Question

Three resonant frequencies of a string are 90, 150 and 210 Hz.

(i) Find the highest possible fundamental frequency of vibration of this string

(ii) Which harmonics of the fundamental are the given frequencies?

(iii) Which overtones are these frequencies?

• A. 30Hz, harmonics - 3rd, 5th, 7th, overtones - 2nd, 4th, 6th
• B. 60 Hz, harmonics - 3rd, 5th, 7th, overtones - 2nd, 4th, 6th
• C. 30 Hz, harmonics - 2nd, 4th, 6th, overtones, 3rd, 5th, 7th
• D. None of these

Answer 1

The correct option is A

30Hz, harmonics - 3rd, 5th, 7th, overtones - 2nd, 4th, 6th

Given frequencies are $f_1=90Hz,f_2=150Hz,f_3=120Hz$

The highest possible fundamental frequency is the highest common factor of the three frequency which is 30 Hz.

So the fundamental frequency is 30 Hz

$f_1=3\times 30$ Hz3rd harmonic and 2nd overtone

$f_2=5\times 30$ Hz5th harmonic and 4th overtone

$f_3=7\times 30$ Hz7th harmonic and 6th overtone

[In case of fixed string nth harmonic is (n - 1)th overtone]