Which of the following is incorrect for a vibrating string, in `n^(th)` harmonic mode of vibration

Options

A. The number of nodes are (n + 1)
B. The number of antinodes are n
C. The frequency of vibration in vn
D. The number of nodes are n.

Answer

To determine which statement is incorrect regarding a vibrating string in the `n^(th)` harmonic mode of vibration, we will analyze each option based on the properties of harmonics in a vibrating string.

### Step-by-Step Solution:

1. **Understanding Harmonics**:
- In the `n^(th)` harmonic mode, the string vibrates in such a way that it forms `n` loops. Each loop corresponds to one complete wave cycle.

2. **Identifying Nodes and Antinodes**:
- **Nodes**: Points along the string where the amplitude of vibration is zero. In the `n^(th)` harmonic, there are `n + 1` nodes. This is because there is one node at each end of the string and `n - 1` nodes in between.
- **Antinodes**: Points where the amplitude is maximum. In the `n^(th)` harmonic, there are `n` antinodes, which are located between the nodes.

3. **Analyzing Each Option**:
- **Option 1**: The number of nodes is `n + 1`.
- This is correct because for `n` loops, there are `n + 1` nodes.
- **Option 2**: The number of antinodes is `n`.
- This is correct as there are `n` antinodes in the `n^(th)` harmonic.
- **Option 3**: The frequency of vibration is `ν * n`, where `ν` is the fundamental frequency.
- This is also correct since the frequency of the `n^(th)` harmonic is indeed `n` times the fundamental frequency.
- **Option 4**: The number of nodes is `n`.
- This is incorrect because, as established, the number of nodes is `n + 1`, not `n`.

4. **Conclusion**:
- The incorrect statement regarding a vibrating string in the `n^(th)` harmonic mode is that the number of nodes is `n`. The correct answer is that the number of nodes is `n + 1`.

### Final Answer:
The incorrect statement is: "The number of nodes is `n`."

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