A periodic variation in loudness (intensity) due to superposition of two sound notes of slightly different frequencies sounded at the same time is called beats.
The time interval between successive maxima or minima of sound at a given place is called the period of beats.
The number of beats produced per unit time is called the beat frequency.
Consider two sound waves of equal amplitude (A) and slightly different frequencies `n_(1)` and `n_(2)` (with `n_(1) gt n_(2)`) propagating through the same part of the medium in the same direction. These waves can be represented by the equations `y_(1)=A sin 2pi n_(1)t` and `y_(2)=A sin 2pin_(2)t` at x=0, where y denotes the displacement of the particle of the medium from its mean position.
By the principle of superposition of waves, the resultant displacement of the particle of the medium at the point at which the two waves arrive simultaneously is the algebraic sum
`y=y_(1)+y_(2)=A sin 2pi n_(1)t+A sin 2pi n_(2)t`
Now, `sin C+sin D=2 sin ((C+D)/(2))cos((C-D)/(2))`
`:. y=2A sin [2pi((n_(1)+n_(2))/(2))t]. cos[2pi((n_(1)-n_(2))/(2))t]`
`=2A cos[2pi((n_(1)-n_(2))/(2))t].sin[2pi((n_(1)-n_(2))/(2))t]`
Let `R=2A cos[2pi((n_(1)-n_(2))/(2))t]` and `n=(n_(1)+n_(2))/(2)`
`:. y=R sin 2pi nt`
The above equation shows that the resultant motion has amplitude |R| which changes periodically with time. Period of beats is the period of waxing (maximum intensity of sound) or the period of waning (minimum intensity of sound).
The intensity of sound is directly proportional to the square of the amplitude of the wave. It is maximum when |R| becomes maximum `(=2A) :R=-+2A`.
`:. cos[2pi((n_(1)-n_(2))/(2))t]=-+1`
`:. 2pi((n_(1)-n_(2))/(2))t=0, pi,2pi,3pi`,....
`:. t=0, (1)/(n_(1)-n_(2)),(2)/(n_(1)-n_(2)),(3)/(n_(1)-n_(2))`,...
`:.` Period of beats =period of waxing `=(1)/(n_(1)-n_(2))`
`:.` Beat frequency `=(1)/("period of beats")=n_(1)-n_(2)`
[Note : The intensity of sound is minimum (waning) when R=0.
`:. cos [2pi((n_(1)-n_(2))/(2))t]=0`
`:. 2pi((n_(1)-n_(2))/(2))t=(pi)/(2),(3pi)/(2),(5pi)/(2)`,.......
`:. t=(1)/(2(n_(1)-n_(2))),(3)/(2(n_(1)-n_(2))),(5)/(2(n_(1)-n_(2)))`,.....,
`:.` Period of beats=period of waning`=(1)/(n_(1)-n_(2))`
`:.` Beat frequency `=(1)/("period of beats")=n_(1)-n_(2)`
Thus, waxing and waning occur alternately and the period of waning equals that of waxing.]
