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{{Sound measurements}}
'''Sound intensity''' or '''acoustic intensity''' ('''''I''''') is defined as the [[sound power]] '''''P<sub>ac</sub>''''' per unit area ''A''. The usual context is the noise measurement of sound [[intensity (physics)|intensity]] in the air at a listener's location as a sound energy quantity.
 
Sound intensity is not the same physical quantity as [[sound pressure]]. Hearing is directly sensitive to sound pressure which is [http://www.sengpielaudio.com/calculator-soundlevel.htm related to sound intensity]. In consumer audio electronics, the level differences are called "intensity" differences, but sound intensity is a specifically defined quantity and cannot be sensed by a simple microphone.
 
== Acoustic intensity ==
The intensity is the product of the sound pressure and the particle velocity
:<math>
\vec{I} = p \vec{v}
</math>
 
Notice that both <math>\vec{v}</math> and <math>\vec{I}</math> are [[Vector (geometric)|vectors]], which means that both have a ''direction'' as well as a magnitude. The direction of the intensity is the average direction in which the energy is flowing. For instantaneous [[acoustic pressure]] ''p''<sub>inst</sub>(''t'') and [[particle velocity]] <math>\vec{v}</math>(''t'') the average acoustic intensity during time ''T'' is given by
:<math>
I = \frac{1}{T} \int_{0}^{T}p_\mathrm{inst}(t) v(t)\,dt
</math>
 
The [[SI]] unit of intensity is  W/m<sup>2</sup> ([[watt]] per [[square metre]]). For a plane progressive wave we have:
:<math>
I = \frac{p^2}{Z} = Z v^2 = \xi^2 \omega^2 Z = \frac{a^2 Z}{\omega^2} = E c = \frac{P_{ac}}{A}
</math>
 
where:
 
{| class="wikitable"
! Symbol !! Unit !! Meaning
|-
! ''p''
| [[pascal (unit)|pascal]] || [[Root mean square|RMS]] [[sound pressure]]
|-
! ''f''
| [[hertz]] || [[frequency]]
|-
! ''ξ''
| m, [[metre]] || [[particle displacement]]
|-
! ''c''
| [[metre|m]]/[[second|s]] || [[speed of sound]]
|-
! ''v''
| [[Meter per second|m/s]] || [[particle velocity]]
|-
! ω = 2π''f''
| [[radian]]s/[[second|s]] || [[angular frequency]]
|-
! ''ρ''
| [[kilogram|kg]]/[[Metre|m]]<sup>3</sup> || [[density of air]]
|-
! ''Z = c ρ''
| [[Newton (unit)|N]]·[[second|s]]/[[Metre|m]]³ || characteristic [[acoustic impedance]]
|-
! ''a''
| [[Metre|m]]/[[second|s]]² || [[particle acceleration]]
|-
! ''I''
| [[Watt|W]]/m² || sound intensity
|-
! ''E''
| [[Watt|W]]·[[second|s]]/[[Metre|m]]³ || [[sound energy density]]
|-
! ''P''<sub>ac</sub>
| W, [[watt]] || [[sound power]] or [[acoustic power]]
|-
! ''A''
| [[Metre|m]]² || [[area]]
|}
 
== Spatial expansion ==
For a spherical sound source, the intensity in the radial direction as a function of distance ''r'' from the centre of the source is:
:<math>
I_r =  \frac{P_{ac}}{A} = \frac{P_{ac}}{4 \pi r^2} \,
</math>
 
Here, ''P''<sub>ac</sub> (upper case) is the sound power and ''A'' the surface area of a sphere of radius ''r''. Thus the sound intensity decreases with 1/''r''<sup>2</sup> the distance from an acoustic point source, while the sound pressure decreases only with 1/''r'' from the distance from an acoustic point source after the 1/''r''-distance law.
:<math>
I \propto {p^2} \propto \dfrac{1}{r^2} \,
</math>
 
:<math>
\dfrac{I_2}{I_1} = \dfrac{{r_1}^2}{{r_2}^2} \,
</math>
 
:<math>
I_2 = I_{1} \dfrac{{r_1}^2}{{r_2}^2} \,
</math>
<math>I_1\,</math> = sound intensity at close distance <math>r_1\,</math><br>
<math>I_2\,</math> = sound intensity at far distance <math>r_2\,</math>
 
Hence
:<math>
p \propto \dfrac{1}{r} \,
</math>
 
where ''p'' (lower case) is the [[Root mean square|RMS]] sound pressure (acoustic pressure).
 
== Sound intensity level ==
'''Sound intensity level''' or '''acoustic intensity level''' is a [[logarithm]]ic measure of the sound intensity (measured in  W/m<sup>2</sup>), in comparison to a reference level.
 
The measure of a ratio of two sound intensities is
:<math>
L_\mathrm{I}=10\, \log_{10}\left(\frac{I_1}{I_0}\right)\ \mathrm{dB} \,
</math>
 
where ''I''<sub>1</sub> and ''I''<sub>0</sub> are the intensities.
 
The sound intensity level is given the letter "L<sub>I</sub>" and is measured in "dB". The decibel is a [[dimensionless]] quantity.
 
If I<sub>0</sub> is the standard reference sound intensity
:<math>
I_0 = \;10^{-12} \, \mathrm{W/{m}^{2}} \,
</math>
 
(W = watt), then instead of "dB SPL" we use "dB SIL".  (SIL = sound intensity level). The reference value is defined such that a plane wave propagating in a free field has the same value of SPL and SIL as the ratio of the reference pressure squared to the reference intensity is approximately equal to the characteristic impedance of air.<ref>Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.</ref> In an anechoic chamber, which approximates a free field, the SIL can be taken as being equal to the SPL. This fact is exploited to measure sound power in anechoic conditions.
 
== References ==
{{reflist}}
 
== External links ==
{{External links|date=December 2012}}
*[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/calculator-soundlevel.htm Conversion: Sound intensity level to sound intensity and vice versa]
*[http://www.sengpielaudio.com/calculator-ak-ohm.htm Ohm's law as acoustic equivalent  - calculations]
*[http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf Relationships of acoustic quantities associated with a plane progressive acoustic sound wave - pdf]
*[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?]
 
{{DEFAULTSORT:Sound Intensity}}
[[Category:Physical quantities]]
[[Category:Sound measurements]]

Latest revision as of 00:31, 20 December 2014

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