Find the frequencies of the first three harmonics of a piano string of length 1.5m; if the velocity of the wave on the string is 120ms−¹?

A. 40Hz, 80Hz, 120Hz
B. 80Hz, 160Hz, 240Hz
C. 180Hz, 360Hz, 540Hz
D. 369Hz, 180Hz, 90Hz

Correct Answer:

Option A – 40Hz, 80Hz, 120Hz

Explanation

In vibrations in string, the fundamental frequency happens to be the first harmonic, and is given as

f₀ = V/2l.

Where V = Velocity of the sound wave, I = Length of the string.

The second harmonic is the first overtone given by

f₁ = V/l = 2f₀ ;

and the third harmonic happens to be the second overtone, given by

f₂ = 3V/2l = 2f₀.

Thus if V = 120m/s and l = 1.5m
Thus f₁ = 2f₀ = 2 x 40 = 80Hz
Thus f₂ = 3f₀ = 3 x 40 = 120Hz
Thus we have in order, 40Hz, 80Hz, 120Hz