4.4 Resonance tube

1.1 Principle of superposition

2.1 Relation between phase difference $\left( \phi \right)$ and path difference $\left( {\Delta x} \right)$
2.2 Interference of waves from coherent sources
2.3 Interference of waves from incoherent sources
2.4 Reflection and transmission of a wave
2.5 Motion of wave during reflection
2.6 Expression for the reflection and transmission of wave

3.1 Transverse stationary wave on a stretched string
3.2 Vibrations in a stretched string
3.3 Melde's Experient
3.4 Resonance

4.1 Open organ pipe
4.2 Closed organ pipe
4.3 End correction
4.4 Resonance tube
4.5 Energy in a stationary wave

Resonance tube consists of a long vertical glass tube $T$. A meter scale $S$ (graduated in $mm$) is fixed adjacent to this tube.

The zero of the scale coincides with the upper end of the tube. The lower end of the tube $T$ is connected to a reservoir $R$ of water through a U shaped pipe $P$.

The water level in the tube can be adjusted by adjustable screws attached with the reservoir.

The vertical adjustment of the tube can be made with the help of leveling screws. For fine adjustment of water level in the tube, the pinchcock is used.

A tuning fork of known frequency is vibrated over the open end of the resonance tube and the water level is adjusted so that the sound coming from the tube becomes maximum. Then the length of the air column is read on the scale attached.

The minimum length of the air column for which the resonance takes place corresponds to the fundamental mode of vibration.

A displacement node is formed at the surface of water (which acts as a closed end of the air column) and a displacement antinode formed near the open end. The displacement antinode is formed slight above the open end because of the air pressure from outside.

So, for the first resonance, the length $L_1$ of the air column in the resonance tube is given by,$${L_1} + e = \frac{\lambda }{4}\quad ...(i)\quad $$ where $e$ is the end correction

Now, the length of the air column is increased to little less than 3 times of $L$. The water level is again adjusted, so that the loudness of the sound coming from the tube becomes maximum again. The length of the air column is noted on the scale. In this second resonance, the air column vibrates in the second harmonic or first overtone. $${L_2} + e = \frac{{3\lambda }}{4}\quad ...(ii)\quad $$ Subtracting equation $(ii)$ by $(i)$ we get, $$\begin{equation} \begin{aligned} {L_2} - {L_1} = \frac{\lambda }{2} \\ \lambda = 2\left( {{L_2} - {L_1}} \right)\quad ...(iii) \\\end{aligned} \end{equation} $$ Also, from equation $(i)$ & $(iii)$ we get, $$e = \left( {\frac{{{L_2} - 3{L_1}}}{2}} \right)$$