Acoustic impedance

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In acoustics, there are three different special definitions of impedance - as resistances that counteract the propagation of vibrations in a specific environment. The properties of the propagation medium , obstacles, transitions to other propagation media as well as objects, surfaces or areas with certain acoustic properties have an influence on the impedance .

General information on impedance

→ Main article : Wave impedance

The impedance is a complex quantity that is composed of the resistance  R (real part) and the reactance  X (imaginary part):

${\displaystyle {\underline {Z}}=R+iX}$

The reciprocal of the impedance is called admittance  Y:

${\displaystyle {\underline {Y}}={{\underline {Z}}^{-1}}={\frac {1}{\underline {Z}}}}$

Acoustic field impedance

→ Main article : Characteristic acoustic impedance

The acoustic field impedance Z _F , also known as the characteristic sound impedance , describes the resistance that is opposed to the sound propagation in the (free) sound field . It results from the quotient of sound pressure  p and sound velocity  v :

${\displaystyle {\underline {Z_{F}}}={\frac {\ {\underline {p}}\ }{\ {\underline {v}}\ }}\qquad \left[{\frac {{\text{N}}\,{\text{s}}}{{\text{m}}^{3}}}={\frac {\text{kg}}{{\text{m}}^{2}\,{\text{s}}}}\right]}$

Sound pressure and sound velocity, and thus also the acoustic field impedance, are described here as complex quantities that  depend on the frequency f:

${\displaystyle {\underline {p}}(f)=|p(f)|e^{i\varphi _{p}(f)}}$

${\displaystyle {\underline {v}}(f)=|v(f)|e^{i\varphi _{v}(f)}}$

${\displaystyle \Rightarrow {\underline {Z_{F}}}(f)={\frac {|p(f)|}{|v(f)|}}e^{i(\varphi _{p}(f)-\varphi _{v}(f))}}$

If the sound pressure and sound velocity are in phase ( ), the acoustic field impedance is a real quantity. ${\displaystyle \varphi _{p}=\varphi _{v}}$

In the free sound field, the acoustic field impedance is determined by the properties of the propagation medium:

${\displaystyle Z_{F}=\rho \cdot c}$

With

• ρ: density
• c: speed of sound .

The greater the difference between two materials (e.g. air, water) in their field impedance, the greater the proportion of sound energy that is reflected when sound waves hit an interface (e.g. from air on water) ; the other part is let through . For the example mentioned, the impedance of water is about 3000 times higher than that of air, which means that most of the sound energy is reflected. (This is why we can hear all the noises that arise in the water well underwater , but we hardly perceive any noises that come from the air.)

Acoustic flow impedance

The acoustic flow impedance Z _A , also simply referred to as acoustic impedance, describes the resistance that is opposed to the propagation of sound in pipes . It results from the quotient of sound pressure and sound flow  q :

${\displaystyle {\underline {Z_{A}}}={\frac {\ {\underline {p}}\ }{\ {\underline {q}}\ }}\qquad \left[{\frac {{\text{N}}\,{\text{s}}}{{\text{m}}^{5}}}={\frac {\text{kg}}{{\text{m}}^{4}\,{\text{s}}}}\right]}$

Acoustic flow impedance, sound pressure and sound flow are described here as complex quantities that  depend on the frequency and the phase angle φ:

${\displaystyle {\underline {p}}(f)=\left|p(f)\right|e^{i\varphi _{p}(f)}}$

${\displaystyle {\underline {q}}(f)=\left|q(f)\right|e^{i\varphi _{q}(f)}}$

${\displaystyle \Rightarrow {\underline {Z_{A}}}(f)={\frac {\left|p(f)\right|}{\left|q(f)\right|}}e^{i\left(\varphi _{p}(f)-\varphi _{q}(f)\right)}}$

If the sound pressure and sound flow are in phase, the acoustic flow impedance is a real quantity.

Mechanical impedance

The mechanical impedance Z _M describes the resistance that the propagation of mechanical vibrations z. B. of loudspeaker membranes , microphones , ossicles or mechanical filters is opposed. It results from the quotient of force  F and speed  v :

${\displaystyle {\underline {Z_{M}}}={\frac {\ {\underline {F}}\ }{\ {\underline {v}}\ }}\qquad \left[{\frac {{\text{N}}\,{\text{s}}}{\text{m}}}={\frac {\text{kg}}{\text{s}}}\right]}$

Mechanical impedance, force and speed are described here as complex quantities that depend on frequency and phase angle:

${\displaystyle {\underline {F}}(f)=|F(f)|e^{i\varphi _{F}(f)}}$

${\displaystyle {\underline {v}}(f)=|v(f)|e^{i\varphi _{v}(f)}}$

${\displaystyle \Rightarrow {\underline {Z_{M}}}(f)={\frac {|F(f)|}{|v(f)|}}e^{i(\varphi _{F}(f)-\varphi _{v}(f))}}$

If force and speed are in phase, the mechanical impedance is a real quantity.

See also

• Acoustic radiation resistance