A stretched string of length L , fixed at both ends can sustain stationary waves of wavelength `lamda` Which of the following value of wavelength is not possible ?

Options

A. 2L
B. 4L
C. L
D. `L/2`

Answer

To determine which wavelength is not possible for a stretched string of length \( L \) fixed at both ends, we need to analyze the conditions for stationary waves on such a string.

### Step-by-Step Solution:

1. **Understanding Stationary Waves**: A string fixed at both ends can support stationary waves. The length of the string must be an integer multiple of half wavelengths (\( \frac{\lambda}{2} \)).

2. **Formulating the Condition**: For a string of length \( L \), the relationship between the length of the string and the wavelength is given by:
\[ L = n \cdot \frac{\lambda}{2} \]
where \( n \) is a positive integer (1, 2, 3,...).

3. **Rearranging the Equation**: From the above equation, we can express the wavelength \( \lambda \) in terms of the length \( L \) and the integer \( n \):
\[ \lambda = \frac{2L}{n} \]

4. **Possible Wavelengths**: As \( n \) takes positive integer values, the possible wavelengths can be calculated:
- For \( n = 1 \): \( \lambda = 2L \)
- For \( n = 2 \): \( \lambda = L \)
- For \( n = 3 \): \( \lambda = \frac{2L}{3} \)
- For \( n = 4 \): \( \lambda = \frac{L}{2} \)
- And so on...

5. **Identifying the Maximum Wavelength**: The maximum wavelength that can be sustained is when \( n = 1 \), which gives \( \lambda = 2L \). As \( n \) increases, the wavelength decreases.

6. **Conclusion**: Since the possible wavelengths are \( 2L, L, \frac{2L}{3}, \frac{L}{2}, \) etc., we can see that a wavelength of \( 4L \) is not possible, as it exceeds the maximum wavelength that can be sustained by the string.

### Final Answer:
The value of wavelength that is not possible is \( 4L \).