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Beats
The term beats is used to describe an effect due to the interference of two waves (or oscillations) of very nearly (but not quite) equal frequencies.
Musicians often use this phenomenon to tune instruments.
If two strings of, for example, a guitar, which are very close in pitch (frequency) are sounded together, the resulting sound has periodic variations in volume (loudness).
These variation in volume, beats, are not heard if the two strings vibrate at exactly the same frequency.
Thus, by changing the tension in one string until beats are not heard, you can tune the instrument.

The diagrams below represent graphs of displacement against time for waves of slightly different frequencies, f1 and f2 (f2 > f1)

At t = 0, the two oscillations are in phase with each other.
At t = tA, they are in anti-phase and at t = tP they are again in phase.
So, at time t = 0 and t = tp the sum of the amplitudes of the two oscillations will produce a large amplitude oscillation.
At t = tA a low amplitude oscillation will result (going to zero if the two amplitudes are equal, as shown here)
The next diagram represents the sum of the two waves.

Beats can occur in all kinds of waves but if the graphs represent sound waves, then we would hear a loud sound at t = 0 and t = tP but a quiet sound near t = tA.

This phenomenon can be observed in any situation where we have two periodic variations of different frequencies.
For example, two masses on two different springs, as shown in this animation.

Relation between f1  f2 and F the Beat Frequency
Recall that the relation between frequency and time period is

T1 is the time period of wave 1, T2 is the time period of wave 2 and T is the time period of the beats.

If there are N time periods of wave 1 between t = 0 and t = tP, then there will be (N+1) time periods of wave 2.
Therefore, we can write
 and

Eliminating N from these two equations gives

so

which means that

Thus the beat frequency is simply equal to the difference between the two frequencies.
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