# Sound intensity

Sound measurements
Characteristic
Symbols
Sound pressure  p, SPL
Particle velocity  v, SVL
Particle displacement  ξ
Sound intensity  I, SIL
Sound power  P, SWL
Sound energy  W
Sound energy density  w
Sound exposure  E, SEL
Acoustic impedance  Z
Speed of sound  c
Audio frequency  AF
Transmission loss  TL

Sound intensity or acoustic intensity is defined as the sound power per unit area. The SI unit of sound intensity is the watt per square metre (W/m2). The usual context is the noise measurement of sound intensity in the air at a listener's location as a sound energy quantity.[1]

Sound intensity is not the same physical quantity as sound pressure. Hearing is directly 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.

## Mathematical definition

Sound intensity, denoted I, is defined by

$\mathbf I = p \mathbf v$

where

Both I and v are 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

$\langle \mathbf I\rangle = \frac{1}{T}\int_0^T p(t) \mathbf v(t)\,\mathrm{d}t.$

## Relationships with other quantities

Sound intensity in the direction of the sound force is related to sound power:

$I = \frac{P}{A},$

where

• P is the sound power;
• A is the area.

Sound intensity in the direction of the sound force is related sound energy density:

$I = cw,$

where

## Inverse-square law

Further information: 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

$I(r) = \frac{P}{A(r)} = \frac{P}{4 \pi r^2},$

where

• P is the sound power;
• A(r) is the area of a sphere of radius r.

Thus sound intensity decreases as 1/r2 from the centre of the sphere:

$I(r) \propto \frac{1}{r^2}.$

This relationship is an inverse-square law.

## Sound intensity level

For other uses, see Sound level.

Sound intensity level (SIL) or acoustic intensity level is a logarithmic measure of the sound intensity of a sound relative to a reference value.
Sound intensity level, denoted LI and measured in dB, is defined by[2]

$L_I = \frac{1}{2} \ln\!\left(\frac{I}{I_0}\right)\!~\mathrm{Np} = \log_{10}\!\left(\frac{I}{I_0}\right)\!~\mathrm{B} = 10 \log_{10}\!\left(\frac{I}{I_0}\right)\!~\mathrm{dB},$

where

• I is the sound intensity;
• I0 is the reference sound intensity;
• 1 Np = 1 is the neper;
• 1 B = (1/2) ln(10) is the bel;
• 1 dB = (1/20) ln(10) is the decibel.

The commonly used reference sound intensity in air is[3]

$I_0 = 1~\mathrm{pW/m^2}.$

The proper notations for sound intensity level using this reference are LI /(1 pW/m2) or LI (re 1 pW/m2), but the notations dB SIL, dB(SIL), dBSIL, or dBSIL are very common, even if they are not accepted by the SI.[4]

The reference sound intensity I0 is defined such that a progressive plane wave has the same value of sound intensity level (SIL) and sound power level (SPL), since

$I \propto p^2.$

The equality of SIL and SPL requires that

$\frac{I}{I_0} = \frac{p^2}{p_0^2},$

where p0 = 20 μPa is the reference sound pressure.

For a progressive spherical wave,

$\frac{p}{v} = z_0,$

where z0 is the characteristic specific acoustic impedance. Thus,

$I_0 = \frac{p_0^2 I}{p^2} = \frac{p_0^2 pv}{p^2} = \frac{p_0^2}{z_0}.$

In air at ambient temperature, z0 = 410 Pa·s/m, hence the reference value I0 = 1 pW/m2.[5]

In an anechoic chamber, which approximates a free field (no reflection), the SIL can be taken as being equal to the SPL. This fact is exploited to measure sound power in anechoic conditions.

## 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.[6]

## References

1. ^ "Sound Intensity". Retrieved 22 April 2015.
2. ^ "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.
3. ^ Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.
4. ^ 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), SP811 PDF
5. ^ Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.
6. ^ Sound Intensity (Theory)