Question

If we study the vibration of a pipe open at both ends, then the following statement is not true
A. Open end with be the anti-node
B. Odd harmonics of the fundamental frequency will be generated
C. All harmonics of the fundamental frequency will be generated.
D. Pressure change will be maximum at both ends

Answer 1

Hint: In these cases, the air inside the tube or pipe vibrates and the air particles vibrate parallel to the walls of the pipe and it results in longitudinal standing waves of different harmonics are generated inside the pipe. These waves reflect from the walls of the, in case of close ended pipes. These ends create nodes and antinodes in the standing waves. The open end of the pipe is at atmospheric pressure, since it is in direct contact with the atmosphere.

Complete step by step answer:
There are many instruments’ that use the principle of vibrating air columns’ inside a pipe or tube. The flute, trombone etc being some of them. In these cases, the air inside the tube or pipe vibrates and the air particles vibrate parallel to the walls of the pipe and it results in longitudinal standing waves of different harmonics are generated inside the pipe. These waves reflect from the walls of the, in case of close ended pipes.
In the question the pipe mentioned is open ended. The waves formed will be as shown in the diagram.

As can be clearly seen from the diagram the open ends are anti nodes. So option A is true.
The harmonics formed will have the form
\[f=\dfrac{nv}{\lambda }\], where n is the number of harmonics.
n can take values of all positive integers. So, all harmonics of the fundamental frequency will be generated. Option B and C are true.
Since the pipes are open ended the ends will have constant pressure which will be equal to the atmospheric pressure. So, option D is not true.

Therefore Option D is our answer.

Note: The waves formed in pipes are results of the change in pressure due to the displacement of air particles. This displacement of air particles and pressure are opposite to each other, this is due to the fact that air particles move where pressure is steady and pressure fluctuates rapidly where particles are not allowed to move freely. These displacement and pressure are important for solving problems as they give indication towards the nature of standing waves formed.