Question
Define beats. Give the analytical treatment of beats.
Solution
When two waves of nearly equal frequencies, travelling in the same direction superimpose, the intensity of sound at a point changes with time periodically.
Beats are the periodic variation in the intensity of sound.
The time elapsed between two successive loud sounds or two faint sounds are called beat period.
Reciprocal of the beat period is called beat frequency.
Analytical treatment of beats:
Let two waves be represented by,
...(1)
...(2)
Beats are the periodic variation in the intensity of sound.
The time elapsed between two successive loud sounds or two faint sounds are called beat period.
Reciprocal of the beat period is called beat frequency.
Analytical treatment of beats:
Let two waves be represented by,
...(1) ...(2)
If y is the resultant displacement due to superimposition, then 
where,
A = 2r cos 
(v1 – v2)t is the amplitude of the resultant wave.
As intensity is directly proportional to the square of amplitude, therefore, the intensity is maximum when amplitude is maximum.
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That is at times,
intensity will be maximum.
Therefore, the interval between two successive loud sounds is,
Therefore, Beat frequency, b = v1 – v2
i.e. Beat frequency is equal to the difference of frequencies of two interfering harmonic waves.
