# Acoustic impedance

Sound measurements
Characteristic
Symbol
Sound pressure  p · SPL
Particle velocity  v · SVL
Particle displacement  ξ
Sound intensity  I · SIL
Sound power  Pac
Sound power level  SWL
Sound energy
Sound exposure  E
Sound exposure level  SEL
Sound energy density  E
Sound energy flux  q
Acoustic impedance  Z
Speed of sound
Audio frequency  AF

Acoustic impedance indicates how much sound pressure is generated by the vibration of molecules of a particular acoustic medium at a given frequency. Acoustic impedance Z (or sound impedance) is frequency (f) dependent. It is very useful, for example, for describing the behaviour of musical wind instruments. Mathematically, it is the sound pressure p divided by the particle velocity v and the surface area S, through which an acoustic wave of frequency f propagates. If the impedance is calculated for a range of excitation frequencies the result is an impedance curve. Planar, single-frequency traveling waves have acoustic impedances equal to the characteristic impedance divided by the surface area, where the characteristic impedance is the product of longitudinal wave velocity and density of the medium. Acoustic impedance can be expressed in either its constituent units (pressure per velocity per area) or in rayls per square meter.

$Z = \frac{p}{vS} \,$

Note that sometimes vS is referred to as the volume velocity.

The specific acoustic impedance z (also called shock impedance) is the ratio of sound pressure p to particle velocity v at a single frequency and is expressed in rayls. Therefore

$z = \frac{p}{v} = ZS =\rho D \,$

where D is the sound wave velocity and ρ is the density.

Distinction has to be made between:

• the characteristic acoustic impedance $Z_0$ of a medium, usually air (compare with characteristic impedance in transmission lines).
• the impedance $Z$ of an acoustic component, like a wave conductor, a resonance chamber, a muffler or an organ pipe.

## Characteristic acoustic impedance

The characteristic (acoustic) impedance of a medium is an inherent property of a medium:[1]

${Z_0= \rho_0 c_0}$

Here ${Z_0}$ is the characteristic impedance, measured in Rayls, and ${\rho_0}$ and ${c_0}$ are the density and speed of sound in the unperturbed medium (i.e. when there are no sound waves travelling in it).

In a viscous medium, there will be a phase difference between the pressure and velocity, so the specific acoustic impedance $\underline{Z}$ will be different from the characteristic acoustic impedance ${Z_0}$.

The characteristic impedance of air at room temperature is about 415 Pa·s/m or N·s/m³. By comparison the sound speed and density of water are much higher, resulting in an impedance of 1.5 MPa·s/m, about 3400 times higher. This differences leads to important differences between room acoustics or atmospheric acoustics on the one hand, and underwater acoustics on the other.

Effect of temperature on properties of air
Temperature T in °C Speed of sound c in m·s−1 Density of air ρ in kg·m−3 Acoustic impedance Z in N·s·m−3
+35 351.88 1.1455 403.2
+30 349.02 1.1644 406.5
+25 346.13 1.1839 409.4
+20 343.21 1.2041 413.3
+15 340.27 1.2250 416.9
+10 337.31 1.2466 420.5
+5 334.32 1.2690 424.3
0 331.30 1.2922 428.0
−5 328.25 1.3163 432.1
−10 325.18 1.3413 436.1
−15 322.07 1.3673 440.3
−20 318.94 1.3943 444.6
−25 315.77 1.4224 449.1

## Specific impedance of acoustic components

The specific acoustic impedance $Z$ of an acoustic component (in Rayls, i.e. Pa·s/m or N·s/m3) is the ratio of sound pressure $p$ to particle velocity $v$ at its connection point. Specific acoustic impedance is defined as:[1]

${\underline{Z}(\mathbf{r},\omega) = \frac{\underline{p}(\mathbf{r},\omega)}{\underline{v}(\mathbf{r},\omega)}}$

where $\underline{Z}, \underline{p}$ and $\underline{v}$ are the specific acoustic impedance, pressure and particle velocity phasors, $\mathbf{r}$ is the position and $\omega$ is the frequency. The phasors are used because, in general, the different terms will not be in phase.

If they are in phase, the following equations hold:

$Z = \frac{p}{v} = \frac{I}{v^2} = \frac{p^2}{I} \,$

where $I$ is the sound intensity (W/m²).

## Complex impedance

In general, a phase relation exists between the pressure and the matter velocity. The complex impedance is defined as

$Z = R + iX$

where

R is the resistive part, and
X is the reactive part of the impedance

The resistive part represents the energy transfer of an acoustic wave. The pressure and motion are in phase, so work is done on the medium ahead of the wave.

The reactive part represents pressure that is out of phase with the motion and causes no average energy transfer. For example, a closed bulb connected to an organ pipe will have air moving into it and pressure, but they are out of phase so no net energy is transmitted into it. While the pressure rises, air moves in, and while it falls, it moves out, but the average pressure when the air moves in is the same as that when it moves out, so the power flows back and forth but with no time averaged energy transfer. The electrical analogy for this is a capacitor connected across a power line. Current flows through the capacitor but it is out of phase with the voltage, so no net power is transmitted into it.

The phase of the impedance is then given by

$\angle Z = \tan^{-1} \left(\frac{X}{R}\right)$