Acoustic impedance

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

Acoustic impedance is the measure of the opposition that a system presents to an acoustic flow when an acoustic pressure is applied to it.

In quantitative terms, it is the ratio of the complex acoustic pressure applied to a system to the resulting complex acoustic volume flow rate through a surface perpendicular to the direction of this acoustic pressure at its point of application. There is a close analogy with electrical impedance, the ratio of the complex electrical voltage applied to a system to the resulting complex electrical current intensity.

Mathematical definitions[edit]

Acoustic impedance[edit]

Acoustic impedance is the complex representation (also called analytic representation) of acoustic resistance. It is the ratio of the complex acoustic pressure applied to a system to the resulting complex acoustic volume flow rate through a surface perpendicular to the direction of this acoustic pressure at its point of application.
Acoustic impedance, denoted Z and measured in Pa·m−3·s or in raylMKS·m−2, is given by:

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

where:

  • p is the complex acoustic pressure, measured in Pa;
  • Q is the complex acoustic volume flow rate, measured in m3·s−1;
  • R is the complex acoustic resistance, measured in Pa·m−3·s;
  • R is the acoustic resistance, measured in Pa·m−3·s;
  • X is the acoustic reactance, measured in Pa·m−3·s.

The symbol raylMKS is used to indicate the MKS rayl,[1] abbreviated henceforth as "rayl".

Acoustic admittance, denoted Y and measured in Pa−1·m3·s−1 or in rayl−1·m2,[1] is given by:

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

where:

  • Z is the acoustic impedance, measured in Pa·m−3·s;
  • G = 1/R is the complex acoustic conductance, measured in Pa−1·m3·s−1;
  • G = R/(R2 + X2) is the acoustic conductance, measured in Pa−1·m3·s−1;
  • B = −X/(R2 + X2) is the acoustic susceptance, measured in Pa−1·m3·s−1.

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, as well, 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[edit]

Specific acoustic impedance is the complex representation of specific acoustic resistance. It is the ratio of the complex acoustic pressure applied to a system to the resulting complex particle velocity in the direction of this acoustic pressure at its point of application.
Specific acoustic impedance, denoted z and measured in Pa·m−1·s or in rayl,[1] is given by:

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

where:

  • p is the complex acoustic pressure, measured in Pa;
  • v is the complex particle velocity, measured in m·s−1;
  • r is the complex specific acoustic resistance, measured in Pa·m−1·s;
  • r is the specific acoustic resistance, measured in Pa·m−1·s;
  • x is the specific acoustic reactance, measured in Pa·m−1·s.

Specific acoustic admittance, denoted y and measured in Pa−1·m·s−1 or in rayl−1,[1] is given by:

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

where:

  • z is the specific acoustic impedance, measured in Pa·m−1·s;
  • g = 1/r is the complex specific acoustic conductance, measured in Pa−1·m·s−1;
  • g = r/(r2 + x2) is the specific acoustic conductance, measured in Pa−1·m·s−1;
  • b = −x/(r2 + x2) is the specific acoustic susceptance, measured in Pa−1·m·s−1.

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

Relationship[edit]

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

Q = \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 rate, so provided that the wave is only one-dimensional, it yields:

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

Characteristic acoustic impedance[edit]

Characteristic specific acoustic impedance[edit]

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 traveling 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(\mathbf r,\, t) = \xi(x,\, t)

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

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

from which can be derived:

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

and:

p(\mathbf r,\, t) = -\rho c^2 \frac{\partial \xi}{\partial x}(\mathbf 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:


\begin{cases}
p(\mathbf r,\, t) = -\rho c^2\, f'(x - ct)\\
v(\mathbf r,\, t) = -c\, f'(x - ct)
\end{cases}

or:


\begin{cases}
p(\mathbf r,\, t) = -\rho c^2\, g'(x + ct)\\
v(\mathbf r,\, t) = 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.

The absolute value of this specific acoustic impedance is often called characteristic specific acoustic impedance and denoted z0:

z_0 = \rho c.

z0 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 z0, 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)
Characteristic specific acoustic impedance
z0 (Pa·m−1·s)
+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

Characteristic acoustic impedance[edit]

For a one dimensional wave passing through an aperture with area A:

Z = \frac{z}{A},

so if the wave is a progressive plane wave, then:

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

The absolute value of this acoustic impedance is often called characteristic acoustic impedance and denoted Z0:

Z_0 = \frac{z_0}{A} = \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.

See also[edit]

References[edit]

  1. ^ a b c d Cobbold, Richard S. C. (2007). Foundations of Biomedical Ultrasound. New York: Oxford University Press. pp. 41–42. ISBN 978-0-19-516831-0. 
  • Kinsler, Lawrence; Frey, Austin; Coppens, Alan; Sanders, James (2000), Fundamentals of Acoustics, New York: John Wiley & Sons, Inc., ISBN 0-471-84789-5 .

External links[edit]