Consider a cylindrical pipe of length l open at both the ends. When sound waves are sent down the air column in a cylindrical open pipe, they are reflected at the open ends without a change of phase. Interference between the incident and reflected waves under appropriate conditions sets up stationary waves in the air column.
The stationary waves in the air column in this case are subject to the two boundary conditions that there must be an antinode at each open end.
Taking into account the end correction e at each of the open ends, the resonating length of the air column is L = l + 2e.
Let v be the speed of sound in air. In the simplest mode of vibration, the fundamental mode or first harmonic, there is a node midway between the two antinodes at the open ends. The distance between two consecutive antinodes is λ/2, where λ is the wavelength of sound. The corresponding wavelength λ and the fundamental frequency n are
In the next higher mode, the first overtone, there are two nodes and three antinodes. The corresponding wavelength λ1 and frequency n1
i.e., twice the fundamental. Therefore, the first overtone is the second harmonic.
In the second overtone, there are three nodes and four antinodes. The corresponding wavelength \(\lambda\)2 and frequency n2 are
or thrice the fundamental. Therefore, the second overtone is the third harmonic.
Therefore, in general, the frequency of the pth overtone (p = 1, 2, 3, …) is
np = (p + 1)n … (4)
i.e., the pth overtone is the (p + 1)th harmonic. Equations (1), (2) and (3) show that allowed frequencies in an air column in a pipe open at both ends are n, 2n, 3n, …. That is, all the harmonics are present as overtones.
