Sound intensity

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 energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound c
Audio frequency AF

Sound intensity or acoustic intensity (I) is defined as the sound power P_ac per unit area A. The usual context is the noise measurement of sound intensity in the air at a listener's location.

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 p_inst(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:

Symbol	Units	Meaning
p	pascals	RMS sound pressure
f	hertz	frequency
ξ	m, metres	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, watts	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, P_ac (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 "L_I" 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).

Note 1^ : The term "intensity" is used exclusively for the measurement of sound in watts per unit area.
To describe the strength of sound in terms other than strict intensity, one can use "magnitude" "strength", "amplitude", or "level" instead.

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 is not valuable in music recording.

External links

• How many decibels is twice as loud? Sound level change and the respective factor of sound pressure or sound intensity
• Acoustic Intensity
• Conversion: Sound intensity level to sound intensity and vice versa
• Ohm's law as acoustic equivalent - calculations
• Relationships of acoustic quantities associated with a plane progressive acoustic sound wave - pdf
• Table of Sound Levels - Corresponding Sound Intensity and Sound Pressure