10.3 Wave impedance

Impedance is the ratio of a driving quantity to the response it produces — pressure to velocity, force to velocity, voltage to current. For a wave it is not an independent property but a consequence of the wave equation itself: the two fields that make up a wave are locked together in a fixed ratio set by the medium. This lesson derives that ratio and generalises it.

Acoustic impedance, derived

In a plane sound wave the pressure and the particle velocity are not free to vary independently; the equation of motion ties them.

▶ Z = ρ₀c from linearised Euler Derivation

Take a plane wave $p' = P_0\,e^{i(kx-\omega t)}$ and $u' = U_0\,e^{i(kx-\omega t)}$ . The linearised Euler equation relates them:

\rho_0\frac{\partial u'}{\partial t} = -\frac{\partial p'}{\partial x} \;\;\Longrightarrow\;\; \rho_0(-i\omega)U_0 = -(ik)P_0 \;\;\Longrightarrow\;\; U_0 = \frac{k}{\omega\rho_0}P_0.

The acoustic dispersion is $\omega = ck$ , so $k/\omega = 1/c$ , giving

\frac{P_0}{U_0} = \rho_0 c \equiv Z.

where

$Z$: specific acoustic impedance Pa·s/m
$\rho_0$: equilibrium density of the medium kg/m³
$c$: speed of sound m/s

The specific acoustic impedance $Z = \rho_0 c$ is a property of the medium alone — it depends on the density and sound speed, not on the wave’s amplitude or frequency.

ρ₀1.20

c343

P₀1.00 Pa

Z = ρ₀ c411.6

Density ρ₀ = 1.20 kg/m³ Wave speed c = 343 m/s Pressure amplitude P₀ = 1.00 Pa

Substituting a plane-wave ansatz into linearised Euler (ρ₀ ∂u′/∂t = −∂p′/∂x) together with the wave dispersion ω = ck gives U₀ = P₀ / (ρ₀ c) ≡ P₀ / Z. The impedance Z = ρ₀c is a *medium property*, not a wave property: it depends on ρ₀ and c, not on the specific wave. Water has Z ≈ 1.5 × 10⁶, air ≈ 410 — a factor-of-3500 mismatch that the middle-ear ossicles are built to bridge.

The numbers matter for what follows. Water has $Z\approx 1.5\times10^6\,\text{Pa·s/m}$ ; air has $Z\approx 410$ . Their ratio, some 3500, is a severe impedance mismatch, and — as the reflection lesson will show — it is why almost no airborne sound crosses into water, and why any device transferring sound between such mismatched media needs an impedance-matching stage.

Mechanical impedance, generalised

The same idea applies to any linear system where a sinusoidal force $F$ drives a sinusoidal velocity $v$ : the mechanical impedance is $Z_\text{mech} = F/v$ . For a damped, driven harmonic oscillator,

Z_\text{mech}(\omega) \;=\; b \;+\; i\!\left(\omega m - \frac{k}{\omega}\right).

where

$b$: damping coefficient N·s/m
$m$: mass kg
$k$: stiffness N/m

b (damping)0.50

m (mass)1.00

k (stiffness)4.00

ω₀ = √(k/m)2.00

Damping b = 0.50 Mass m = 1.00 Stiffness k = 4.00

Below ω₀ the stiffness term k/ω dominates and the impedance is *capacitive* (phase −90°); above ω₀ the mass term ωm dominates and the impedance is *inductive* (+90°). At ω₀ the reactive parts cancel and only the damping b remains. The complex impedance is the operative model for the basilar membrane in [Hearing Ch 4.3](/hearing/cochlea/traveling-wave).

Three frequency regimes divide the response, split at the natural frequency $\omega_0 = \sqrt{k/m}$ :

Stiffness-controlled, $\omega\ll\omega_0$ : the $-k/\omega$ term dominates, $|Z|\sim k/\omega$ , and the velocity lags the force by $90^\circ$ .
Damping-controlled, $\omega\approx\omega_0$ : the mass and stiffness reactances cancel, leaving $|Z| = b$ real; force and velocity are in phase, and the system absorbs power most efficiently — resonance.
Mass-controlled, $\omega\gg\omega_0$ : the $\omega m$ term dominates, $|Z|\sim\omega m$ , and the velocity leads by $90^\circ$ .

This impedance is the frequency-domain fingerprint of every driven damped oscillator, and its resonant cancellation of reactances is the same phenomenon that governs electrical $RLC$ circuits and acoustic resonators alike.