Question

Three sound waves of equal amplitudes have frequencies $\left( {v - 1} \right),{\text{ }}v,{\text{ }}\left( {v + 1} \right)$ . They are supposed to give beats. The number of beats produced per second will be:
A) 2
B) 1
C) 4
D) 3

Answer 1

Hint The beats due to the superposition of sound waves are caused by the difference in frequencies of two different waves. We need to count the number of possible beats considering every possible combination of 2 waves from all 3 waves. The waves with the maximum frequency difference will create the most number of beats per second.

Formula used: In this solution we will be using the following formula,
$n = {f_1} - {f_2}$ where $n$ is the number of beats caused by the superposition of waves of frequency ${f_1}$ and ${f_2}$ .

Complete step by step answer
Beats are produced due to the superposition of sound waves with varying frequencies. When two sound waves with different frequencies superimpose, they interfere with each other and create regions of maximum and minimum amplitude which are called beats.
We’ve been given three sound waves of equal amplitudes have frequencies $\left( {v - 1} \right),{\text{ }}v,{\text{ }}\left( {v + 1} \right)$ . Then, the possible beats can be caused by a combination of any two waves from three given waves. This can be calculated using the formula $n = {f_1} - {f_2}$
So, the combination of waves with frequency $v - 1$ and $v$ will generate
$n = v - (v - 1) \\\ = 1$
Similarly, the beats formed by the difference of two frequencies $v$ and $\left( {v + 1} \right)$ will be
$n = (v + 1) - v \\\ = 1$
The number of beats caused by frequencies $v - 1$ and $v + 1$ will be
$n = (v + 1) - (v - 1) \\\ = 2$
Since all the 3 frequencies are consecutive, the maximum number of beats per second will be formed by the difference in frequencies of the waves with the highest and the lowest frequencies.
Hence the maximum number of beats generated per second will be 2 which correspond to option (A).

Note
Beats per second is the difference in frequencies of two waves. Since the frequencies in question are consecutive, the difference of frequencies of the wave with the highest and the lowest frequency will decide the number of beats that will be generated. So, in our case, the beats generated by waves with frequency $v - 1$ and $v + 1$ will decide the number of beats generated.