Acoustic impedance

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

Acoustic impedance is the measure of the opposition that a system presents to the acoustic flow resulting of an acoustic pressure applied to the system. There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electrical flow resulting of an electrical voltage applied to the system.

Mathematical definitions[edit]

Acoustic impedance[edit]

For a linear time-invariant system, the relationship between the acoustic pressure applied to the system and the resulting acoustic volume flow rate through a surface perpendicular to the direction of that pressure at its point of application is given by:

p(t) = [R * Q](t),

or equivalently by:

Q(t) = [G * p](t),

where

  • p is the acoustic pressure, measured in Pa;
  • Q is the acoustic volume flow rate, measured in m3/s;
  • * is the convolution operator;
  • R is the acoustic resistance in the time domain, measured in Pa·m−3·s or in rayl·m−2;
  • G = R −1 is the acoustic conductance in the time domain, measured in Pa−1·m3·s−1 or in rayl−1·m2 (R −1 is the convolution inverse of R).

The symbol rayl is used to indicate the MKS rayl.

Acoustic impedance, denoted Z and measured in Pa·m−3·s or in rayl·m−2,[1] is the Laplace transform, or the Fourier transform, or the analytic representation of time domain acoustic resistance:

Z(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[R](s) = \frac{\mathcal{L}[p](s)}{\mathcal{L}[Q](s)},
Z(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[R](\omega) = \frac{\mathcal{F}[p](\omega)}{\mathcal{F}[Q](\omega)},
Z(t) \stackrel{\mathrm{def}}{{}={}} R_\mathrm{a}(t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(Q^{-1}\right)_\mathrm{a}\right]\!(t),

where

  • \mathcal L is the Laplace transform operator;
  • \mathcal F is the Fourier transform operator;
  • subscript "a" is the analytic representation operator;
  • where Q −1 is the convolution inverse of Q.

Acoustic resistance, denoted R and measured in Pa·m−3·s or in rayl·m−2, and acoustic reactance, denoted X and measured in Pa·m−3·s or in rayl·m−2, are the real part and imaginary part of acoustic impedance respectively:

Z(s) = R(s) + iX(s),
Z(\omega) = R(\omega) + iX(\omega),
Z(t) = R(t) + iX(t),

where

  • i is the imaginary unit;
  • in Z(s), R(s) is not the Laplace transform of the time domain acoustic resistance R(t), Z(s) is;
  • in Z(ω), R(ω) is not the Fourier transform of the time domain acoustic resistance R(t), Z(ω) is;
  • in Z(t), R(t) is the time domain acoustic resistance and X(t) is the Hilbert transform of the time domain acoustic resistance R(t), according to the definition of the analytic representation.

Inductive acoustic reactance, denoted XL and measured in Pa·m−3·s or in rayl·m−2, and capacitive acoustic reactance, denoted XC and measured in Pa·m−3·s or in rayl·m−2, are the positive part and negative part of acoustic reactance respectively:

X(s) = X_L(s) - X_C(s),
X(\omega) = X_L(\omega) - X_C(\omega),
X(t) = X_L(t) - X_C(t).

Acoustic admittance, denoted Y and measured in Pa−1·m3·s−1 or in rayl−1·m2,[1] is the Laplace transform, or the Fourier transform, or the analytic representation of time domain acoustic conductance:

Y(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[G](s) = \frac{1}{Z(s)} = \frac{\mathcal{L}[Q](s)}{\mathcal{L}[p](s)},
Y(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[G](\omega) = \frac{1}{Z(\omega)} = \frac{\mathcal{F}[Q](\omega)}{\mathcal{F}[p](\omega)},
Y(t) \stackrel{\mathrm{def}}{{}={}} G_\mathrm{a}(t) = Z^{-1}(t) = \frac{1}{2}\!\left[Q_\mathrm{a} * \left(p^{-1}\right)_\mathrm{a}\right]\!(t),

where Z −1 and p −1 are the convolution inverses of Z and p respectively.

Acoustic conductance, denoted G and measured in Pa−1·m3·s−1 or in rayl−1·m2, and acoustic susceptance, denoted B and measured in Pa−1·m3·s−1 or in rayl−1·m2, are the real part and imaginary part of acoustic admittance respectively:

Y(s) = G(s) + iB(s),
Y(\omega) = G(\omega) + iB(\omega),
Y(t) = G(t) + iB(t),

where

  • in Y(s), G(s) is not the Laplace transform of the time domain acoustic conductance G(t), Y(s) is;
  • in Y(ω), G(ω) is not the Fourier transform of the time domain acoustic conductance G(t), Y(ω) is;
  • in Y(t), G(t) is the time domain acoustic conductance and B(t) is the Hilbert transform of the time domain acoustic conductance G(t), according to the definition of the analytic representation.

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]

For a linear time-invariant system, the relationship between the acoustic pressure applied to the system and the resulting particle velocity in the direction of that pressure at its point of application is given by:

p(t) = [r * v](t),

or equivalently by:

v(t) = [g * p](t),

where

  • p is the acoustic pressure, measured in Pa;
  • v is the particle velocity, measured in m/s;
  • r is the specific acoustic resistance in the time domain, measured in Pa·m−1·s or in rayl;
  • g = r −1 is the specific acoustic conductance in the time domain, measured in Pa−1·m·s−1 or in rayl−1 (r −1 is the convolution inverse of r).

Specific acoustic impedance, denoted z and measured in Pa·m−1·s or in rayl,[1] is the Laplace transform, or the Fourier transform, or the analytic representation of time domain specific acoustic resistance:

z(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[r](s) = \frac{\mathcal{L}[p](s)}{\mathcal{L}[v](s)},
z(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[r](\omega) = \frac{\mathcal{F}[p](\omega)}{\mathcal{F}[v](\omega)},
z(t) \stackrel{\mathrm{def}}{{}={}} r_\mathrm{a}(t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(v^{-1}\right)_\mathrm{a}\right]\!(t),

where v −1 is the convolution inverse of v.

Specific acoustic resistance, denoted r and measured in Pa·m−1·s or in rayl, and specific acoustic reactance, denoted x and measured in Pa·m−1·s or in rayl, are the real part and imaginary part of specific acoustic impedance respectively:

z(s) = r(s) + ix(s),
z(\omega) = r(\omega) + ix(\omega),
z(t) = r(t) + ix(t),

where

  • in z(s), r(s) is not the Laplace transform of the time domain specific acoustic resistance r(t), z(s) is;
  • in z(ω), r(ω) is not the Fourier transform of the time domain specific acoustic resistance r(t), z(ω) is;
  • in z(t), r(t) is the time domain specific acoustic resistance and x(t) is the Hilbert transform of the time domain specific acoustic resistance r(t), according to the definition of the analytic representation.

Specific inductive acoustic reactance, denoted xL and measured in Pa·m−1·s or in rayl, and specific capacitive acoustic reactance, denoted xC and measured in Pa·m−1·s or in rayl, are the positive part and negative part of specific acoustic reactance respectively:

x(s) = x_L(s) - x_C(s),
x(\omega) = x_L(\omega) - x_C(\omega),
x(t) = x_L(t) - x_C(t).

Specific acoustic admittance, denoted y and measured in Pa−1·m·s−1 or in rayl−1,[1] is the Laplace transform, or the Fourier transform, or the analytic representation of time domain specific acoustic conductance:

y(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[g](s) = \frac{1}{z(s)} = \frac{\mathcal{L}[v](s)}{\mathcal{L}[p](s)},
y(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[g](\omega) = \frac{1}{z(\omega)} = \frac{\mathcal{F}[v](\omega)}{\mathcal{F}[p](\omega)},
y(t) \stackrel{\mathrm{def}}{{}={}} g_\mathrm{a}(t) = z^{-1}(t) = \frac{1}{2}\!\left[v_\mathrm{a} * \left(p^{-1}\right)_\mathrm{a}\right]\!(t),

where z −1 and p −1 are the convolution inverses of z and p respectively.

Specific acoustic conductance, denoted g and measured in Pa−1·m·s−1 or in rayl−1, and specific acoustic susceptance, denoted b and measured in Pa−1·m3·s−1 or in rayl−1·m2, are the real part and imaginary part of specific acoustic admittance respectively:

y(s) = g(s) + ib(s),
y(\omega) = g(\omega) + ib(\omega),
y(t) = g(t) + ib(t),

where

  • in y(s), g(s) is not the Laplace transform of the time domain acoustic conductance g(t), y(s) is;
  • in y(ω), g(ω) is not the Fourier transform of the time domain acoustic conductance g(t), y(ω) is;
  • in y(t), g(t) is the time domain acoustic conductance and b(t) is the Hilbert transform of the time domain acoustic conductance g(t), according to the definition of the analytic representation.

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 = v dt, then the volume of medium passing through is dV = A dx, so:

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

Provided that the wave is only one-dimensional, it yields:

Z(s) = \frac{\mathcal{L}[p](s)}{\mathcal{L}[Q](s)} = \frac{\mathcal{L}[p](s)}{A \mathcal{L}[v](s)} = \frac{z(s)}{A},
Z(\omega) = \frac{\mathcal{F}[p](\omega)}{\mathcal{F}[Q](\omega)} = \frac{\mathcal{F}[p](\omega)}{A \mathcal{F}[v](\omega)} = \frac{z(\omega)}{A},
Z(t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(Q^{-1}\right)_\mathrm{a}\right]\!(t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(\frac{v^{-1}}{A}\right)_\mathrm{a}\right]\!(t) = \frac{z(t)}{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

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

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}

By definition of specific acoustic impedance z, it gives:

z(\mathbf{r},\, s) = \frac{\mathcal{L}[p](\mathbf{r},\, s)}{\mathcal{L}[v](\mathbf{r},\, s)} = \pm \rho c,
z(\mathbf{r},\, \omega) = \frac{\mathcal{F}[p](\mathbf{r},\, \omega)}{\mathcal{F}[v](\mathbf{r},\, \omega)} = \pm \rho c,
z(\mathbf{r},\, t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(v^{-1}\right)_\mathrm{a}\right]\!(\mathbf{r},\, t) = \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 1.145,5 403.2
30 349.02 1.164,4 406.5
25 346.13 1.183,9 409.4
20 343.21 1.204,1 413.3
15 340.27 1.225,0 416.9
10 337.31 1.246,6 420.5
5 334.32 1.269,0 424.3
0 331.30 1.292,2 428.0
−5 328.25 1.316,3 432.1
−10 325.18 1.341,3 436.1
−15 322.07 1.367,3 440.3
−20 318.94 1.394,3 444.6
−25 315.77 1.422,4 449.1

Characteristic acoustic impedance[edit]

For a one dimensional wave passing through an aperture with area A, Z = z/A, so if the wave is a progressive plane wave, then:

Z(\mathbf{r},\, s) = \pm \frac{\rho c}{A},
Z(\mathbf{r},\, \omega) = \pm \frac{\rho c}{A},
Z(\mathbf{r},\, t) = \pm \frac{\rho c}{A}.

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

See also[edit]

Notes[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. 

References[edit]

  • 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]