Sound intensity

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Sound measurements
Characteristic
Symbol
 Sound pressure  p · SPL
 Particle velocity  v · SVL
 Particle displacement  ξ
 Sound intensity  I · SIL
 Sound power  P · SWL
 Sound energy  W
 Sound energy density  w
 Sound exposure  E · SEL
 Sound energy flux  Q
 Acoustic impedance  Z
 Speed of sound  c
 Audio frequency  AF

Sound intensity or acoustic intensity is defined as the sound power per unit area. 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.

Mathematical definition[edit]

Sound intensity, denoted I and measured in W·m−2, is given by:

\mathbf I = p \mathbf v

where:

Both I and v are vectors, which means that both have a direction as well as a magnitude. The direction of sound intensity is the average direction in which energy is flowing.

The average sound intensity during time T is given by:

\langle \mathbf I\rangle = \frac{1}{T}\int_0^T p(t) \mathbf v(t)\,\mathrm{d}t.

Equations in terms of other measurements[edit]

Sound intensity can be related to other sound measurements:

I = \frac{p^2}{\mathfrak{R}(z)} = \mathfrak{R}(z) v^2 = \frac{P}{A} = c w.

For sine waves with angular frequency ω, the amplitude of the sound intensity can be related to those of the particle displacement and the particle acceleration:

I_\mathrm{m}(\mathbf r) = \omega^2 z_\mathrm{m}(\mathbf r) \xi_\mathrm{m}(\mathbf r)^2 = \frac{z_\mathrm{m}(\mathbf r) a_\mathrm{m}(\mathbf r)^2}{\omega^2}.
Symbol Unit Meaning
c m·s−1 speed of sound
v m·s−1 particle velocity
z Pa·m−1·s specific acoustic impedance
A m2 area
p Pa sound pressure
P W sound power
I W·m−2 sound intensity
w J·m−3 sound energy density
ω rad·s−1 angular frequency
ξ m particle displacement
a m·s−2 particle acceleration

Spatial expansion[edit]

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}{A} = \frac{P}{4 \pi r^2}

Here, P 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}

where:

  • I1 is the sound intensity at close distance r1;
  • I2 is the sound intensity at far distance r2.

Hence

p \propto \dfrac{1}{r}

where p is the RMS sound pressure.

Sound intensity level[edit]

For other uses, see Sound level.

Sound intensity level (SIL) or sound velocity level is a logarithmic measure of sound intensity in comparison to a reference level.
Sound intensity level, denoted LI and measured in dB, is given by:

L_I =10\, \log_{10}\left(\frac{I}{I_0}\right)~\mathrm{dB}

where:

  • I is the sound intensity, measured in W·m−2;
  • I0 is the reference sound intensity, measured in W·m−2.

If I0 is the standard reference sound intensity

I_0 = 10^{-12}~\mathrm{W}\cdot\mathrm{m}^{-2},

then instead of dB SPL, dB SIL (sound intensity level) are used. 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.[2] 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[edit]

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.[3]

References[edit]

  1. ^ "Sound Intensity - HyperPhysics."
  2. ^ Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.
  3. ^ Sound Intensity (Theory)

External links[edit]