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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 ω 0 + ν and ω 0 - ν (for example, ω 0 = 2 π × 10 3 r a d / sec and ν = 2 π rad / sec ). The average frequency is ω 0 , and the difference frequency is 2 ν . What you hear is the sum of the two tones:

x ( t ) = A 1 cos [ ( ω 0 + ν ) t + φ 1 ] + A 2 cos [ ( ω 0 - ν ) t + φ 2 ] .

The first tone has amplitude A 1 and phase φ 1 ; the second has amplitude A 2 and phase φ 2 . We will assume that the two amplitudes are equal to A . Furthermore, whatever the phases, we may write them as

φ 1 = φ + ψ and φ 2 = φ - ψ φ = 1 2 ( φ 1 + φ 2 ) and ψ = 1 2 ( φ 1 - φ 2 ) .
Figure one is a set of three sinusoidal functions graphed along a horizontal axis. Two are identical in wavelength and amplitude, but inverted so that their peaks and troughs look like they are enclosing the third function, and they meet four times at the horizontal axis when they progress in the opposite directions. The third sinusoidal function is bound by these two larger first functions, changing in amplitude based on the boundaries of those functions, but retaining a short, constant wavelength throughout the graph. Figure one is a set of three sinusoidal functions graphed along a horizontal axis. Two are identical in wavelength and amplitude, but inverted so that their peaks and troughs look like they are enclosing the third function, and they meet four times at the horizontal axis when they progress in the opposite directions. The third sinusoidal function is bound by these two larger first functions, changing in amplitude based on the boundaries of those functions, but retaining a short, constant wavelength throughout the graph.
Beating between Tones

Recall our trick for representing x ( t ) as a complex phasor:

x ( t ) = A Re e j [ ( ω 0 + ν ) t + φ + ψ ] + e j [ ( ω 0 - ν ) t + φ - ψ ] = A Re e j ( ω 0 t + φ ) [ e j ( ν t + ψ ) + e - j ( ν t + ψ ) ] = 2 A Re e j ( ω 0 t + φ ) cos ( ν t + ψ ) = 2 A cos ( ω 0 t + φ ) cos ( ν t + ψ ) .

This is an amplitude modulated wave, wherein a low frequency signal with beat frequency ν rad/sec modulates a high frequency signal with carrier frequency ω 0 rad/sec. Over short periods of time, the modulating term cos ( ν t + ψ ) remains essentially constant while the carrier term cos ( ω 0 t + φ ) turns out many cycles of its tone. For example, if t runs from 0 to 2 π 10 ν (about 0.1 seconds in our example), then the modulating wave turns out just 1/10 cycle while the carrier turns out 1 ¯ 0 ν ¯ ω Δ cycles (about 100 in our example). Every time ν t changes by 2 π radians, then the modulating term goes from a maximum (a wax) through a minimum (a wane) and back to a maximum. This cycle takes

ν t = 2 π t = 2 π ν seconds ,

which is 1 second in our example. In this 1 second the carrier turns out 1000 cycles.

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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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