Chapter 13
Sound Intensity
13.1 Intensity
Let’s now look at another property of sound: itsloudness. The loudness of sound is just the intensity of the
sound waves, in watts per square meter. The sound intensityIis the powerPof the sound source (e.g. a
loudspeaker), divided by the area over which this power is spread. For example, if the source of sound waves
is an isotropic point source, then spherical sound waves are emitted equally in all directions. At a distance
rfrom the source, the source’s power will be spread over the surface of a sphere of radiusr, so the sound
intensity at distancerwill be
ID
P
A
D
P
4r^2: (13.1)
13.2 Decibels
Our ears are capable of hearing sounds over a tremendous range of intensities. It has been said that if our ears
were any more sensitive than they are, we would be able to hear the sound of individual air molecules hitting
our eardrums. But we can also hear very loud sounds, like from a jet engine. In order to accommodate this
large range of intensities, our ears tend to respondlogarithmicallyto sounds; this has motivated the creation
of alogarithmicloudness scale, where sound level is proportional to the logarithm of the intensity.
Simply taking the logarithm of the intensity doesn’t work dimensionally, though—when you take the
logarithm of a quantity, it should be dimensionless. We therefore take the logarithm of aratioof intensities
to get thesound level:
BDlog 10I
I 0
; (13.2)
whereBis the sound level in units ofbels(B) (named after Alexander Graham Bell),Iis the sound intensity,
andI 0 D 10 ^12 W/m^2 is called thethreshold of hearing, and is roughly the lowest-intensity sound that an
average person can hear. Theactualsoftest audible sound varies from person to person, changes with age,
and is also a function of frequency. But for the purpose of defining the bel, we always use 10 ^12 W/m^2 for
I 0. Also, notice that by convention, thecommon(base 10) logarithm is used in defining the bel.
In practice, the bel is rarely used; the more common unit is^1 / 10 bel, or thedecibel(dB). The sound level
in decibels (ˇ)isgivenby
ˇD 10 log 10