Question

An open tube is in resonance with string ( frequency of vibration of tube is nₒ) . If tube is dipped in water so that 75% of the length of tube is inside water, then the ratio of frequency of tube to the string now will be,
\[\begin{align}
  & \text{A}\text{. }\dfrac{l}{2} \\
 & \text{B}\text{. 2} \\
 & \text{C}\text{. }\dfrac{2}{3} \\
 & \text{D}\text{. }\dfrac{3}{2} \\
\end{align}\]

Answer 1

Hint: Fundamental frequency for open tube concept is used to solve the above problem.

Formula used: Fundamental frequency for open tube \[{{\eta }_{0}}=\dfrac{v}{2l}\] where, v=velocity of sound and l = length of the tube.
And, frequency for the tube which is closed on one side and open on the other side = \[{{\eta }_{0}}=\dfrac{v}{4l}\]

Complete step by step solution:
 The fundamental frequency for open tube is given by,
 \[{{\eta }_{0}}=\dfrac{v}{2l}...............\left( i \right)\], when we are assuming that tube is open.
When the tube is dipped inside water, it becomes closed on one side and open on the other side, therefore, length of the tube
available as closed tube is ,

   \[{{l}_{1}}=\left( 1-\dfrac{75}{100} \right)\text{x }l\]where, l₁is the length for the part of the tube closed on one side and open on the other side.
Therefore, \[{{l}_{1}}=\left( \dfrac{25}{100} \right)\text{x }l=\dfrac{l}{4}\]
Now, we know that frequency for the tube which is closed on one side and open on the other side is given by,
\[\eta =\dfrac{v}{4{{l}_{1}}}=\dfrac{v}{4\text{x}\left( \dfrac{l}{4} \right)}=\dfrac{v}{l}................\left( ii \right)\]
Let us compare equations (i) and (ii) we get,
\[\begin{align}
  & \eta =2\text{x }{{\eta }_{0}} \\
 & \therefore \dfrac{\eta }{{{\eta }_{0}}}=2 \\
\end{align}\]

So, the answer is option B.

Additional information: We can create a standing wave in a tube, which is open at both ends, and in a tube which is open on one end and closed at the other end. The open and closed ends in a tube reflect waves differently. The closed end of the tube is an antinode in the longitudinal displacement. The open end of a tube approximately is a node in the longitudinal displacement.


Note: The longest standing wave in a tube of length L with two open ends has displacement pressure nodes at both ends which is called the fundamental harmonic and second harmonics are there for two open ends and third harmonics are for tubes which are closed on one side and open on the other side.