7.1 Concept of acoustic impedance

Understanding Acoustic Impedance

Definition of acoustic impedance

Acoustic impedance quantifies how much a medium resists the propagation of sound waves. More precisely, it's the ratio of acoustic pressure to particle velocity at a point in the medium.

Think of it this way: when a sound wave travels through a material, it creates pressure variations and causes tiny particles to move back and forth. Some materials "push back" against this motion more than others. That resistance is acoustic impedance.

• A medium's impedance depends on its density and elasticity. Dense, stiff materials like steel have very high impedance, while soft, flexible materials like rubber have much lower impedance.
• At boundaries between materials with different impedances (like an air-water interface), sound waves partially reflect and partially transmit. This is why impedance matters so much for real-world acoustics.

One common misconception: higher impedance doesn't mean sound travels more slowly. Sound actually travels faster in steel than in air, even though steel has far higher impedance. Impedance describes resistance to the wave's propagation, not the wave's speed directly.

Calculation of acoustic impedance

The characteristic acoustic impedance of a medium is calculated with:

$Z = \rho c$

where:

• $Z$ = acoustic impedance
• $\rho$ = density of the medium (kg/m³)
• $c$ = speed of sound in the medium (m/s)

The unit is the Rayl (Pa·s/m, equivalent to kg/(m²·s)).

Here are some typical values to build intuition:

Notice that water's impedance is roughly 3,500 times that of air. This enormous mismatch is why so much sound reflects at the air-water boundary.

Environmental conditions also affect impedance. Temperature changes alter both density and sound speed: warmer air is less dense and carries sound slightly faster, shifting the impedance value. Pressure variations in gases have a similar effect, though typically smaller.

Types of acoustic impedance

There are two types you need to distinguish:

Characteristic impedance ($Z_0 = \rho c$) is an inherent property of the medium itself. It doesn't change with frequency or position. It applies to idealized conditions: plane waves traveling through a uniform, lossless medium. This is the type you'll use most often in impedance calculations.

Specific acoustic impedance is the local ratio of sound pressure to particle velocity at a particular point in a sound field. Unlike characteristic impedance, it can vary with frequency and position. In complex sound fields (where waves reflect, diffract, or interfere), the pressure-to-velocity ratio at any given point may differ from the medium's characteristic value.

Both are measured in Rayls. For a simple plane wave in a lossless medium, specific impedance equals characteristic impedance. They diverge when the sound field gets more complex.

Effects of acoustic impedance

When sound hits a boundary between two media with different impedances, some energy reflects and some transmits. The greater the impedance mismatch, the more energy reflects.

The reflection coefficient $R$ and transmission coefficient $T$ for normal incidence at a boundary are:

$R = \frac{Z_2 - Z_1}{Z_2 + Z_1}$

$T = \frac{2Z_2}{Z_2 + Z_1}$

where $Z_1$ is the impedance of the first medium and $Z_2$ is the impedance of the second.

• If $Z_1 = Z_2$, then $R = 0$ and all energy transmits. This is perfect impedance matching.
• If $Z_2 \gg Z_1$ (or vice versa), $R$ approaches $\pm 1$ and most energy reflects. This explains why sound barely penetrates from air into glass.

Standing waves can form when impedance discontinuities cause repeated reflections within a bounded space. Organ pipes are a classic example: the open or closed ends create impedance mismatches that reflect waves back and forth, producing resonance.

Impedance matching is the practice of minimizing reflection at boundaries. Applications include:

• Room acoustics: wall treatments and absorbers are designed to reduce impedance mismatch so sound energy is absorbed rather than reflected
• Musical instruments: the bell of a brass instrument gradually transitions impedance from the narrow bore to open air, improving sound radiation
• Ultrasound imaging: coupling gel is applied between the transducer and skin to eliminate the air gap, since air's impedance is drastically different from tissue