Question

You have just replaced the first string on your guitar and wish to tune it to the correct frequency of 329.6 Hz (the note E4). To do this, you pluck both this string and the sixth (E2) string, which you know is properly tuned to 82.4 Hz. You hear one beat every 2.50 s. How far out of tune (in Hz) is the first string? (Ans: 0.4 Hz)

Answer 1

Frequencies for a standing wave on a string is given by the expression

where, n = 1,2,3........

The harmonics of the string are the integer multiples of its fundamental frequency. So for sixth string, harmonics are given as

So if the guitar is properly tuned, the fundamental frequency of the first string is the same as the fourth harmonic frequency of the sixth string. It is given that there is one beat per 2.50 s, so the beat frequency i.e. the number of beats per second is

Now, the beat frequency is also given by

So the first string is out of tune by 0.400 Hz.