__Beats__: Beats are produced when two harmonic waves with equal amplitude but slightly different frequencies interfere. This occurs, for example, when two tuning forks with slightly different frequencies are sounded together.

Due to superposition of two such waves at all points where they interfere, sometimes we get maximum displacement and sometimes minimum. As a result intensity alternately increases and decreases periodically. This periodic, ups and downs in intensity called, Beats.

The time between two consecutive minimas or between two consecutive maximas is called T_{beat }(Period of beat) and $n= \frac{1}{T_{beat}}$ is the corresponding Beat frequency.

Let the two waves of frequencies n_{1} and n_{2} meet at some point. The corresponding periods are T_{1} & T_{2} with T_{1} < T_{2} .

If the two waves start out in phase at time t = 0, they will again be in phase when the first wave has gone through exactly one more cycle. This will happen at a value of t equal to T_{beat }.

Let X be the number of cycles of the first wave in time T_{beat} , then the number of cycles of the second wave in the same time is (X-1) and we have the relation.

T_{beat} = X T_{1} = (X-1)T_{2}

Eliminating X between these two equations we find.

$ \displaystyle T_{beat} = \frac{T_1 T_2}{T_2 – T_1 } $

The reciprocal of the beat period is the beat frequency so-

$ \displaystyle n_{beat} = \frac{1}{T_{beat}} = \frac{T_2 – T_1}{T_1 T_2}$

$ \displaystyle n_{beat} = (\frac{1}{T_1} -\frac{1}{T_2}) $

$ \displaystyle n_{beat} = (n_1 – n_2) $

=> T_{1} < T_{2} ; n_{1} > n_{2}

Here we see that after time t = T_{beat} both the waves are in same phase.

The displacement due to both the waves were +A (A = amplitude) or –A [i.e resultants of two is +2A or – 2A or maxima] they will again be in this position after T_{beat} and if the displacement of the two is +A and –A (i.e. resultant is zero), they will again be in the same position, after time t = T_{beat} and thus we get a series of maximas and minimas at the same place with frequency n1~ n2.

As the persistence of ear is about (1/10) s beats will be heard by the ear only if beat frequency < 10 Hz.