Question

An open pipe of length ‘I’ vibrates in a fundamental mode. The pressure variation is the maximum at:
(A) $\dfrac{1}{4}$ from ends
(B) middle of the pipe
(C) ends of the pipe
(D) at $\dfrac{1}{8}$ from the end of the pipe

Answer 1

Hint:It is always important to remember that when an open pipe vibrates in fundamental mode, the ends of the pipe are antinodes and the middle part of the pipe is node. Pressure variation can then be found accordingly.

Step by step answer:
The pressure variation diagram for an open pipe in fundamental mode is as follows:

Through this diagram it is clear that the pressure variation is maximum at the antinodes and minimum at the nodes. Thus, pressure variation of an open pipe in fundamental mode is the maximum at the middle of the pipe. Option (B) is the correct answer.

Additional Information:In an open pipe, when a compressed wave reaches the far end, the air at that point is, for an instant, at a pressure greater than the atmospheric pressure. In an open end pipe, the air inside can vibrate with maximum freedom and so it can suddenly expand into the surrounding air. Thus, the pressure diminishes quickly and thus, the pressure inside becomes less than that of the air in the surroundings. This causes a sudden refraction at the ends of the pipe. A rarefied wave is then set up which passes back along the pipe.

Note:The oscillation of the air column with lowest frequency mode is known as fundamental mode. Fundamental mode is also known as the first harmonic. An open pipe is in resonance in its fundamental mode. In the resonance condition, the wavelength and amplitude of the vibrating air column cannot change and remains constant. However, the frequency changes when the gases inside are changed as the velocity of sound in different media are different.