The Open Door Web Site  
Waves
The diagrams below
represent graphs of displacement against time for waves of
slightly different frequencies, f_{1} and f_{2}.
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. 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. 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, .....................NT_{1} = T .......and ............(N+1)T_{2} = T Eliminating N from these two equations and remembering that frequency = 1/Time period, we find that the beat frequency, f, is given by

