Page 62 - Master Handbook of Acoustics

Building on the preceding discussion (and referring again to Fig. 3-1), we observe that sound from a
2
point source travels outward spherically. We also note that the area of a sphere is 4πr . Therefore, the
area of any small segment on the surface of the sphere also varies as the square of the radius. This

means that the sound intensity (sound power per unit area) decreases as the square of the radius. This
is an inverse square law. The intensity of a point-source sound in a free field is inversely
proportional to the square of the distance from the source. In other words, intensity is proportional to
2
1/r . More specifically:

where I =
intensity of sound per unit area
W =

power of source
r =
distance from source (radius)

In this equation, since W and 4π are constants, we see that doubling the distance from r to 2r reduces
the intensity I to I/4; this is because at twice the distance, the sound passes through an area that is four
times the previous area. Likewise, tripling the distance reduces the intensity to I/9, and quadrupling
the distance reduces intensity to I/16. Similarly, halving the distance from 2r to r increases the

intensity to 4I.

Sound Pressure in the Free Field

The intensity of sound (power per unit area) is a difficult parameter to measure. However, sound

pressure is easily measured, for example, by using ordinary microphones. When using sound
pressure, the free-field equation must be modified. Because sound intensity is proportional to the
square of sound pressure, the inverse square law (for sound intensity) becomes the inverse distance
law (for sound pressure). In other words, sound pressure is inversely proportional to distance r. In
particular:

where P =
sound pressure
k =

a constant
r =
distance from source (radius)

For every doubling of distance r from the sound source, sound pressure will be halved (not
quartered). In Fig. 3-2, the sound-pressure level in decibels is plotted against distance. This