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, f_{1 }and f_{2}
(f_{2} > f_{1}) 





At t = 0, the two oscillations are in phase
with each other. 

At t = t_{A}, they are in antiphase and at t = t_{P} they are
again in phase. 

So, at time t = 0 and t = t_{p} the sum of the
amplitudes of the two oscillations will produce a large amplitude
oscillation. 

At t = t_{A} 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 = t_{P} but a quiet sound near t = t_{A}. 



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 f_{1}
f_{2} and F the Beat
Frequency 

Recall that the relation between frequency and time period is 



T_{1} is the time period of wave 1, T_{2} 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 = t_{P},
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. 
