Digital Sound & Music: Concepts, Applications, & Science, Chapter 4, last updated 6/25/2013
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4.3.4 The Mathematics of the Inverse Square Law and
PAG Equations
The inverse square law says, in essence, that for two points at distance r0 and r1 from a point
sound source, where , the sound intensity diminishes by ( ) dB. To derive the
inverse square law mathematically, we can use the formula for the surface area of a sphere,
, where is the radius of the sphere. Notice that in Figure 4.18, the radius of the sphere is
also the distance from the sound source to the surface of that sphere. Recall that intensity is
defined as power per unit area that is, power proportional to the area over which it is spread.
As the sound gets farther from the source, it spreads out over a larger area. At any distance r
from the source, where I is intensity and P is the power at the source. Notice that if
you increase the radius of the sphere by a factor of , gets smaller by a factor of . Thus, is
proportional to the inverse of , which can be stated mathematically as . We can state
this more completely as
( *
where is the intensity of the sound at the first location,
is the intensity of the sound at the second location,
is the initial distance from the sound,
and is the new distance from the sound.
Equation 4.16 Ratio of sound intensity comparing one location to another
We usually represent intensities in decibels, so let’s convert to decibels applying the definition of
dBSIL.
( ( * )
( *
Thus
( ) dB
where is the intensity of the sound at the first location in decibels,
is the intensity of the sound at the second location in decibels,
is the initial distance from the sound,
and is the new distance from the sound
Equation 4.17
Recall that when you subtract dBSIL from dBSIL, you get dB.
Based on the inverse square law, it is easy to prove if you double the distance from the
sound, you get about a 6 dB decrease (as listed in Table 4.5).
( ) ( *
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