Mathematical Treatment of Beats

Suppose we have two waves of equal amplitudebut slightly different frequenciesand

Wave 1 has a displacement as a function of time given by

Wave 2 has a displacement as a function of time given by

Then

Use the identityto write the sum of sines as a product. We obtain

This is of the formwhere

The intensity of each wave isbut the intensity of the wave resulting from the interference of the two waves is

Whenthere is constructive interference and the amplitude of the resulting wave will be a maximum.

Whenthere is destructive interference and the amplitude of the resulting wave will be zero.

The sum of the waves is shown below as a function of time. Notice that within the wave there is a rapidly oscillating wave. The frequency of this wave will be the sum of the frequencies of the two waves.