<

## Multiple Sources of Sound

The final thing we need to discuss is how we get the total loudness of
sound when more than one source produces sound power. Before we can
do this, we need to make a subtle distinction in combining sound
waves. It is possible to arrange two different sound waves so that they
have a *fixed phase relationship*.
This condition is called *coherence*. In this case, there can be
either *constructive interference* (in other words the amplitudes
of the two waves add when the crests and troughs of the two waves
align) or *destructive interference* (when the crest of one wave
aligns with the trough of the other and vice versa). There is
interference between sound waves only when the crests and troughs of
the waves have a fixed relationship. In general, this is very hard to
arrange and requires that the source of the two interfering waves is
identical.
When we have the coherent addition of two sounds, resulting in
constructive interference, we must use the relationship:

sound level, L = 20 log(p / p_{0})

to find the amplitude of the individual
sound waves, if the level is known. Once we have the amplitude, we
ADD the amplitudes from the constructively interfering sounds, to find
the amplitude of the wave when the two sounds are played together. We
can then convert this to a level with the above equation, using the
total amplitude of the sound wave.
More generally, sounds from different sources have no
*phase relationship*, meaning that the alignment of
crests and troughs between the two waves is random: sometimes they
align and sometimes they don't. In this case, there is
*no interference* between the two sound waves. What happens when
the two incoherent sounds are played together, is that the *intensities*
of the individual sounds add together. This is actually easier to
understand than interference. If you're at a party and one stereo
system has a power output of 20 W and a second stereo of equal power
is turned on, then the total sound power is just the sum of the
powers from the two separate stereos.
Let's see how this works by looking at an example. Let's say that the
sound intensity from one person talking is

I_{1} = 10^{-6}W/m^{2}.

The sound intensity level from the person's voice is

L_{I} (1) = 10 log ( I_{1} /
I_{0}) = 60 dB

(prove this for yourself, remembering that I_{0} = 10^{-12} W/m^{2} !). If a second person talks and produces sound with the same intensity, then the total intensity you hear is just the sum of the two intensitities: I_{total} = 2 I_{1}. The total sound intensity level is
L_{I} (total) = 10 log ( I_{total} / I_{0} ) = 10 log ( 2 I_{1} / I_{0} ).

Now, we make use of a property of logarithms, log(A x B) = log A + log B to write
L_{I} (total) = 10 log (2) + 10 log ( I_{1} / I_{0} ) = 3 dB + L_{I} (1)

When two equal sound intensitities are added, the sound intensity level compared to the level from a single source is increased by 10 log (2) = 3 dB!
Return to P105 Course Schedule

Last updated: `27 Oct 1999`

Comments: bland@indiana.edu