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A source of unknown frequency gives 4 beats / s, when sounded with a source of known frequency 250 Hz. The second harmonic of the source of unknown frequency gives five beats per second, when sounded with a source of frequency 513 Hz. The unknown frequency is:

A source of unknown frequency gives 4 beats / s, when sounded with a source of known frequency 250 Hz. The second harmonic of the source of unknown frequency gives five beats per second, when sounded with a source of frequency 513 Hz. The unknown frequency is: Correct Answer 254 Hz

CONCEPT:

  • The beat of a Sound: Beats are the periodic and repeating fluctuations heard in the intensity of a sound when two sound waves of very similar frequencies interfere with one another.
  • Beat Frequency: The beat frequency is always equal to the difference in frequency of the two notes that interfere to produce the beats. The beat frequency for the first harmonic is given as

b = | f - f' |

  • The beat Frequency for the second Harmonic is given as 

b = | 2f - f' |

CALCULATION:

Let beat frequency be f

Then for the first Harmonic, the beat frequency = 4

frequency of source = f' = 250Hz

4 = | f - 250 |

If we consider f > 250

f - 250 = 4

f = 254 --- (1)

Let beat frequency be f

Then for the second Harmonic, the beat frequency = 5

Source Frequency = 513 Hz

5 = |2 f - 513|

2f = 518

f = 518 / 2 = 259 Hz --- (2)

This is different from the result first

Now, if we consider 513 > 2f

513 - 2f = 5

2f = 508

f = 508 /2 = 254 Hz --- (3)

So, we can see condition (1) and Condition (3) Matched. So, the required frequency is 254 Hz.

Important Points

  • The beat frequency is given as | f - f' |. We need to check both conditions for f > f' and f < f' for both harmonics.
  • We will be finding a common frequency from both harmonics. 

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