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Sound intensity

From Wikipedia, the free encyclopedia
Power carried by sound waves
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

Sound intensity, also known asacoustic intensity, is defined as the power carried by sound waves per unit area in a direction perpendicular to that area, also called thesound power density and thesound energy flux density.[2] TheSI unit of intensity, which includes sound intensity, is thewatt per square meter (W/m2). One application is thenoise measurement of soundintensity in the air at a listener's location as asound energy quantity.[3]

Sound intensity is not the same physical quantity assound pressure. Human hearing is sensitive to sound pressure which is 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.

Sound intensity level is a logarithmic expression of sound intensity relative to a reference intensity.

Mathematical definition

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Sound intensity, denotedI, is defined byI=pv{\displaystyle \mathbf {I} =p\mathbf {v} }where

BothI andv arevectors, which means that both have adirection as well as a magnitude. The direction of sound intensity is the average direction in which energy is flowing.

The average sound intensity during timeT is given by⟨I⟩=1T∫0Tp(t)v(t)dt.{\displaystyle \langle \mathbf {I} \rangle ={\frac {1}{T}}\int _{0}^{T}p(t)\mathbf {v} (t)\,\mathrm {d} t.}For a plane wave[4],I=2π2ν2δ2ρc{\displaystyle \mathrm {I} =2\pi ^{2}\nu ^{2}\delta ^{2}\rho c}Where,

Inverse-square law

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Further information:Inverse-square law

For aspherical sound wave, the intensity in the radial direction as a function of distancer from the centre of the sphere is given byI(r)=PA(r)=P4πr2,{\displaystyle I(r)={\frac {P}{A(r)}}={\frac {P}{4\pi r^{2}}},}where

Thus sound intensity decreases as 1/r2 from the centre of the sphere:I(r)∝1r2.{\displaystyle I(r)\propto {\frac {1}{r^{2}}}.}

This relationship is aninverse-square law.

Sound intensity level

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For other uses, seeSound level.

Sound intensity level (SIL) oracoustic intensity level is thelevel (alogarithmic quantity) of the intensity of a sound relative to a reference value.

It is denotedLI, expressed innepers,bels, ordecibels, and defined by[5]LI=12ln⁡(II0)Np=log10⁡(II0)B=10log10⁡(II0)dB,{\displaystyle L_{I}={\frac {1}{2}}\ln \left({\frac {I}{I_{0}}}\right)\mathrm {Np} =\log _{10}\left({\frac {I}{I_{0}}}\right)\mathrm {B} =10\log _{10}\left({\frac {I}{I_{0}}}\right)\mathrm {dB} ,}where

  • I is the sound intensity;
  • I0 is thereference sound intensity;
    • 1 Np = 1 is theneper;
    • 1 B =⁠1/2⁠ ln(10) is thebel;
    • 1 dB =⁠1/20⁠ ln(10) is thedecibel.

The commonly used reference sound intensity in air is[6]I0=1 pW/m2=1 ⋅ 10−12 W/m2.{\displaystyle I_{0}=1~\mathrm {pW/m^{2}} =1~\cdot ~10^{-12}~\mathrm {W/m^{2}} .}

being approximately the lowest sound intensity hearable by an undamaged human ear under room conditions.The proper notations for sound intensity level using this reference areLI /(1 pW/m2) orLI (re 1 pW/m2), but the notationsdB SIL,dB(SIL), dBSIL, or dBSIL are very common, even if they are not accepted by the SI.[7]

The reference sound intensityI0 is defined such that a progressiveplane wave has the same value of sound intensity level (SIL) andsound pressure level (SPL), sinceI∝p2.{\displaystyle I\propto p^{2}.}

The equality of SIL and SPL requires thatII0=p2p02,{\displaystyle {\frac {I}{I_{0}}}={\frac {p^{2}}{p_{0}^{2}}},}wherep0 = 20 μPa is the reference sound pressure.

For aprogressive spherical wave,pc=z0,{\displaystyle {\frac {p}{c}}=z_{0},}wherez0 is thecharacteristic specific acoustic impedance. Thus,I0=p02Ip2=p02pcp2=p02z0.{\displaystyle I_{0}={\frac {p_{0}^{2}I}{p^{2}}}={\frac {p_{0}^{2}pc}{p^{2}}}={\frac {p_{0}^{2}}{z_{0}}}.}

In air at ambient temperature,z0 = 410 Pa·s/m, hence the reference valueI0 = 1 pW/m2.[8]

In ananechoic chamber which approximates a free field (no reflection) with a single source, measurements in thefar field in SPL can be considered to be equal to measurements in SIL. This fact is exploited to measure sound power in anechoic conditions.

Measurement

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Sound intensity is defined as the time averaged product of sound pressure and acoustic particle velocity.[9] Both quantities can be directly measured by using a sound intensityp-u probe comprising a microphone and aparticle velocity sensor, or estimated indirectly by using ap-p probe that approximates the particle velocity by integrating the pressure gradient between two closely spaced microphones.[10]

Pressure-based measurement methods are widely used in anechoic conditions for noise quantification purposes. The bias error introduced by ap-p probe can be approximated by[11]I^np−p≃In−φpeprms2kΔrρc=In(1−φpekΔrprms2/ρcIr),{\displaystyle {\widehat {I}}_{n}^{p-p}\simeq I_{n}-{\frac {\varphi _{\text{pe}}\,p_{\text{rms}}^{2}}{k\Delta r\rho c}}=I_{n}\left(1-{\frac {\varphi _{\text{pe}}}{k\Delta r}}{\frac {p_{\text{rms}}^{2}/\rho c}{I_{r}}}\right),}whereIn{\displaystyle I_{n}} is the “true” intensity (unaffected by calibration errors),I^np−p{\displaystyle {\hat {I}}_{n}^{p-p}} is the biased estimate obtained using ap-p probe,prms{\displaystyle p_{\text{rms}}} is the root-mean-squared value of the sound pressure,k{\displaystyle k} is the wave number,ρ{\displaystyle \rho } is the density of air,c{\displaystyle c} is the speed of sound andΔr{\displaystyle \Delta r} 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 ofp-p intensity probes in environments with high levels of background noise or reflections.

On the other hand, the bias error introduced by ap-u probe can be approximated by[11]I^np−u=12Re⁡{PV^n∗}=12Re⁡{PVn∗e−jφue}≃In+φueJn,{\displaystyle {\hat {I}}_{n}^{p-u}={\frac {1}{2}}\operatorname {Re} \left\{{P{\hat {V}}_{n}^{*}}\right\}={\frac {1}{2}}\operatorname {Re} \left\{{PV_{n}^{*}e^{-j\varphi _{\text{ue}}}}\right\}\simeq I_{n}+\varphi _{\text{ue}}J_{n}\,,}whereI^np−u{\displaystyle {\hat {I}}_{n}^{p-u}} is the biased estimate obtained using ap-u probe,P{\displaystyle P} andVn{\displaystyle V_{n}} are the Fourier transform of sound pressure and particle velocity,Jn{\displaystyle J_{n}} is the reactive intensity andφue{\displaystyle \varphi _{\text{ue}}} is thep-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.[11] 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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  1. ^"Sound Energy Terms and Definitions".
  2. ^IEC 801-21-38[1]
  3. ^"Sound Intensity". Retrieved22 April 2015.
  4. ^"Intensity".The Physics Hypertextbook. Retrieved21 September 2025.
  5. ^"Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units",IEC 60027-3 Ed. 3.0, International Electrotechnical Commission, 19 July 2002.
  6. ^Ross Roeser, Michael Valente,Audiology: Diagnosis (Thieme 2007), p. 240.
  7. ^Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel",Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811PDF
  8. ^Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.
  9. ^Fahy, Frank (2017).Sound Intensity. CRC Press.ISBN 978-1138474192.OCLC 1008875245.
  10. ^Jacobsen, Finn (2013-07-29).Fundamentals of general linear acoustics. Wiley.ISBN 9781118346419.OCLC 857650768.
  11. ^abcJacobsen, Finn; de Bree, Hans-Elias (2005-09-01)."A comparison of two different sound intensity measurement principles"(PDF).The Journal of the Acoustical Society of America.118 (3):1510–1517.Bibcode:2005ASAJ..118.1510J.doi:10.1121/1.1984860.ISSN 0001-4966.S2CID 56449985.

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