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).

( ) ( *