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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 / p0)


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

I1 = 10-6W/m2.


The sound intensity level from the person's voice is

LI (1) = 10 log ( I1 / I0) = 60 dB


(prove this for yourself, remembering that I0 = 10-12 W/m2 !). 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: Itotal = 2 I1. The total sound intensity level is

LI (total) = 10 log ( Itotal / I0 ) = 10 log ( 2 I1 / I0 ).

Now, we make use of a property of logarithms, log(A x B) = log A + log B to write

LI (total) = 10 log (2) + 10 log ( I1 / I0 ) = 3 dB + LI (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!

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Last updated: 27 Oct 1999
Comments: bland@indiana.edu