Define beats. Give the analytical treatment of beats.

When two waves of nearly equal frequencies, travelling in same direction superimpose, the intensity of sound at a point changes with time periodically. The periodic variation in intensity of sound is called beats.

One loud sound followed by one faint sound or one faint sound followed by one loud sound constitute a beat.
Time lapsed between two successive loud sounds or two faint sounds is called beat-period and reciprocal of beat period is called beat frequency.

If y is the resultant displacement due to superimposition, then

   
    
   
where A = 2r cos  (v₁ – v₂)t is the amplitude of resultant wave. As intensity is directly proportional to the square of amplitude, therefore, intensity is maximum when A is maximum.
i.e.       
or                
or                              
i.e. 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.