are broad-band. A tuning fork is $Q \sim 10^3$. An organ pipe is $Q \sim 10^2$. A struck rubber ball is $Q \sim 1$. Your ear canal’s first resonance is $Q \sim 2$ — broad-band, because the canal is lossy and short.

Lorentzian shape

The power absorption curve near resonance has the universal Lorentzian form

This functional form appears so often in physics — wherever a damped resonance is driven by a sinusoid — that it is worth memorising. The peak height is set by $1/\gamma$; the FWHM is $\omega_0 / Q = 2\gamma$. Halving the damping doubles the peak and halves the bandwidth.

Bandwidth–time tradeoff

There is no free lunch. A high-$Q$ resonator that responds to a narrow frequency band also takes a long time to reach that response — the ring-up time is $\sim Q/\omega_0$. This is the same uncertainty-principle tradeoff as in the Fourier transform: a narrow filter in $\omega$ is a wide impulse response in $t$.

This will return as a central design tension in chapter 8 (spectrograms and short-time Fourier transforms) and in chapter 4 of the hearing book (cochlear filter banks).

What we use this for

• Estimating the bandwidth of acoustic resonators: tubes, rooms, instruments.
• Understanding why high-$Q$ resonances ring on after the driving stops.
• Reading impedance plots: a peak with width $\Delta\omega$ has $Q = \omega_0/\Delta\omega$.
• The Fourier-domain meaning of resonance, which we will see explicitly in chapter 8.