Digital Sound & Music: Concepts, Applications, & Science, Chapter 2, last updated 6/25/2013
Figure 2.15 Adding waves
We‟re able to hear multiple sounds simultaneously in our environment because sound
waves can be added. Another interesting consequence of the addition of sound waves results
from the fact that waves have phases. Consider two sound waves that have exactly the same
frequency and amplitude, but the second wave arrives exactly one half cycle after the first that
out-of-phase, as shown in Figure 2.16. This could happen because the second sound
wave is coming from a more distant loudspeaker than the first. The different arrival times result
in phase-cancellations as the two waves are summed when they reach the listener's ear. In this
case, the amplitudes are exactly opposite each other, so they sum to 0.
Figure 2.16 Combining waves that are 180 out-of-phase
2.2.3 Sound Analysis
We showed in the previous section how we can add frequency components to create a complex
sound wave. The reverse of the sound synthesis process is sound analysis, which is the
determination of the frequency components in a complex sound wave. In the 1800s, Joseph
Fourier developed the mathematics that forms the basis of frequency analysis. He proved that
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