View source for Acoustic impedance

{{Short description|Opposition that a system presents to an acoustic pressure}}
{{Sound measurements}}

'''Acoustic impedance''' and '''specific acoustic impedance''' are measures of the opposition that a system presents to the acoustic flow resulting from an [[acoustic pressure]] applied to the system. The [[International System of Units|SI unit]] of acoustic impedance is the pascal-second per cubic metre (symbol Pa·s/m<sup>3</sup>), or in the [[MKS system of units|MKS system]] the [[rayl]] per square metre (Rayl/m<sup>2</sup>), while that of specific acoustic impedance is the pascal-second per metre (Pa·s/m), or in the MKS system the rayl (Rayl).<ref name=Kinsler>{{cite book |vauthors=Kinsler L, Frey A, Coppens A, Sanders J |year=2000 |title=Fundamentals of Acoustics |publisher=Wiley |location=Hoboken |isbn=0-471-84789-5 }}</ref> There is a [[Mechanical–electrical analogies#Impedance analogies|close analogy]] with [[electrical impedance]], which measures the opposition that a system presents to the [[electric current]] resulting from a [[voltage]] applied to the system.

== Mathematical definitions==
=== Acoustic impedance ===
For a [[LTI system theory|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:{{citation needed|date=March 2019}}
: <math>p(t) = [R * Q](t),</math>
or equivalently by
: <math>Q(t) = [G * p](t),</math>
where
* ''p'' is the acoustic pressure;
* ''Q'' is the acoustic volume flow rate;
* <math>*</math> is the [[convolution]] operator;
* ''R'' is the '''acoustic resistance in the ''time domain''''';
* ''G'' = ''R''{{i sup|−1}} is the '''acoustic conductance in the ''time domain''''' (''R''{{i sup|−1}} is the convolution inverse of ''R'').

'''Acoustic impedance''', denoted ''Z'', is the [[Laplace transform]], or the [[Fourier transform]], or the [[Analytic signal|analytic representation]] of ''time domain'' acoustic resistance:<ref name=Kinsler />
: <math>Z(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[R](s) = \frac{\mathcal{L}[p](s)}{\mathcal{L}[Q](s)},</math>
: <math>Z(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[R](\omega) = \frac{\mathcal{F}[p](\omega)}{\mathcal{F}[Q](\omega)},</math>
: <math>Z(t) \stackrel{\mathrm{def}}{{}={}} R_\mathrm{a}(t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(Q^{-1}\right)_\mathrm{a}\right]\!(t),</math>
where
* <math>\mathcal L</math> is the Laplace transform operator;
* <math>\mathcal F</math> is the Fourier transform operator;
* subscript "a" is the analytic representation operator;
* ''Q''{{i sup|−1}} is the convolution inverse of ''Q''.

'''Acoustic resistance''', denoted ''R'', and '''acoustic reactance''', denoted ''X'', are the [[real part]] and [[imaginary part]] of acoustic impedance respectively:{{citation needed|date=March 2019}}
: <math>Z(s) = R(s) + iX(s),</math>
: <math>Z(\omega) = R(\omega) + iX(\omega),</math>
: <math>Z(t) = R(t) + iX(t),</math>
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 ''X''<sub>''L''</sub>, and '''capacitive acoustic reactance''', denoted ''X''<sub>''C''</sub>, are the [[positive part]] and [[negative part]] of acoustic reactance respectively:{{citation needed|date=March 2019}}
: <math>X(s) = X_L(s) - X_C(s),</math>
: <math>X(\omega) = X_L(\omega) - X_C(\omega),</math>
: <math>X(t) = X_L(t) - X_C(t).</math>

'''Acoustic admittance''', denoted ''Y'', is the Laplace transform, or the Fourier transform, or the analytic representation of ''time domain'' acoustic conductance:<ref name=Kinsler />
: <math>Y(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[G](s) = \frac{1}{Z(s)} = \frac{\mathcal{L}[Q](s)}{\mathcal{L}[p](s)},</math>
: <math>Y(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[G](\omega) = \frac{1}{Z(\omega)} = \frac{\mathcal{F}[Q](\omega)}{\mathcal{F}[p](\omega)},</math>
: <math>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),</math>
where
* ''Z''{{i sup|−1}} is the convolution inverse of ''Z''; 
* ''p''{{i sup|−1}} is the convolution inverse of ''p''.

'''Acoustic conductance''', denoted ''G'', and '''acoustic susceptance''', denoted ''B'', are the real part and imaginary part of acoustic admittance respectively:{{citation needed|date=March 2019}}
: <math>Y(s) = G(s) + iB(s),</math>
: <math>Y(\omega) = G(\omega) + iB(\omega),</math>
: <math>Y(t) = G(t) + iB(t),</math>
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 the pressure that is out of phase with the motion and causes no average energy transfer.{{citation needed|date=March 2019}} 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.{{citation needed|date=March 2019}} A further electrical analogy is a capacitor connected across a power line: current flows through the capacitor but it is out of phase with the voltage, so [[AC power|no net power]] is transmitted into it.

=== Specific acoustic impedance ===
For a [[LTI system theory|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
: <math>p(t) = [r * v](t),</math>
or equivalently by:
: <math>v(t) = [g * p](t),</math>
where
* ''p'' is the acoustic pressure;
* ''v'' is the particle velocity;
* ''r'' is the '''specific acoustic resistance in the ''time domain''''';
* ''g'' = ''r''{{i sup|−1}} is the '''specific acoustic conductance in the ''time domain''''' (''r''{{i sup|−1}} is the convolution inverse of ''r'').{{citation needed|date=March 2019}}

'''Specific acoustic impedance''', denoted ''z'' is the Laplace transform, or the Fourier transform, or the analytic representation of ''time domain'' specific acoustic resistance:<ref name=Kinsler />
: <math>z(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[r](s) = \frac{\mathcal{L}[p](s)}{\mathcal{L}[v](s)},</math>
: <math>z(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[r](\omega) = \frac{\mathcal{F}[p](\omega)}{\mathcal{F}[v](\omega)},</math>
: <math>z(t) \stackrel{\mathrm{def}}{{}={}} r_\mathrm{a}(t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(v^{-1}\right)_\mathrm{a}\right]\!(t),</math>
where ''v''{{i sup|−1}} is the convolution inverse of ''v''.

'''Specific acoustic resistance''', denoted ''r'', and '''specific acoustic reactance''', denoted ''x'', are the real part and imaginary part of specific acoustic impedance respectively:{{citation needed|date=March 2019}}
: <math>z(s) = r(s) + ix(s),</math>
: <math>z(\omega) = r(\omega) + ix(\omega),</math>
: <math>z(t) = r(t) + ix(t),</math>
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 ''x''<sub>''L''</sub>, and '''specific capacitive acoustic reactance''', denoted ''x''<sub>''C''</sub>, are the positive part and negative part of specific acoustic reactance respectively:{{citation needed|date=March 2019}}
: <math>x(s) = x_L(s) - x_C(s),</math>
: <math>x(\omega) = x_L(\omega) - x_C(\omega),</math>
: <math>x(t) = x_L(t) - x_C(t).</math>

'''Specific acoustic admittance''', denoted ''y'', is the Laplace transform, or the Fourier transform, or the analytic representation of ''time domain'' specific acoustic conductance:<ref name=Kinsler />
: <math>y(s) \stackrel{\mathrm{def}}{{}={}} \mathcal{L}[g](s) = \frac{1}{z(s)} = \frac{\mathcal{L}[v](s)}{\mathcal{L}[p](s)},</math>
: <math>y(\omega) \stackrel{\mathrm{def}}{{}={}} \mathcal{F}[g](\omega) = \frac{1}{z(\omega)} = \frac{\mathcal{F}[v](\omega)}{\mathcal{F}[p](\omega)},</math>
: <math>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),</math>
where
* ''z''{{i sup|−1}} is the convolution inverse of ''z'';
* ''p''{{i sup|−1}} is the convolution inverse of ''p''.

'''Specific acoustic conductance''', denoted ''g'', and '''specific acoustic susceptance''', denoted ''b'', are the real part and imaginary part of specific acoustic admittance respectively:{{citation needed|date=March 2019}}
: <math>y(s) = g(s) + ib(s),</math>
: <math>y(\omega) = g(\omega) + ib(\omega),</math>
: <math>y(t) = g(t) + ib(t),</math>
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 and extensive properties|intensive property]] of a particular ''medium'' (e.g., the ''z'' of air or water can be specified); on the other hand, acoustic impedance ''Z'' is an [[intensive and extensive properties|extensive property]] of a particular ''medium and geometry'' (e.g., the ''Z'' of a particular duct filled with air can be specified).{{citation needed|date=March 2019}}

=== Acoustic ohm ===
The '''acoustic ohm''' is a unit of measurement of acoustic impedance. The SI unit of pressure is the pascal and of flow is cubic metres per second, so the acoustic ohm is equal to 1&nbsp;Pa·s/m<sup>3</sup>.{{fact|date=January 2025}}

The acoustic ohm can be applied to fluid flow outside the domain of acoustics.  For such applications a '''hydraulic ohm''' with an identical definition may be used.  A hydraulic ohm measurement would be the ratio of hydraulic pressure to hydraulic volume flow.

=== Relationship ===
For a ''one-dimensional'' wave passing through an aperture with area ''A'', the acoustic volume flow rate ''Q'' is the volume of medium passing per second through the aperture; if the acoustic flow moves a distance {{nowrap|1=d''x'' = ''v'' d''t''}}, then the volume of medium passing through is {{nowrap|1=d''V'' = ''A'' d''x''}}, so:{{citation needed|date=March 2019}}
: <math>Q = \frac{\mathrm{d}V}{\mathrm{d}t} = A \frac{\mathrm{d}x}{\mathrm{d}t} = A v.</math>
If the wave is one-dimensional, it yields
: <math>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},</math>
: <math>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},</math>
: <math>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}.</math>

== Characteristic acoustic impedance ==
=== Characteristic specific acoustic impedance ===
The constitutive law of nondispersive linear acoustics in one dimension gives a relation between stress and strain:<ref name=Kinsler />
: <math>p = -\rho c^2 \frac{\partial \delta}{\partial x},</math>
where
* ''p'' is the [[sound pressure|acoustic pressure]] in the medium;
* ''ρ'' is the [[density|volumetric mass density]] of the medium;
* ''c'' is the [[speed of sound|speed of the sound]] waves traveling in the medium;
* ''δ'' is the [[particle displacement]];
* ''x'' is the space variable along the direction of propagation of the sound waves.

This equation is valid both for fluids and solids. In
* [[fluids]], {{nowrap|1=''ρc''<sup>2</sup> = ''K''}} (''K'' stands for the [[bulk modulus]]);
* [[solids]], {{nowrap|1=''ρc''<sup>2</sup> = ''K'' + 4/3 ''G''}} (''G'' stands for the [[shear modulus]]) for [[longitudinal wave]]s and {{nowrap|1=''ρc<sup>2</sup>'' = ''G''}} for [[transverse wave]]s.{{citation needed|date=March 2019}}

[[Newton's laws of motion|Newton's second law]] applied locally in the medium gives:<ref>{{cite book |vauthors=Attenborough K, Postema M|title=A pocket-sized introduction to acoustics |date=2008 |publisher=University of Hull|location=Kingston upon Hull |url=https://hal.archives-ouvertes.fr/hal-03188302/document| isbn=978-90-812588-2-1|doi=10.5281/zenodo.7504060}}</ref>
: <math>\rho \frac{\partial^2 \delta}{\partial t^2} = -\frac{\partial p}{\partial x}.</math>
Combining this equation with the previous one yields the one-dimensional [[wave equation]]:
: <math>\frac{\partial^2 \delta}{\partial t^2} = c^2 \frac{\partial^2 \delta}{\partial x^2}.</math>
The ''plane waves''
: <math>\delta(\mathbf{r},\, t) = \delta(x,\, t)</math>
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'':{{citation needed|date=March 2019}}
: <math>\delta(\mathbf{r},\, t) = f(x - ct) + g(x + ct)</math>
from which can be derived
: <math>v(\mathbf{r},\, t) = \frac{\partial \delta}{\partial t}(\mathbf{r},\, t) = -c\big[f'(x - ct) - g'(x + ct)\big],</math>
: <math>p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \delta}{\partial x}(\mathbf{r},\, t) = -\rho c^2 \big[f'(x - ct) + g'(x + ct)\big].</math>
For ''progressive'' plane waves:{{citation needed|date=March 2019}}
: <math>
\begin{cases}
p(\mathbf{r},\, t) = -\rho c^2\, f'(x - ct)\\
v(\mathbf{r},\, t) = -c\, f'(x - ct)
\end{cases}
</math>
or
: <math>
\begin{cases}
p(\mathbf{r},\, t) = -\rho c^2\, g'(x + ct)\\
v(\mathbf{r},\, t) = c\, g'(x + ct).
\end{cases}
</math>
Finally, the specific acoustic impedance ''z'' is
: <math>z(\mathbf{r},\, s) = \frac{\mathcal{L}[p](\mathbf{r},\, s)}{\mathcal{L}[v](\mathbf{r},\, s)} = \pm \rho c,</math>
: <math>z(\mathbf{r},\, \omega) = \frac{\mathcal{F}[p](\mathbf{r},\, \omega)}{\mathcal{F}[v](\mathbf{r},\, \omega)} = \pm \rho c,</math>
: <math>z(\mathbf{r},\, t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(v^{-1}\right)_\mathrm{a}\right]\!(\mathbf{r},\, t) = \pm \rho c.</math>{{citation needed|date=March 2019}}
The [[absolute value]] of this specific acoustic impedance is often called '''characteristic specific acoustic impedance''' and denoted ''z''<sub>0</sub>:<ref name=Kinsler />
: <math>z_0 = \rho c.</math>
The equations also show that
: <math>\frac{p(\mathbf{r},\, t)}{v(\mathbf{r},\, t)} = \pm \rho c = \pm z_0.</math>

=== Effect of temperature ===
{{unreferenced section|date=March 2019}}
{{original research|section|date=March 2019}}
Temperature acts on speed of sound and mass density and thus on specific acoustic impedance.
{{Temperature effect}}

=== Characteristic acoustic impedance ===
For a ''one dimensional'' wave passing through an aperture with area ''A'', {{nowrap|1=''Z'' = ''z''/''A''}}, so if the wave is a progressive plane wave, then:{{citation needed|date=March 2019}}
: <math>Z(\mathbf{r},\, s) = \pm \frac{\rho c}{A},</math>
: <math>Z(\mathbf{r},\, \omega) = \pm \frac{\rho c}{A},</math>
: <math>Z(\mathbf{r},\, t) = \pm \frac{\rho c}{A}.</math>
The [[absolute value]] of this acoustic impedance is often called '''characteristic acoustic impedance''' and denoted ''Z''<sub>0</sub>:<ref name=Kinsler />
: <math>Z_0 = \frac{\rho c}{A}.</math>
and the characteristic specific acoustic impedance is
: <math>\frac{p(\mathbf{r},\, t)}{Q(\mathbf{r},\, t)} = \pm \frac{\rho c}{A} = \pm Z_0.</math>

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, and the usually reflections from the other end of the pipe, whether open or closed, are the sum of waves travelling from one end to the other.<ref name=":0">{{Cite book|vauthors=Rossing TD, Fletcher NH|title=Principles of Vibration and Sound|date=2004|publisher=Springer|isbn=978-1-4757-3822-3|edition=2nd|location=Heidelberg|oclc=851835364}}</ref> (It is possible to have no reflections when the pipe is very long, because of the long time taken for the reflected waves to return, and their attenuation through losses at the pipe wall.<ref name=":0" />) Such reflections and resultant standing waves are very important in the design and operation of musical wind instruments.<ref>{{Cite book|vauthors=Fletcher NH, Rossing TD|title=The physics of musical instruments|date=1998|isbn=978-0-387-21603-4|edition=2nd|publisher=Springer|location=Heidelberg|oclc=883383570}}</ref>

== See also ==
{{div col}}
* [[Acoustic attenuation]]
* [[Earthquake bomb]]
* [[Impedance analogy]]
* [[Mechanical impedance]]
{{div col end}}

== References ==
{{reflist}}

== External links ==
* [http://www.animations.physics.unsw.edu.au/jw/sound-wave-equation.htm The Wave Equation for Sound]
* [http://newt.phys.unsw.edu.au/jw/z.html What Is Acoustic Impedance and Why Is It Important?]

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[[Category:Acoustics]]
[[Category:Physical quantities]]
[[Category:Sound]]
[[Category:Sound measurements]]