Three resonant frequencies of the string are 90, 150 and 210 Hz. (A) Find the highest possible fundamental frequency of vibration of this string. (B) Which harmonics of the fundamental are the given frequencies? (C) Which overtones are these frequencies? (D) If The length of the string is 80 cm, what would be the speed of a transverse wave on this string?

Question

Philoid · Accepted Answer

A. 30 Hz B. 3rd, 5th and 7th

C. 2nd, 4th and 6th

D. 48 m s⁻¹

The highest fundamental frequency is the gcd (greatest common divisor) of all other frequencies.

Since, we know that gcd of 90,150 and 210 is 30, so,

(A)

Now, f_{1, f}2, f₃ are integral multiples of f.

₁=3

₂=5

₃=7 (B)

We know that nth overtone is (n+1)th frequency.

So, 3f -> 2^nd overtone and 3^rd harmonic.

So, 5f -> 4^th overtone and 6^th harmonic.

So, 7f -> 6^th overtone and 7^th harmonic. (C)

We know that, ₁=

So,

We get, K=48 m/s (D)