Question

When a standing wave is set up in a pipe which is open from one end, which of the following statements is true?

Options

A
The open end of the pipe is a displacement antinode.
B
Wavelength = length string / number of nodes
C
The shape of the string at any instant shows a symmetry about the midpoint of the string
D
Frequency = number of nodes x fundamental frequency

Explanation

For a standing wave set up in a pipe that is open at one end, the correct statement is:

A. Sum of the number of antinodes and the number of nodes is always odd.

In a pipe open at one end, standing waves can form due to the reflection of waves at the closed end and the formation of antinodes and nodes along the length of the pipe. An antinode is a point of maximum displacement, while a node is a point of zero displacement. When a standing wave is set up in the pipe, there will always be one node at the closed end (due to fixed boundary conditions) and varying numbers of nodes and antinodes along the length of the pipe, depending on the mode of vibration. In this case, the sum of the number of antinodes and the number of nodes will always be an odd number. This is because there is always one more node than antinode in a standing wave in a pipe open at one end.

When a standing wave is set up in a pipe that is open from one end, the following statement is true:

The open end of the pipe is an antinode (a point of maximum displacement) of the standing wave, and the closed end of the pipe is a node (a point of zero displacement) of the standing wave, but more specifically, the closed end is a displacement node and a pressure antinode, while the open end is a displacement antinode and a pressure node.

However, a more simplified answer would be:

The open end of the pipe is a displacement antinode.

This is because the open end of the pipe allows the air molecules to move freely, creating a point of maximum displacement (antinode) of the standing wave. The closed end, on the other hand, is a point of zero displacement (node) because the air molecules are unable to move due to the closed end.

Answer: The open end of the pipe is a displacement antinode.

B. Wavelength = length string / number of nodes: This statement is not applicable to a pipe open at one end. The wavelength in such a pipe is determined by the fundamental frequency and the length of the pipe.

C. The shape of the string at any instant shows symmetry about the midpoint of the string: This statement refers to a standing wave on a string, not a pipe open at one end.

D. Frequency = number of nodes x fundamental frequency: The frequency of a standing wave in a pipe open at one end is determined by the speed of sound, the length of the pipe, and the mode of vibration. It is not directly related to the number of nodes.