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{{Short description|Power carried by sound waves}}
{{Sound measurements}}
'''Sound intensity''', also known as '''acoustic intensity''', is defined as the power carried by sound waves per unit area in a direction perpendicular to that area. The [[International System of Units|SI unit]] of intensity, which includes sound intensity, is the [[watt]] per square meter (W/m<sup>2</sup>). One application is the noise measurement of sound [[intensity (physics)|intensity]] in the air at a listener's location as a sound energy quantity.<ref name="“GeorgiaStateUniversity">{{cite web|title=Sound Intensity|url=http://hyperphysics.phy-astr.gsu.edu/hbase/sound/intens.html|
Sound intensity is not the same physical quantity as [[sound pressure]]. Human hearing is
[[#Sound intensity level|Sound intensity level]] is a logarithmic expression of sound intensity relative to a reference intensity.
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==Mathematical definition==
Sound intensity, denoted '''I''', is defined by
where
Both '''I''' and '''v''' are [[Vector (geometric)|vectors]], which means that both have a ''direction'' as well as a magnitude. The direction of sound intensity is the average direction in which energy is flowing.
The average sound intensity during time ''T'' is given by
For a plane wave {{Citation needed|reason=This is a special case of the above expression, link to the derivation is needed|date=August 2022}},
Where,
==Inverse-square law==
{{Further|Inverse-square law}}
For a ''spherical'' sound wave, the intensity in the radial direction as a function of distance ''r'' from the centre of the sphere is given by
where
Thus sound intensity decreases as 1/''r''<sup>2</sup> from the centre of the sphere:
This relationship is an ''inverse-square law''.
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'''Sound intensity level''' (SIL) or '''acoustic intensity level''' is the [[level (logarithmic quantity)|level]] (a [[logarithmic quantity]]) of the intensity of a sound relative to a reference value.
It is denoted ''L''<sub>''I''</sub>, expressed in [[
where
The commonly used reference sound intensity in air is<ref>Ross Roeser, Michael Valente, ''Audiology: Diagnosis'' (Thieme 2007), p. 240.</ref>
being approximately the lowest sound intensity hearable by an undamaged human ear under room conditions.
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The reference sound intensity ''I''<sub>0</sub> is defined such that a progressive plane wave has the same value of sound intensity level (SIL) and [[sound pressure level]] (SPL), since
The equality of SIL and SPL requires that
where {{nobreak|1=''p''<sub>0</sub> = 20 μPa}} is the reference sound pressure.
For a ''progressive'' spherical wave,
where ''z''<sub>0</sub> is the [[Acoustic impedance#Characteristic specific acoustic impedance|characteristic specific acoustic impedance]]. Thus,
In air at ambient temperature, {{nobreak|1=''z''<sub>0</sub> = 410 Pa·s/m}}, hence the reference value {{nobreak|1=''I''<sub>0</sub> = 1 pW/m<sup>2</sup>}}.<ref>Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.</ref>
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==Measurement==
Sound intensity is defined as the time averaged product of sound pressure and acoustic particle velocity.<ref>{{Cite book | title=
Pressure-based measurement methods are widely used in anechoic conditions for noise quantification purposes. The bias error introduced by a ''p-p'' probe can be approximated by<ref name=":0">{{Cite journal |last1=Jacobsen|first1=Finn |last2=de Bree|first2=Hans-Elias |date=2005-09-01 |title=A comparison of two different sound intensity measurement principles |journal=The Journal of the Acoustical Society of America |volume=118 |issue=3 |pages=1510–1517 |doi=10.1121/1.1984860 |bibcode=2005ASAJ..118.1510J |s2cid=56449985 |issn=0001-4966 |url=https://backend.orbit.dtu.dk/ws/files/4449916/Jacobsen.pdf}}</ref>
<math display="block">\widehat{I}^{p-p}_n \simeq I_n - \frac{\varphi_{\text{pe}}\,p_{\text{rms}}^2}{{k\Delta r \rho c}}=I_n \
▲<math>\widehat{I}^{p-p}_n \simeq I_n - \frac{\varphi_{\text{pe}}\,p_{\text{rms}}^2}{{k\Delta r \rho c}}=I_n \biggl( 1-\frac{\varphi_{\text{pe}}}{{k\Delta r}}\frac{p_{\text{rms}}^2/ \rho c}{I_r}\biggr) \, ,</math>
where <math>I_n</math>is the “true” intensity (unaffected by calibration errors), <math>\hat{I}^{p-p}_n</math> is the biased estimate obtained using a ''p-p'' probe, <math>p_{\text{rms}}</math>is the root-mean-squared value of the sound pressure, <math>k</math> is the wave number, <math>\rho</math> is the density of air, <math>c</math> is the speed of sound and <math>\Delta r</math> is the spacing between the two microphones. This expression shows that phase calibration errors are inversely proportional to frequency and microphone spacing and directly proportional to the ratio of the mean square sound pressure to the sound intensity. If the pressure-to-intensity ratio is large then even a small phase mismatch will lead to significant bias errors. In practice, sound intensity measurements cannot be performed accurately when the pressure-intensity index is high, which limits the use of ''p-p'' intensity probes in environments with high levels of background noise or reflections.
On the other hand, the bias error introduced by a ''p-u'' probe can be approximated by<ref name=":0" />
<math display="block">\hat{I}^{p-u}_n = \frac{1}{2} \
where <math>\hat{I}^{p-u}_n</math> is the biased estimate obtained using a ''p-u'' probe, <math>P</math> and <math>V_n</math> are the Fourier transform of sound pressure and particle velocity, <math>J_n </math>is the reactive intensity and <math>\varphi_{\text{ue}} </math>is the ''p-u'' phase mismatch introduced by calibration errors. Therefore, the phase calibration is critical when measurements are carried out under near field conditions, but not so relevant if the measurements are performed out in the far field.<ref name=":0" />
▲<math>\hat{I}^{p-u}_n=\frac{1}{2} \text{Re}\{{P\hat{V}^*_n}\}=\frac{1}{2} \text{Re}\{{P V^*_n \text{e}^{-\text{j}\varphi_{\text{ue}}} }\} \simeq I_n + \varphi_{\text{ue}} J_n \, ,</math>
▲where <math>\hat{I}^{p-u}_n</math> is the biased estimate obtained using a ''p-u'' probe, <math>P</math> and <math>V_n</math> are the Fourier transform of sound pressure and particle velocity, <math>J_n </math>is the reactive intensity and <math>\varphi_{\text{ue}} </math>is the ''p-u'' phase mismatch introduced by calibration errors. Therefore, the phase calibration is critical when measurements are carried out under near field conditions, but not so relevant if the measurements are performed out in the far field<ref name=":0" />. The “reactivity” (the ratio of the reactive to the active intensity) indicates whether this source of error is of concern or not. Compared to pressure-based probes, ''p-u'' intensity probes are unaffected by the pressure-to-intensity index, enabling the estimation of propagating acoustic energy in unfavorable testing environments provided that the distance to the sound source is sufficient.
==References==
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==External links==
*[http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave]
*[http://www.sengpielaudio.com/TableOfSoundPressureLevels.htm Table of Sound Levels. Corresponding Sound Intensity and Sound Pressure]
*[http://www.acoustical-consultants.com/noise-vibration-acoustical-related-resources/sound-intensity-noise-measurements/ What Is Sound Intensity Measurement and Analysis?]
{{Authority control}}
[[Category:Acoustics]]
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