Question

Two tones of frequencies \[{n_1}\] and \[{n_2}\] are sounded together. The beats can be heard distinctly when
A. \[10 < ({n_1} - {n_2}) < 20\]
B. \[5 < ({n_1} - {n_2}) > 20\]
C. \[5 < ({n_1} - {n_2}) < 20\]
D. \[0 < ({n_1} - {n_2}) < 10\]

Answer 1

Hint: As we know that beat is produced when two waves of nearby frequencies superimpose when both travel in the same path. The beat frequency is defined as the number of beats produced per second.

Formula used
The formula for beat frequency is given as,
\[n = {n_1} - {n_2}\]
Where \[{n_1}\] and \[{n_2}\] is the frequency of two waves.

Complete step by step solution:
Given two tones of frequencies sounded together are \[{n_1}\]and \[{n_2}\]. As we know that the efficiency of human ears to listen to a sound must be at least one beat per 0.1 second otherwise if it is less than 0.1 second then we are not able to listen to that sound. So to listen to the 1 beat, the time must be 0.1 sec.
i.e., 1 beat = 0.1 sec
Also, we can write 10 beats = 1 sec. It shows in 1 sec there are a total 10 beats. This is the maximum efficiency.

If time decreases from 0.1 sec (like 0.01 sec), the efficiency of the human ear will also decrease. As the number of beats per second is given by the difference of frequencies. If \[{n_1}\]and \[{n_2}\] are two tones of frequencies sounded together. So (\[{n_1} - {n_2}\]) produces the beat and has to be less than 10 or greater than 0. Therefore, the beats can be heard distinctly when \[0 < ({n_1} - {n_2}) < 10\].

Hence option D is the correct answer.

Note: The formula for beat frequency is the difference in frequency of the two superimposed waves. Due to this there is a cause in a periodic variation of intensity of the resultant wave. The phenomenon of the beats can take place in both longitudinal waves as well as transverse waves.