Question

If a note x of unknown frequency produces 8 beats/sec with a source of 250Hz and 12beats/sec with a source of 270Hz. The frequency of the source X is
\[\begin{align}
  & \text{A)258Hz} \\
 & \text{B)242Hz} \\
 & \text{C)262Hz} \\
 & \text{D)282Hz} \\
\end{align}\]

Answer 1

Hint: Here, the sound waves from x overlap with two other sources of different frequencies, and produce beats. Beat frequency of a sound wave is the absolute value of difference of the two frequencies. Here, beat frequency or no of beats per second is given for two conditions. We need to find the frequency of x, which is superimposed with other frequencies.
Formula used:
\[\text{Unknown frequency = frequency }\!\!\pm\!\!\text{ number of beats}\]

Complete step by step solution:
Beats are produced, when two sound waves with different frequencies come across each other and their amplitude gets either added or subtracted for a given time period.
In the first case,
A source x of unknown frequency produces \[8\text{ beats/sec}\]with a source of \[\text{250Hz}\]
Then,
Number of beats per second \[=8\]
We have,
\[\text{Unknown frequency = frequency }\!\!\pm\!\!\text{ number of beats}\]
Then,
\[\text{Frequency of x= 250 }\!\!\pm\!\!\text{ 8 = 258Hz or 242Hz}\]
In the second case,
Source x of unknown frequency produces \[\text{12 beats/sec}\]with a source of \[\text{270Hz}\]
Then,
Number of beats per second \[=12\]
Therefore,
\[\text{Frequency of x= 270 }\!\!\pm\!\!\text{ 12 = 282Hz or 258Hz}\]
Thus, frequency of source x is \[\text{258Hz}\]

Hence, answer is option A.

Additional information:
When two waves of nearly equal frequencies travelling along the same direction in a medium superimpose upon each other, beats are produced. The number of beats produced per second is known as beat frequency. Beat frequency is equal to the difference in frequencies of two waves. The amplitude of the resultant sound regularly rises and falls at a point. When the intensity rises to its maximum it is known as waxing of sound, when it falls to minimum, it is called as waning of sound.

Note:
If two frequencies \[{{f}_{1}}\] and \[{{f}_{2}}\] are superimposed, then their beat frequency is given by,
\[{{f}_{beat}}=\left| {{f}_{1}}-{{f}_{2}} \right|\]
Since, a negative frequency would not make any sense, the beat frequency is
considered as the absolute value of the difference between the two frequencies.
Regular frequency of a wave resulting from superimposition and the beat frequency are two different frequencies. Beat frequency describes the frequency of beats, while the actual frequency of a wave resulting from superimposition, is the average of the frequencies of two waves.