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 of finding the shape of a string of length L that is clamped at both ends and undergoes transverse vibrations, we can analyze the conditions required for the displacement of the string at its endpoints.

### Step-by-Step Solution:

1. **Understanding Boundary Conditions**:
The string is clamped at both ends, which means that the displacement of the string at both ends (x = 0 and x = L) must be zero. Therefore, we have:
- At x = 0, y(0, t) = 0
- At x = L, y(L, t) = 0

2. **General Form of the Solution**:
The general form of the transverse vibration of a string can be expressed as:
\[ y(x, t) = A f(x) g(t) \]
where \( f(x) \) is a function of position and \( g(t) \) is a function of time.

3. **Applying the Boundary Conditions**:
- For \( y(0, t) = 0 \):
This implies that \( f(0) = 0 \).
- For \( y(L, t) = 0 \):
This implies that \( f(L) = 0 \).

4. **Choosing the Function \( f(x) \)**:
A common choice for \( f(x) \) that satisfies the boundary conditions is:
\[ f(x) = \sin\left(\frac{n \pi x}{L}\right) \]
where \( n \) is an integer. This function equals zero at both \( x = 0 \) and \( x = L \).

5. **Choosing the Function \( g(t) \)**:
The time-dependent part \( g(t) \) can be either \( \cos(\omega t) \) or \( \sin(\omega t) \). Thus, we can write:
\[ g(t) = \cos(\omega t) \quad \text{or} \quad g(t) = \sin(\omega t) \]

6. **Formulating the Complete Solution**:
Therefore, the complete solution for the shape of the string can be expressed as:
\[ y(x, t) = A \sin\left(\frac{n \pi x}{L}\right) \cos(\omega t) \quad \text{(Option A)} \]
or
\[ y(x, t) = A \sin\left(\frac{n \pi x}{L}\right) \sin(\omega t) \quad \text{(Option B)} \]

7. **Evaluating Other Options**:
- For \( y(x, t) = A \cos\left(\frac{n \pi x}{L}\right) \cos(\omega t) \): This does not satisfy the boundary conditions since at \( x = 0 \), \( y(0, t) = A \) which is not zero.
- Therefore, this option is incorrect.

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