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 solve the problem, we need to analyze the conditions under which a stretched string fixed at both ends can sustain stationary waves. The fundamental principle here is that the length of the string must be an integer multiple of half the wavelength of the wave.

### Step-by-Step Solution:

1. **Understanding the Wave on a Stretched String**:
A string fixed at both ends can support standing waves. The length of the string \( L \) must be equal to an integer multiple of half the wavelength \( \lambda \).

2. **Fundamental Frequency (First Harmonic)**:
For the fundamental mode (first harmonic), the string supports one half-wavelength:
\[ L = \frac{\lambda}{2} \]
Rearranging gives:
\[ \lambda = 2L \]

3. **First Overtone (Second Harmonic)**:
For the first overtone (second harmonic), the string supports one full wavelength:
\[ L = \lambda \]
Thus:
\[ \lambda = L \]

4. **Second Overtone (Third Harmonic)**:
For the second overtone (third harmonic), the string supports one and a half wavelengths:
\[ L = \frac{3\lambda}{2} \]
Rearranging gives:
\[ \lambda = \frac{2L}{3} \]

5. **General Formula for Harmonics**:
In general, for the \( n \)-th harmonic, the relationship can be expressed as:
\[ L = \frac{n\lambda}{2} \]
Therefore:
\[ \lambda = \frac{2L}{n} \]
where \( n \) is a positive integer (1, 2, 3,...).

6. **Possible Values of Wavelength**:
From the above relationships, we can conclude that the possible wavelengths for the string are:
- 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...

7. **Identifying the Impossible Wavelength**:
The maximum wavelength that can be sustained is \( 2L \) (for the fundamental mode). Any wavelength greater than \( 2L \) cannot be sustained by the string. Therefore, if the options provided include a wavelength greater than \( 2L \), that would be the impossible value.

### Conclusion:
Based on the analysis, the wavelength that is not possible for a stretched string of length \( L \) fixed at both ends is any wavelength greater than \( 2L \).