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 exposure  E · SEL
 Sound energy density  w
 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 SI unit of sound intensity is W·m-2. 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.

Sound intensity is the product of sound pressure and particle velocity:

\vec I = p \vec v.

Both \vec I and \vec 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. For instantaneous sound pressure p and particle velocity \vec v, the average sound intensity during time T is given by:

I = \frac{1}{T} \int_0^T p(t) 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}(\vec r) = \omega^2 z_\mathrm{m}(\vec r) \xi_\mathrm{m}(\vec r)^2 = \frac{z_\mathrm{m}(\vec r) a_\mathrm{m}(\vec 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_{ac}}{A} = \frac{P_{ac}}{4 \pi r^2} \,

Here, Pac (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[edit]

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 "LI" and is measured in "dB". The decibel is a dimensionless quantity.[2]

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). 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.[3] 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.[4]

References[edit]

  1. ^ "Sound Intensity - HyperPhysics."
  2. ^ "ISO9614-1 Acoustics - Determination of sound power levels of noise sources using sound intensity"
  3. ^ Sound Power Measurements, Hewlett Packard Application Note 1230, 1992.
  4. ^ Sound Intensity (Theory)

External links[edit]