Question

A cylindrical tube, open at both ends has a fundamental frequency $f$ in air. If the tube is dipped vertically in water so that half of it is dipped, then then the fundamental frequency of the air column is now?
A) f
B) 2f
C) f/2
D) 3f/4

Answer 1

Hint: These musical instruments either open at one end, open at both the ends uses the concept of standing waves to produce sounds by means of vibrations in air columns. When air is pushed inside them, air starts vibrating with some frequency which is the characteristics of the source. If resonance takes place then the amplitude of the vibrations increases.

Complete step by step solution:
Initially for an open pipe, fundamental frequency is given as $$v_0={\displaystyle \frac{v}{4l}}$$, where $v$ is the speed of the sound in the medium and $l$ is the length of the organ pipe. We know that the speed of the sound changes with the change in the medium. On dipping in water, the tube becomes closed at that end and that too of half the length, so,
$$\begin{array}{rl}& v^{\prime }={\displaystyle \frac{v}{4l}}\\ & \Rightarrow {\displaystyle \frac{v}{4{\displaystyle \frac{l}{2}}}}\\ & \Rightarrow {\displaystyle \frac{v}{2l}}\\ & \Rightarrow f\end{array}$$

So, the correct answer is “Option A”.

Note:
The length of the column determines the frequencies. Also, shorter the length of the column, higher will be the pitch of the sound obtained. A cylindrical pipe closed at both ends is rarely used deliberately in music acoustics. The frequency is measured in units of per second or hertz. The length is to be taken in metres. The speed of the sound to be taken in m/s.