Question

A string fixed at both ends has a length of 1 m. With a frequency of 44 kHz (fourth overtone), standing waves are produced. Which harmonics will be audible to a human? (The frequency range for human hearing is 20 Hz to 20 kHz)

Answer 1

The frequency of the nth-harmonic of a standing wave is an integer multiple of the fundamental frequency of the string, according to:

`f_n = nf_1` (1)
where
`f_1` is the fundamental frequency.

We know the frequency of the 4th overtone (44 kHz), which corresponds to the 5th harmonic (n=5). Therefore, we can find the fundamental frequency of the string by rearranging equation (1):

`f_1 = (f_5)/(5)= (44 kHz)/(5)=8.8 kHz`

And by using the same formula we can find the following harmonics:

`f_2 = 2 f_1 = 2 \cdot 8.8 kHz =17.6 kHz`

`f_3 = 3f_1 = 3 \cdot 8.8 kHz =26.4 kHz`

`f_4 = 4 f_1 = 4 \cdot 8.8 kHz =35.2 kHz`

And since the audible range for human goes from 20 Hz to 20 kHz, we see that onlyt the first two harmonics (8.8 kHz and 17.6 kHz) will be audible to a human.