Sound intensity

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Sound measurements
Sound pressure p
Particle velocity v
Particle velocity level (SVL)
(Sound velocity level)
Particle displacement ξ
Sound intensity I
Sound intensity level (SIL)
Sound power P_ac
Sound power level (SWL)
Sound energy density E
Sound energy flux q
Surface S
Acoustic impedance Z
Speed of sound c v • d • e

The sound intensity, I, (acoustic intensity) 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. For instantaneous acoustic pressure p_inst(t) and particle velocity v(t) the average acoustic intensity during time T is given by

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

Notice that both v(t) 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. The SI units of intensity are W/m² (watts per square metre).

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.

$I \sim {p^2} \sim \dfrac{1}{r^2} \,$

$\dfrac{I_1}{I_2} = \dfrac{{r_2}^2}{{r_1}^2} \,$

$I_1 = I_{2} \cdot {r_{2}^2} \cdot \dfrac{1}{{r_1}^2} \,$

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

Hence

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

The sound intensity I in W/m² of a plane progressive wave is:

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

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
P_ac	W, watts	sound power or acoustic power
A	m²	area

Sound intensity level, L_I, is the magnitude of sound intensity, expressed in logarithmic units (decibels).

$L_I=10 \log_{10} \frac {|I|}{I_o}$ (dB-SIL),

where I_o is the reference intensity, 10^-12 W/m²

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 stereo the level differences have been called "intensity" differences, but sound intensity is a specifically defined quantity and cannot be sensed by a simple microphone, nor would it be valuable in music recording if it could.