What is beat? Obtain the equation of no. of beats produced in unit time.

Explanation

Beat is an interesting phenomenon produced by interference of two harmonic waves of close frequency values.
The phenomenon of wavering of sound intensity when two waves of nearly same frequencies and amplitudes travelling in the same direction are superimposed on each other is called beats.
Beat is produced due to increase in intensity of sound once and decrease once.
No. of beats in one second (unit time) is called frequency. Frequency of beat is difference of two rearer frequencies.
Mathematical method of beat : Equation of transverse harmonic wave is,
` y (x,t) = a sin [kx - omega t + phi]`
Harmonic waves are required for beat hence by taking s in place of `y (x,t)`
and by taking `x =0 and phi = (pi)/(2) ,` then `s = a sin (k xx 0 - omega t + (pi)/(2))`
`therefore s = a cos omega t [ because sin "" (pi)/(2) - theta = cos theta ]`
Displacement of two harmonic waves of equal displacement are similar angular frequencies
`omega _(1) , omega _(2) (omega _(1) gt omega _(2)) ` are ,
`s _(1) = a cos omega _(1) t " "...(1)`
`s _(2) = a cos omega _(2) t " "...(2)`
From the principle of superposition, displacement of resultant wave,
`s = s _(1) +s _(2)`
`therefore s = a cos omega _(1) + a cos omega _(2) t `
`=a [ 2 cos ((omega _(1)- omega _(2))/( 2 )) t (cos "" (omega _(1) + omega _(2))/(2))t]`
`[ because 2 cos C cos D = cos "" (C -D)/( 2 ) cos "" (C+D)/( 2)] = 2 a cos omega _(b) t cos omega _(a) t" "...(3)`
where `omega _(a) = (omega _(1) + omega _(2))/(2) and omega _(b) = (omega _(1) - omega _(2))/(2)`
`omega _(b)` is angular frequency of beat and `omega _(a)` is angular frequency of amplitude.
If we take `|omega _(1) - omega _(2) | lt lt omega _(1) or omega _(2) and omega _(1) gt gt omega _(b),` thne it can be considered that resultant wave is oscillating with average angular frequency `omega _(a).`
Amplitude of resultant wave `cos omega _(b) t ` varies with time.
For maximum amplitude `|cos omega _(b) t| =1.` Hence, intensity of resultant wave increases and decreases as per `2 omega _(b) = omega _(1) -omega _(2),`
But `omega = 2 pi v,`
Frequency of beat can be as below,
`2pi v _("beat") = 2pi v _(1) - 2pi v _(2)`
`therefore v _("beat") = v _(1) -v _(2)`
Thus, amplitude of beat becomes maximum and zero for `v _(1) - v _(2)` times.
For clear feel of beat of sound `v _(1) - v _(2)` shoudl not be greater then, 6 to 7. Becaue if increase and decrease is more than 6 to 7 times, then that sound will be heard but beat will not be felt.