A standing wave is produced on a string on a string clamped at one end and free at the other. The length of the string

Options

A. must be an integral multiple `lamda/4`
B. must be an integral multiple of `lamda/2`
C. must be an integral multiple of `lamda`
D. may be an integral multiple of `lamda/2`

Answer

To solve the problem of determining the length of a string that produces a standing wave when one end is clamped and the other end is free, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a string that is fixed at one end (clamped) and free at the other end. This setup allows for the formation of standing waves.

2. **Identifying the Fundamental Frequency**:
- For a string that is fixed at one end and free at the other, the fundamental frequency (first harmonic) can be described by the equation:
\[ L = \frac{(2n + 1) \lambda}{4} \]
- Here, \(L\) is the length of the string, \(n\) is the harmonic number (0, 1, 2,...), and \(\lambda\) is the wavelength.

3. **Understanding the Harmonics**:
- The harmonic number \(n\) can take values starting from 0. Therefore, the possible lengths of the string for different harmonics can be calculated:
- For \(n = 0\): \(L = \frac{1 \lambda}{4}\)
- For \(n = 1\): \(L = \frac{3 \lambda}{4}\)
- For \(n = 2\): \(L = \frac{5 \lambda}{4}\)
- This shows that the lengths of the string correspond to odd multiples of \(\frac{\lambda}{4}\).

4. **Conclusion**:
- The length of the string must be an odd integral multiple of \(\frac{\lambda}{4}\). Therefore, we can conclude that the length of the string can be expressed as:
\[ L = (2n + 1) \frac{\lambda}{4} \]
- This means that the string can have lengths such as \(\frac{\lambda}{4}\), \(\frac{3\lambda}{4}\), \(\frac{5\lambda}{4}\), etc.

### Final Answer:
The length of the string must be an odd integral multiple of \(\frac{\lambda}{4}\).

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