Step 1: Understanding the Concept:
When an air column vibrates in a pipe closed at one end, stationary waves are formed.
A node is always formed at the closed end because the air particles cannot move longitudinally.
An antinode is always formed at the open end because the air particles have maximum freedom to move.
The different modes of vibration are called harmonics or overtones.
Step 2: Key Formula or Approach:
For a pipe closed at one end of length \( L \):
- Fundamental mode (1st harmonic or 0th overtone): \( L = \frac{\lambda}{4} \)
- 1st overtone (3rd harmonic): \( L = \frac{3\lambda}{4} \)
- 2nd overtone (5th harmonic): \( L = \frac{5\lambda}{4} \)
A node-to-antinode distance is \( \frac{\lambda}{4} \).
Step 3: Detailed Explanation:
The problem states the air column is vibrating in its second overtone.
For the second overtone in a closed pipe, the length of the pipe \( L \) fits \( 5 \) quarter-wavelengths:
\[ L = \frac{5\lambda}{4} = \frac{\lambda}{2} + \frac{\lambda}{2} + \frac{\lambda}{4} \]
We can trace the wave pattern starting from the closed end (node):
- Closed end: Node (N)
- At distance \( \lambda/4 \): Antinode (A)
- At distance \( \lambda/2 \): Node (N)
- At distance \( 3\lambda/4 \): Antinode (A)
- At distance \( \lambda \): Node (N)
- At distance \( 5\lambda/4 \) (open end): Antinode (A)
The pattern along the length of the pipe is: Node - Antinode - Node - Antinode - Node - Antinode.
Counting the components in this pattern:
There are 3 Nodes.
There are 3 Antinodes.
In general, for a closed pipe in its \( p \)-th overtone, the number of nodes is \( p + 1 \) and the number of antinodes is also \( p + 1 \).
Here, for the 2nd overtone (\( p=2 \)), there are \( 2+1 = 3 \) nodes and \( 2+1 = 3 \) antinodes.
Step 4: Final Answer:
The column has three nodes and three antinodes.