Question

In a resonance tube experiment to determine the speed of sound in air, a pipe of diameter $5\,cm$ is used. The column in the pipe resonates with a tuning fork of frequency $480\,Hz$ when the minimum length of the air column is $16\,cm$. Find the speed of sound (in $m{s^{ - 1}}$) in the air column at room temperature.

Answer 1

Hint: Here we have to first convert the unit to S.I unit. Then apply the end correction to find the maximum length. Then we can find the speed of sound in the air column using the minimum and maximum length of the air column. In acoustics, to measure the exact resonant frequency of the pipe, the end adjustment is a small distance applied or attached to the total length of the resonance pipe. The radiation acoustic impedance of a circular piston is a theoretical basis for the end correction computation.

Complete step by step answer:
Given,
Frequency, $f = 480\,Hz$
Diameter, $d = 5\,cm$
Radius, $r = \dfrac{5}{2} = 2.5\,cm = 0.025\,m$

Minimum length of air column is:
${l_1} = 16\,cm = 0.16\,m$
If ${l_2}$ is the maximum length of air column, then
The end correction of the tube will be:
$e = 0.6r$
$
 e = \dfrac{{{l_2} - 3{l_1}}}
{2} \\
 \Rightarrow 0.6 \times 0.025 = \dfrac{{{l_2} - 0.16}}
{2} \\
 \Rightarrow {l_2} = 0.51\,m \\
$
Therefore,
If $v$ is the speed of sound in air column then
Speed of sound in air column,
$
 v = 2f\left( {{l_2} - {l_1}} \right) \\
 = 2 \times 480 \times \left( {0.51 - 0.16} \right) \\
 = 336\,m{s^{ - 1}} \\
$

Additional Information:
A sound wave is a continuous wave in which the wave oscillates in the direction of propagation.
The tuning fork is held by hand just above the opening end of the tube. When a rubber hammer hits the tuning fork, it vibrates and the sound waves are produced. These sound waves pass down the tube and reflect upon touching the surface of the sea. Incoming and reflecting waves interfere, creating standing waves. The sound waves transmitted from the surface of the water change their phase by ${180^ \circ }$ and are thus totally out of phase with the incident sound waves.
In other words, the amplitude of the standing waves must be zero on the surface of the sea. This space point is generally referred to as a node. If the resonance condition is satisfied, the open end of the tube shall have the highest amplitude of standing sound waves and shall be considered an antinode.

Note:
Here we have to first convert all the units to S.I units otherwise the answer would be wrong. Also, in confusion we may apply the diameter in the formula instead of the radius. For an organ pipe, the final adjustment is required since the sound wave reflection takes place only a little above the surface of the organ pipe. For this purpose, the vibrating length of the organ pipe is greater than the real length, so it is important to calculate precise end-of-value correction.