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Perhaps you have heard two slightly mistuned musical instruments play pure tones whose frequencies are close but not equal. If so, you have senseda beating phenomenon wherein a pure tone seems to wax and wane. This waxing and waning tone is, in fact, a tone whose frequency is the average ofthe two mismatched frequencies, amplitude modulated by a tone whose “beat” frequency is half the difference between the two mismatched frequencies. Theeffect is illustrated in [link] . Let's see if we can derive a mathematical model for the beating of tones.
We begin with two pure tones whose frequencies are and (for example, and ). The average frequency is ω 0 , and the difference frequency is . What you hear is the sum of the two tones:
The first tone has amplitude and phase ; the second has amplitude and phase . We will assume that the two amplitudes are equal to . Furthermore, whatever the phases, we may write them as
Recall our trick for representing as a complex phasor:
This is an amplitude modulated wave, wherein a low frequency signal with beat frequency rad/sec modulates a high frequency signal with carrier frequency rad/sec. Over short periods of time, the modulating term remains essentially constant while the carrier term turns out many cycles of its tone. For example, if runs from 0 to (about 0.1 seconds in our example), then the modulating wave turns out just 1/10 cycle while the carrier turns out cycles (about 100 in our example). Every time changes by radians, then the modulating term goes from a maximum (a wax) through a minimum (a wane) and back to a maximum. This cycle takes
which is 1 second in our example. In this 1 second the carrier turns out 1000 cycles.
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