End correction in pipes: Rayleigh found out that when the air in an organ pipe vibrates, the reflection of the sound waves takes place a little above the rim of the organ pipe because the air is free to vibrate at the open end. Due to this reason, an antinode is formed above the open end of the pipe. The vibrating length of the air column is greater than the actual length of the organ pipe. The distance between the actual position of the antinode and the open end of the pipe is called end correction. Therefore, in order to get the accurate value for the frequency of vibration, the end correction is necessary.

The end correction for open pipe is given as λ = 2(l + 2c) and for closed pipe, it is given as λ = 4(l + c), where c is the end correction l is the length of pipe and λ is the wavelength of the wave, also the end correction of pipe is given as 0.3d where d is the diameter of the pipe.

Wave in open pipe: Open pipe is one that is open at both ends. When air is blown into the pipe through one end, a wave travels through the tube to the next end from where it is reflected. Due to the superposition of the incident and reflected waves, a stationary wave is set up in the air in the pipe. Since the two ends are open, they must be the positions of antinodes as shown in the figure.

Fundamental mode (First harmonic): Let the length of the pipe is l and the velocity of sound in air is v. In this mode of vibration, there are antinodes at open ends and one node at the middle of the pipe as shown in Fig (i). Let λ₁ be the wavelength of the wave. Then, 

\(l=\frac{\lambda_1}2\)

Or, λ₁ = 2l

Thus, the frequency of the fundamental mode of the first harmonic is \(f_1=\frac v{\lambda_1}=\frac v{2l}\) ------- (i)

The second mode (First overtone): In this mode of vibration, two antinodes are at the open ends but inside the pipe, there are two nodes and one antinode as shown in Fig. (ii). If λ₂ be the wavelength of the wave, then.

l = λ₂ 

Thus, the frequency of the first overtone or second harmonic is

\(f_2=\frac v{\lambda_2}=\frac vl=2\frac v{2l}=2f_1\)

∴ f₂ = 2 f₁ ----------- (ii)

The third mode (Second overtone): In this mode of vibration, two antinodes are produced at both open ends and inside the pipe; there are three nodes and two antinodes as shown in Fig (iii). If λ₃ be the wavelength of the wave then,

\(l=\frac{3\lambda_3}2\)

Or, \(\lambda_3=\frac{2l}3\)

Thus, the frequency of the second overtone or third harmonic is 

\(f_3=\frac v{\lambda_3}=\frac v{2l/3}=\frac{3v}{2l}=3\frac v{2l}\)

∴ f₃ = 3 f₁ 

In this way, other higher modes of vibration can be obtained and the frequency of nth mode of vibration is f_n =n f₁, where n = 1, 2, 3, ..... is an integer. From this, it is found that frequencies of higher modes of vibration are integral multiple of fundamental frequency f₁. So, all harmonics are possible in an open pipe as f₁, 2f₁, 3f₁, 4f₁,.... i.e., both odd and even harmonics are present.