# Sound intensity

Sound measurements
Characteristic
Symbol
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.[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.

## Sound intensity

The intensity is the product of the sound pressure and the particle velocity

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

Notice that both $\vec{v}$ and $\vec{I}$ 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 $\vec{v}$(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 unit of intensity is W/m2 (watt 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:

Symbol Unit Meaning
p pascal RMS sound pressure
f hertz frequency
ξ m, metre particle displacement
c m/s speed of sound
v m/s particle velocity
ω = 2πf radians/s angular frequency
ρ kg/m3 density of air
Z = c ρ N·s/m³ characteristic acoustic impedance
a m/s² particle acceleration
I W/m² sound intensity
E W·s/m³ sound energy density
Pac W, watt sound power or acoustic power
A m² 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:

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

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. [2]

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). 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.[3] 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.

## 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. [4]

## References

1. ^
2. ^ "ISO9614-1 Acoustics - Determination of sound power levels of noise sources using sound intensity"
3. ^ Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.
4. ^ Sound Intensity (Theory)