# 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 is a complex number that indicates how much acoustic pressure is generated by a given acoustic flow.

## Acoustic impedance

Acoustic impedance is the complex representation (also called analytic representation) of acoustic resistance. It has the symbol Z and is the ratio of complex acoustic pressure p to complex acoustic volume flow (or complex acoustic volume velocity) U. Acoustic impedance can be expressed in either Pa·m−3·s or in rayl·m−2.

$Z = \underline R = R + \mathrm{i} X = \frac{\underline p}{\underline U}$

where:

• R is the acoustic resistance;
• X is the acoustic reactance.

One also defines acoustic admittance as:

$Y = \frac{1}{Z} = \underline G = G + \mathrm{i} B$

where:

• G is the acoustic conductance;
• B is the acoustic susceptance.

There is a close analogy with electrical impedance, the ratio of complex voltage V to complex current I. Acoustic resistance 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.
Acoustic reactance represents the 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.

## Specific acoustic impedance

Specific acoustic impedance is the complex representation (also called analytic representation) of specific acoustic resistance. It has the symbol z and is the ratio of complex acoustic pressure p to complex specific acoustic volume flow (or complex specific acoustic volume velocity or complex particle velocity) v, which is the same as complex acoustic volume flow per unit area. Specific acoustic impedance can be expressed in either Pa·m−1·s or in rayl.

$z = \underline r = r + \mathrm{i} x = \frac{\underline p}{\underline v}$

where:

• r is the specific acoustic resistance;
• x is the specific acoustic reactance.

One also defines specific acoustic admittance as:

$y = \frac{1}{z} = \underline g = g + \mathrm{i} b$

where:

• g is the specific acoustic conductance;
• b is the specific acoustic susceptance.

Specific acoustic impedance, z, is an intensive property of a medium: for instance, the z of air or of water can be specified.
Acoustic impedance Z is the property of a particular geometry and medium: for instance, the Z of a particular duct filled with air can be discussed.

## Specific acoustic impedance for progressive plane waves

The constitutive law of non dispersive linear acoustic in one dimension gives a relation between stress and strain:

$p = -\rho c^2 \frac{\partial \xi}{\partial x}$

where x is the space variable along the direction of propagation of the sound waves, p is the acoustic pressure in the medium, ρ the volumetric mass density of the medium, c the speed of the sound waves propagating in the medium and ξ the particle displacement.

This equation is valid both for fluids and solids. In:

Newton's second law applied locally in the medium gives:

$\rho \frac{\partial^2 \xi}{\partial t^2} = -\frac{\partial p}{\partial x}.$

Combining this equation with the previous one yields the one-dimensional wave equation:

$\frac{\partial^2 \xi}{\partial t^2} = c^2 \frac{\partial \xi^2}{\partial x^2}.$

The plane waves:

$\xi(\vec r,\, t) = \xi(x,\, t)$

that are solutions of this equation are composed of the sum of two progressive plane waves propagating along x with the same speed and in opposite ways:

$\xi(\vec r,\, t) = f(x - ct) + g(x + ct)$

from which can be derived:

$v(\vec r,\, t) = \frac{\partial \xi}{\partial t}(\vec r,\, t) = -c(f'(x - ct) - g'(x + ct))$

and:

$p(\vec r,\, t) = -\rho c^2 \frac{\partial \xi}{\partial x}(\vec r,\, t) = -\rho c^2 (f'(x - ct) + g'(x + ct)).$

Specific acoustic impedance z is defined as the ratio of p to v.
For progressive plane waves:

$p(\vec r,\, t) = \begin{cases} -\rho c^2\, f'(x - ct)\\ \text{or}\\ -\rho c^2\, g'(x + ct). \end{cases}$

and:

$v(\vec r,\, t) = \begin{cases} -c\, f'(x - ct)\\ \text{or}\\ c\, g'(x + ct) \end{cases}$

Since by definition of complex representation:

$\underline p = p + \mathrm{i} \mathcal{H}(p)$

and:

$\underline v = v + \mathrm{i} \mathcal{H}(v)$

with $\mathcal{H}$ the Hilbert transform, this gives:

$z = \frac{\underline p}{\underline v} = \pm \rho c.$

z varies greatly among media, especially between gas and condensed phases. Water is 800 times denser than air and its speed of sound is 4.3 times greater than that of air. So the specific acoustic impedance of water is 3,500 times higher than that of air. This means that a sound in water with a given pressure amplitude is 3,500 times less intense than one in air with the same pressure. This is because the air, with its lower z, moves with a much greater velocity and displacement amplitude than does water. Reciprocally, if a sound in water and another in air have the same intensity, then the pressure is much smaller in air. These variations lead to important differences between room acoustics or atmospheric acoustics on the one hand, and underwater acoustics on the other.

Besides, temperature acts on speed of sound and mass density and thus on specific acoustic impedance.

Effect of temperature on properties of air
Temperature
T (°C)
Speed of sound
c (m·s−1)
Density of air
ρ (kg·m−3)
Specific acoustic impedance
z (Pa·s·m−1)
+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

## Relationship between acoustic impedance and specific acoustic impedance

A one dimensional wave passing through an aperture with area A is now considered. The acoustic volume flow U is the volume passing per second through the aperture. If the acoustic flow moves a distance dx = vdt, then the volume passing through is dV = Adx, so:

$U = \frac{\mathrm{d}V}{\mathrm{d}t} = A \frac{\mathrm{d}x}{\mathrm{d}t} = A v.$

The acoustic impedance Z is the ratio of complex acoustic pressure to complex acoustic volume flow, so provided that the wave is only one-dimensional, it yields:

$Z = \frac{\underline p}{\underline U} = \frac{\underline p}{A \underline v} = \frac{z}{A}.$

If the wave is a progressive plane wave, then:

$Z = \pm \frac{\rho c}{A}$

and the absolute value of this acoustic impedance is often called characteristic acoustic impedance and written Z0:

$Z_0 = \frac{\rho c}{A}.$

If the aperture with area A is the start of a pipe and a plane wave is sent into the pipe, the wave passing through the aperture is a progressive plane wave in the absence of reflections. There are usually reflections from the other end of the pipe, whether open or closed, so there is a sum of waves travelling from one end to the other. The reflections and resultant standing waves are very important in musical wind instruments. It is possible to have no reflections when the pipe is very long, because it then takes a long time for the reflected waves to return and, when it does, they are much attenuated by losses at the wall.