# Sound intensity

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Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power Pac
Sound power level SWL
Sound energy
Sound exposure E
Sound exposure level SEL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound
Audio frequency AF

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Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power Pac
Sound power level SWL
Sound energy
Sound exposure E
Sound exposure level SEL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound
Audio frequency AF

Sound intensity or acoustic intensity (I) is defined as the sound power Pac per unit area A. The usual context is the noise measurement of sound 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 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

$\vec{I} = p \vec{v}$

Notice that both v and I are 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 pinst(t) and particle velocity v(t) the average acoustic intensity during time T is given by

$I = \frac{1}{T} \int_{0}^{T}p_\mathrm{inst}(t) v(t)\,dt$

The SI units of intensity are W/m2 (watts per square metre). For a plane progressive wave we have:

$I = \frac{p^2}{Z} = Z v^2 = \xi^2 \omega^2 Z = \frac{a^2 Z}{\omega^2} = E c = \frac{P_{ac}}{A}$

where:

SymbolUnitsMeaning
ppascalsRMS sound pressure
fhertzfrequency
ξm, metresparticle displacement
cm/sspeed of sound
vm/sparticle velocity
ω = 2πfradians/sangular frequency
ρkg/m3density of air
Z = c ρN·s/m³characteristic acoustic impedance
am/s²particle acceleration
IW/m²sound intensity
EW·s/m³sound energy density
PacW, wattssound power or acoustic power
Am²area

## 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:

$I_r = \frac{P_{ac}}{A} = \frac{P_{ac}}{4 \pi r^2} \,$

Here, Pac (upper case) is the sound power and A the surface area of a sphere of radius r. Thus the sound intensity decreases with 1/r2 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.

$I \propto {p^2} \propto \dfrac{1}{r^2} \,$
$\dfrac{I_2}{I_1} = \dfrac{{r_1}^2}{{r_2}^2} \,$
$I_2 = I_{1} \dfrac{{r_1}^2}{{r_2}^2} \,$

$I_1\,$ = sound intensity at close distance $r_1\,$
$I_2\,$ = sound intensity at far distance $r_2\,$

Hence

$p \propto \dfrac{1}{r} \,$

where p (lower case) is the RMS sound pressure (acoustic pressure).

## Sound intensity level

Sound intensity level or acoustic intensity level is a logarithmic measure of the sound intensity (measured in W/m2), in comparison to a reference level.

The measure of a ratio of two sound intensities is

$L_\mathrm{I}=10\, \log_{10}\left(\frac{I_1}{I_0}\right)\ \mathrm{dB} \,$

where I1 and I0 are the intensities.

The sound intensity level is given the letter "LI" and is measured in "dB". The decibel is a dimensionless quantity.

If I0 is the standard reference sound intensity

$I_0 = \;10^{-12} \, \mathrm{W/{m}^{2}} \,$

(W = watt), then instead of "dB SPL" we use "dB SIL". (SIL = sound intensity level).