The diagrams below represent graphs of displacement against time for waves of slightly different frequencies, f₁ and f₂.
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At t = 0, the two oscillations are in phase with each other. At t = t_A, they are in anti-phase and at t = t_P they are again in phase.

The next diagram represents the sum of the two waves.

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.

These variations in loudness are called beats.

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 = t_P, then there will be (N+1) time periods of wave 2.

Therefore, .....................NT₁ = T .......and ............(N+1)T₂ = T

Eliminating N from these two equations and remembering that frequency = 1/Time period, we find that the beat frequency, f, is given by

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