How will you determine the velocity of sound using resonance air column apparatus ?

Explanation

The resonance air column apparatus is one of the simplest techniques to measure the speed of a sound in air at room temperature. It consists of a cylindrical glass tube of one metre length whose one end A is open and another end B is connected to the water reservior R through a rubber tube as shown in Figure. This cylindrical glass tube is mounted on a vertical stand with a scale attached to it. The tube is partially filled with water and the water level can be adjusted by raising or lowering the water in the reservior R. The surface of the water will act as a closed end and other as the open end. Therefore, it behaves like a closed organ pipe, forming nodes at the surface of water and antinodes at the closed end. When a vibrating tuning fork is brought near the open end of the tube, longitudinal waves are formed inside the air column. These waves move downward as shown in Figure, and reach the surfaces of water and get reflected and produce standing waves. The length of the air column is varied by changing the water level until a loud sound is produced in the air column. At this particular length the frequency of waves in the air column resonance with the frequency of the tuning fork (natural frequency of the tuning fork). At resonance, the frequency of sound waves produced is equal to the frequency of the tuning fork. This will occur only when the length of air column is proportional to `(1/4)th` of the wavelength of the sound waves produced.
Let the first resonance occur at length `L_(1)`, then
`1/4lambda= L_(1)`
`1/4 lambda = L_(1) +e`.......(1)
Now the length of the air column is increased to get the second resonance. Let `L_(2)` be the length at which the second resonance occurs. Again taking end correction into account,
`3/4 lambda =L_(2) +e`..........(2)
In order to avoid end correction, let us take the difference of equation `1/4 lambda =L_(1)` and equation
`f_(1):f_(2) :f_(3):f_(4)......=1:2:3:4......`m, we get
`3/4lambda -1/4lambda =(L_(2)+e) -(L_(1)+e)`
`rArr 1/2 lambda =L_(2)-L_(1) =triangleL`
`rArr lambda = 2Delta L`
The speed of the sound in air at room temperature can be computed by using the formula.
`v=f lambda = 2f Delta L`
Further, to compute the end corrections, we use equation (1) and (2), we get `e(L_(2)-3L_(1))/2`