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==Measurement==
One method of sound intensity measurement involves the use of two microphones located close to each other, normal to the direction of sound energy flow. A signal analyser is used to compute the crosspower between the measured pressures and the sound intensity is derived from (proportional to) the imaginary part of the crosspower.
[[File:Sound intensity pp and pu probes.png|thumb|332x332px|Comparison of a p-p and p-u sound intensity probes.]]
Sound pressure and acoustic particle velocity can be directly acquired using a sound intensity ''p-u'' probe comprising a microphone and a particle velocity sensor (also known as a Microflown), or estimated indirectly by using a ''p-p'' probe to approximate acoustic particle velocity from the gradient between two microphones.
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|last=Jacobsen|first=Finn|last2=de Bree|first2=Hans-Elias|date=2005-09-01|title=A comparison of two different sound intensity measurement principles|url=https://asa.scitation.org/doi/10.1121/1.1984860|journal=The Journal of the Acoustical Society of America|volume=118|issue=3|pages=1510–1517|doi=10.1121/1.1984860|issn=0001-4966}}</ref>
<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, ''k'' is the wave number, <math>\Delta r</math> is the microphone separation distance, <math>\rho</math>is the density of air, and ''c'' is the speed of sound. This expression shows that the effect of a given phase error is inversely proportional to the frequency and the microphone separation distance and is proportional to the ratio of the mean square sound pressure to the sound intensity<ref>{{Cite book|url=http://worldcat.org/oclc/857650768|title=Fundamentals of general linear acoustics|last=Jacobsen, Finn, author.|isbn=9781118346419|oclc=857650768}}</ref>. If this ratio is large then even the small phase errors mentioned earlier will give rise to significant bias errors. In practice, they cannot be utilized 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 sound intensity is simply the time average of the instantaneous product of the pressure and particle velocity signals,<ref>{{Cite book|url=http://worldcat.org/oclc/1008875245|title=SOUND INTENSITY.|last=FAHY, FRANK.|date=2017|publisher=CRC Press|isbn=1138474193|oclc=1008875245}}</ref>
<math>I_n=<p\,u_n>_t=\frac{1}{2} \text{Re}\{{pu^*_n}\}</math>
where <math><>_t</math>indicates time averaging, and the latter expression is based on the complex representation of harmonic variables. The bias error introduced by a ''p-u'' probe can be approximated by<ref name=":0" />
<math>\hat{I}^{p-u}_n=\frac{1}{2} \text{Re}\{{p\hat{u}^*_n}\}=\frac{1}{2} \text{Re}\{{pu^*_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>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 at all critical if the measurements are carried 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. However, it should be noted that ''p-u'' intensity probes are unaffected by the pressure-to-intensity index, enabling the estimation of propagating acoustic energy despite unfavorable testing conditions.
==References==
==External links==
{{External links|date=JulyDecember 20192012}}
*[http://www.sengpielaudio.com/calculator-levelchange.htm How Many Decibels Is Twice as Loud? Sound Level Change and the Respective Factor of Sound Pressure or Sound Intensity]
*[http://ccrma.stanford.edu/~jos/pasp/Acoustic_Intensity.html Acoustic Intensity]
*[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?]
*[https://www.microflown.com/products/standard-probes/ Sound intensity p-u probes]
*[https://www.microflown.com/products/sound-localization-systems/scan-paint-3d/ 3D sound intensity visualization]
[[Category:Acoustics]]
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