The first harmonic (the fundamental mode) occurs at frequency f1 =
The second harmonic occurs at frequency f2 =2 v/2L = 2 f1.
The third harmonic occurs at frequency f3 =3 v/2L = 3 f1.
Resonance occurs when the right-going waves are in phase. Consider two
consecutive right-going waves and call them wave A and wave B. At any
position on the string, wave B travel a distance of 2L more than wave A, where
L is the length of the string. Thus the path difference between them is 2L. For the two waves to be in
phase, their path difference should be an integer multiple of
l 2L = n l n=
1, 2, 3, ...
Thus, resonance occurs at one of the following wavelengths
Using v = f l, the
The same result would be reached if the left-going waves were used.