Three sound waves of equal amplitudes have frequencies (v - 1), v , (v + 1) . They surperpose to give beats. The number of beats produced per second will be :

Options

A. 3
B. 2
C. 1
D. 4

Answer

C

Explanation

According to superposition principle y = `y_(1) + y_(2) + y_(3) `.
y = a sin 2`pi` (v - 1) t + a sin `2 pi v t + a sin 2pi`(v+1) t.
Now a sin `2 pi (v - 1) t + a sin 2pi (v + 1) `
a `[ 2 cos 2 pit sin pi v t] `
`y = a sin 2 pi vt + 2 a cos 2 pi t sin 2 pi v t.`
`y = a (1 + 2 cos 2 pi t ) sin 2 pi v t `
= A sin `2 pi v t`
Where A = `a(1 + 2 cos 2 pi t) ` is resultant amplitude .
Since intensity `prop ("Amplitude")^(2)`
`therefore` Intensity will be maximum if cos `2pi t ` = + 1
`rArr " " 2pi t = 0, 2pi, 4pi, 6pi,.. `etc.
`rArr " " t = 0 , `Is, 2s, 3s, .... etc.
`therefore` Time period of maximum intensity is 1 second so frequency of maximum intenstiy is 1 Hz or beat frequency is 1 Hz.
`rArr ` Choice is (c) .