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The diagrams below
represent graphs of displacement against time for waves of
slightly different frequencies, f1 and f2.
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.
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 = tP but a quiet sound near t = tA.
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 = tP, 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