Sound intensity

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Sound measurements
Sound pressure p
Sound pressure level (SPL)
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
Acoustic impedance Z
Speed of sound c

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^[1]. The SI units are W/m² (watts per square metre).

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

For a spherical sound source, the intensity as a function of distance r is:

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

Here P_ac is the sound power and A the surface of the meant sphere with the radius r. It applies thus:

$I \propto \frac{1}{r^2}$ (6)

$\frac{I_1}{I_2} = \frac{{r_2}^2}{{r_1}^2}$ (7)

$I_1 = I_{2} \cdot {r_{2}^2} \cdot \frac{1}{{r_1}^2}$ (8)

Thus the sound intensity decreases as a sound energy size in the free field (direct field) with 1/r² the distance from an acoustic point source, while the sound pressure decreases as a sound field size only with 1/r from the distance from an acoustic point source after the 1/r-distance law.

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

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

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

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

where:

Symbol	Units	Meaning
p	pascals	sound pressure
f	hertz	frequency
ξ	m, meters	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	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

The amplitude of sound intensity (not sound pressure!) decreases in the free field (direct field) with 1/r² of the distance of a point source.

Sound intensity level, L_I, is the 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.