The air column in a pipe open at both ends vibrates with the frequency of third overtone. It is found that air in th pipe has

Options

A. 4 nodes and 5 antinodes
B. 3 nodes and 3 antinodes
C. 4 nodes and 3 antinodes
D. 3 nodes and 4 antinodes

Answer

To solve the problem, we need to understand the behavior of an air column in a pipe that is open at both ends, particularly when it vibrates in the third overtone.

### Step-by-Step Solution:

1. **Identify the Harmonic**:
- The third overtone corresponds to the fourth harmonic in a pipe open at both ends. This is because the overtone series starts from the fundamental frequency (1st harmonic), and the nth overtone is equal to (n+1)th harmonic.

2. **Understanding Nodes and Antinodes**:
- In a pipe open at both ends, the ends are antinodes (points of maximum amplitude) while the points in between can be nodes (points of minimum amplitude).
- For a pipe open at both ends:
- 1st harmonic (fundamental frequency): 1 node and 2 antinodes
- 2nd harmonic (1st overtone): 2 nodes and 3 antinodes
- 3rd harmonic (2nd overtone): 3 nodes and 4 antinodes
- 4th harmonic (3rd overtone): 4 nodes and 5 antinodes

3. **Drawing the Diagram**:
- To visualize the harmonics, we can sketch the standing wave pattern for the fourth harmonic:
- It will have 4 nodes and 5 antinodes.
- The nodes will be spaced evenly along the length of the pipe, and the antinodes will occur at the open ends and at intervals along the length.

4. **Counting Nodes and Antinodes**:
- From the above understanding, we can count:
- Nodes: 4
- Antinodes: 5

5. **Conclusion**:
- Therefore, in the pipe vibrating in the third overtone (fourth harmonic), there are 4 nodes and 5 antinodes.

### Final Answer:
The air column in the pipe has **4 nodes and 5 antinodes**.