Explain the formation of stationary waves in an air column enclosed in open pipe. Derive the equations for the frequencies of the harmonics produced.

Harmonies : The frequencies which are integral multiples of fundamental frequency are Harmonics.
Overtones : The Frequencies higher than the fundamental frequency are known as overtones. An organ pipe is a cylindrical tube having an air column. If both ends of the tube are open, it is known as an open organ pipe.
i) The first harmonic or the fundamental should at least have an antinode at each end, with a node included between them. Therefore.

$\mathrm{l}=\frac{{\mathrm{\lambda }}_1}{2};{\mathrm{\lambda }}_1=2\mathrm{l}$
Fundamental frequency: ${\mathrm{n}}_1=\frac{\mathrm{v}}{{\mathrm{\lambda }}_1}$
${\mathrm{n}}_1=\frac{\mathrm{v}}{2\mathrm{l}}\dots$(1)
ii) The second harmonic (or) first overtone will have two nodes and three antinodes as shown in figure.

$\mathrm{l}=\frac{2{\mathrm{\lambda }}_2}{2}={\mathrm{\lambda }}_2; \text{ Corresponding, }{\mathrm{n}}_2=\frac{\mathrm{v}}{{\mathrm{\lambda }}_2}$
${\mathrm{n}}_2=\frac{\mathrm{v}}{\mathrm{l}}=2\left(\frac{\mathrm{v}}{2\mathrm{l}}\right);{\mathrm{n}}_2=2{\mathrm{n}}_1\dots \dots \left(2\right)$
iii) Similarly the third harmonic or second overtone will have three nodes and four antinodes are shown in the figure and if λ₃ is wave length.

$\mathrm{l}=\frac{3{\mathrm{\lambda }}_3}{2};{\mathrm{\lambda }}_3=\frac{2\mathrm{l}}{3}$
Frequency of second overtone
$\begin{array}{l}{\mathrm{\lambda }}_3=\frac{2\mathrm{l}}{3};{\mathrm{n}}_3=\frac{\mathrm{v}}{\frac{2\mathrm{l}}{3}}{\mathrm{n}}_3=\frac{3\mathrm{v}}{2\mathrm{l}}\\ {\mathrm{n}}_3=3{\mathrm{n}}_1\dots \dots \left(3\right)\end{array}$
 

Therefore from eq (1), (2), (3) ,

Frequency of r^th harmonic can be calculated as $n_r=r\left(\frac{v}{2l}\right)$

 The fundamental frequency and the next harmonics are in ratio 1 : 2 : 3 : ……..