Sound intensity

(′sau̇nd in′ten·səd·ē)

(acoustics) For a specified direction and point in space, the average rate at which sound energy is transmitted through a unit area perpendicular to the specified direction.

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A fundamental acoustic quantity which describes the rate of flow of acoustic energy through a unit of area perpendicular to the flow direction. The unit of sound intensity is watt per square meter. The intensity is calculated at a field point (x) as a product of acoustic pressure p and particle velocity u. Generally, both p and u are functions of time, and therefore an instantaneous intensity vector is defined by the equation below. $\vec{I}_i(x,t) = p(x,t) \cdot \vec{u}(x,t)$ The time-variable instantaneous intensity, $\vec{I}_i(x,t)$ , which has the same direction as $\vec{u}(x,t)$ , is a nonmoving static vector representing the instantaneous power flow through a point (x). See also Power; Sound pressure.

Many acoustic sources are stable at least over some time interval so that both the sound pressure and the particle velocity in the field of such a source can be represented in terms of their frequency spectra.

The applications of sound intensity were fully developed after a reliable technique for intensity measurement was perfected. Sound intensity measurement requires measuring both the sound pressure and the particle velocity. Very precise microphones for sound-pressure measurements are available.

An application of the intensity technique is the measurement of sound power radiated from sources. The knowledge of the radiated power makes it possible to classify, label, and compare the noise emissions from various pieces of equipment and products and to provide a reliable input into environmental design. See also Sound.

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The average rate of sound energy transmitted in a specified direction through a unit area normal to this direction at the point considered.

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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

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

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.

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