A closed organ pipe is excited to vibrate in the third overtone. If is obertone that there are

Options

A. three nodes and three antinodes
B. three nodes and four antinodes
C. four nodes and three antinodes
D. four nodes and four antinodes

Explanation

To solve the problem of determining how many nodes and antinodes are present in a closed organ pipe vibrating in the third overtone, follow these steps:

### Step-by-Step Solution:

1. **Understanding Closed Organ Pipe Frequencies**:
- A closed organ pipe produces sound waves with specific frequencies. The fundamental frequency (first harmonic) and the overtones are given by the formulas:
- Fundamental frequency (1st harmonic): \( f_1 = \frac{V}{4L} \)
- 1st overtone (2nd harmonic): \( f_2 = \frac{3V}{4L} \)
- 2nd overtone (3rd harmonic): \( f_3 = \frac{5V}{4L} \)
- 3rd overtone (4th harmonic): \( f_4 = \frac{7V}{4L} \)

2. **Identifying the Third Overtone**:
- The third overtone corresponds to the 4th harmonic, which is represented by the frequency \( f_4 = \frac{7V}{4L} \).

3. **Understanding Nodes and Antinodes**:
- In a closed organ pipe, there is one closed end (node) and one open end (antinode). The pattern of nodes and antinodes can be visualized as follows:
- For the fundamental frequency (1st harmonic), there is 1 node and 1 antinode.
- For the first overtone (2nd harmonic), there are 2 nodes and 1 antinode.
- For the second overtone (3rd harmonic), there are 3 nodes and 2 antinodes.
- For the third overtone (4th harmonic), there are 4 nodes and 3 antinodes.

4. **Counting Nodes and Antinodes**:
- In the case of the third overtone (4th harmonic), we find:
- **Nodes**: 4
- **Antinodes**: 3

5. **Final Answer**:
- Therefore, in the third overtone of a closed organ pipe, there are **4 nodes** and **3 antinodes**.