A string of length L is stretched along the x-axis and is rigidly clamped at its two ends. It undergoes transverse vibration. If n an integer, which of the following relations may represent the shape of the string at any time :-

Options

A. `y=A sin ((n pi x)/(L)) cos omega t`
B. `y=A sin((n pi x)/(L))sin omega t`
C. `y=A cos ((n pi x)/(L)) cos omega t`
D. `y=A cos ((n pi x)/(L))sin omega t`

Explanation

To solve the problem, we need to determine which of the given equations can represent the shape of a string undergoing transverse vibrations while being clamped at both ends. The boundary conditions are that the displacement \( y \) of the string must be zero at both ends, \( x = 0 \) and \( x = L \).

### Step-by-Step Solution:

1. **Understanding the Boundary Conditions**:
- The string is clamped at both ends, which means that at \( x = 0 \) and \( x = L \), the displacement \( y \) must be zero:
\[ y(0, t) = 0 \quad \text{and} \quad y(L, t) = 0 \]

2. **Analyzing the First Equation**:
- Let's consider the first equation:
\[ y = A \sin\left(\frac{n \pi x}{L}\right) \cos(\omega t) \]
- Check the boundary conditions:
- At \( x = 0 \):
\[ y(0, t) = A \sin(0) \cos(\omega t) = 0 \]
- At \( x = L \):
\[ y(L, t) = A \sin(n \pi) \cos(\omega t) = 0 \]
- Both conditions are satisfied, so this equation is valid.

3. **Analyzing the Second Equation**:
- Now consider the second equation:
\[ y = A \sin\left(\frac{n \pi x}{L}\right) \sin(\omega t) \]
- Check the boundary conditions:
- At \( x = 0 \):
\[ y(0, t) = A \sin(0) \sin(\omega t) = 0 \]
- At \( x = L \):
\[ y(L, t) = A \sin(n \pi) \sin(\omega t) = 0 \]
- Both conditions are satisfied, so this equation is also valid.

4. **Analyzing the Third Equation**:
- Now consider the third equation:
\[ y = A \cos\left(\frac{n \pi x}{L}\right) \cos(\omega t) \]
- Check the boundary conditions:
- At \( x = 0 \):
\[ y(0, t) = A \cos(0) \cos(\omega t) = A \cos(\omega t) \quad (\text{not } 0) \]
- This does not satisfy the boundary condition at \( x = 0 \), so this equation is invalid.

5. **Analyzing the Fourth Equation**:
- Finally, consider the fourth equation:
\[ y = A \cos\left(\frac{n \pi x}{L}\right) \sin(\omega t) \]
- Check the boundary conditions:
- At \( x = 0 \):
\[ y(0, t) = A \cos(0) \sin(\omega t) = A \sin(\omega t) \quad (\text{not } 0) \]
- This does not satisfy the boundary condition at \( x = 0 \), so this equation is also invalid.

### Conclusion:
The valid equations that represent the shape of the string at any time are:
- \( y = A \sin\left(\frac{n \pi x}{L}\right) \cos(\omega t) \)
- \( y = A \sin\left(\frac{n \pi x}{L}\right) \sin(\omega t) \)

### Final Answer:
The correct options are the first and second equations.