Acoustic impedance

Sound measurements
Sound measurements
Characteristic	Symbols
Sound pressure	p, SPL, LPA
Particle velocity	v, SVL
Particle displacement	δ
Sound intensity	I, SIL
Sound power	P, SWL, LWA
Sound energy	W
Sound energy density	w
Sound exposure	E, SEL
Acoustic impedance	Z
Audio frequency	AF
Transmission loss	TL
	v; t; e;

The acoustic impedance Z (or sound impedance) is a frequency (f) dependent parameter and 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 impedance 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.

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

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

The specific acoustic impedance z is the ratio of sound pressure p to particle velocity v at a single frequency. Therefore

z={\frac {p}{v}}=ZS\,

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

Acoustic impedance of air at different air temperatures and at 1 atm^{[citation needed]}

Effect of temperature on properties of air
Celsius temperature θ [°C]	Speed of sound c [m/s]	Density of air ρ [kg/m³]	Characteristic specific acoustic impedance z₀ [Pa⋅s/m]
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

The characteristic impedance of a medium, such as air, rock or water is a material property:

Z_{0}=\rho \cdot c\,

where

Z₀ is the characteristic acoustic impedance ([M·L^–2·T⁻¹]; N·s/m³ or Pa·s/m)

ρ is the density of the medium ([M·L⁻³]; kg/m³), and

c is the longitudinal wave speed or sound speed ([L·T⁻¹]; m/s)

The characteristic impedance of air at room temperature is about 420 Pa·s/m. 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.

Specific impedance of acoustic components

The specific acoustic impedance z of an acoustic component (in N·s/m³) is the ratio of sound pressure p to particle velocity v at its connection point:

z={\frac {p}{v}}={\frac {I}{v^{2}}}={\frac {p^{2}}{I}}\,

where

p is the sound pressure (N/m² or Pa),

v is the particle velocity (m/s), and

I is the sound intensity (W/m²)

Complex impedance

In general, a phase relation exists between the pressure and the particle 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 various loss mechanisms an acoustic wave experiences such as random thermal motion. For the case of propagation through a duct, wall vibrations and viscous forces at the air/wall interface (boundary layer) can also have a significant effect, especially at high frequencies for the latter. For resistive effects, energy is removed from the wave and converted into other forms. This energy is said to be 'lost from the system'.

The reactive part represents the ability of air to store the kinetic energy of the wave as potential energy since air is a compressible medium. It does so by compression and rarefaction. The electrical analogy for this is the capacitor's ability to store and dump electric charge, hence storing and releasing energy in the electric field between the capacitor plates. For reactive effects, energy is not lost from the system but converted between kinetic and potential forms.

The phase of the impedance is then given by

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

Particle velocity is directly related to pressure at all times - there is neither a time nor spatial "phase" between them. Pressure is defined as a scalar. Velocity is defined with a vector. There is no means to define pressure as a vector - so spatially, it cannot have "phase" with respect to velocity. In the time domain, at any given instant, pressure is defined as being directly proportional to particle velocity. The acoustic wave equation is a mathematical construct to define changes in pressure(and simultaneously, velocity) that are moving with medium at constant velocity. The point of reference for "particle velocity" must be carefully understood before analogies to electrical capacitance or kinetic energy storage can be assigned mathematically and conclusions with respect to "phase" can be drawn.

Impedance and the input impulse response

Impedance is a frequency-domain parameter. The input impulse response (IIR) is a time-domain parameter and is closely related to the impedance via the Fourier transform. Specifically the IIR is defined as the real part of the inverse Fourier transform of the reflection function

\mathrm {IIR} =\mathrm {Re} \left(\mathrm {F.T.} ^{-1}[R(f)]\right)\,

where

R(f)={\frac {Z-Z_{0}}{Z+Z_{0}}}\,

^[1]

is the reflection coefficient

and

T(f)={\frac {2Z_{0}}{Z+Z_{0}}}\,

^[1]

is the transmission coefficient

f is the frequency (Hz)

References

^ ^a ^b Telford, W.M. (1978). Applied Geophysics. New York, U.S.A.: Cambridge University Press. p. 251. ISBN 0521291461. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

[appliedGeo-1] Telford, W.M. (1978). Applied Geophysics. New York, U.S.A.: Cambridge University Press. p. 251. ISBN 0521291461. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[1]

Characteristic acoustic impedance

Specific impedance of acoustic components

Complex impedance

Impedance and the input impulse response

See also

References

External links