The `(x, y)` co-ordinates of the corners of a square plate are `(0, 0)`, `(L, L)` and `(0, L)`. The edges of the plate are clamped and transverse standing waves are set up in it. If `u(x, y)` denotes the displacement of the plate at the point `(x, y)` at some instant of time, the possible expression `(s)` for `u` is (are) `(a = positive constant)`

Options

A. `acos(pix//2L)cos(piy//2L)`
B. `asin(pix//L)sin(piy//L)`
C. `asin(pix//L)sin(2piy//L)`
D. `acos(2pix//L)sin(2piy//L)`

Explanation

To solve the problem of finding the possible expressions for the displacement \( u(x, y) \) of a clamped square plate with corners at coordinates \( (0, 0) \), \( (L, 0) \), \( (L, L) \), and \( (0, L) \), we need to ensure that the displacement at the edges of the plate is zero.

### Step-by-Step Solution:

1. **Understanding the Boundary Conditions**:
- Since the edges of the plate are clamped, the displacement \( u(x, y) \) must be zero along the edges of the square.
- The edges are defined by:
- \( y = 0 \) (bottom edge)
- \( x = L \) (right edge)
- \( y = L \) (top edge)
- \( x = 0 \) (left edge)

2. **Examining Possible Expressions**:
- We need to check each proposed expression for \( u(x, y) \) to see if it satisfies the boundary conditions.

3. **Checking Expression A**:
- \( u(x, y) = a \cos\left(\frac{\pi x}{2L}\right) \cos\left(\frac{\pi y}{2L}\right) \)
- For \( y = 0 \):
\[ u(x, 0) = a \cos\left(\frac{\pi x}{2L}\right) \cos(0) = a \cos\left(\frac{\pi x}{2L}\right) \]
- This is not zero for \( x \) varying from \( 0 \) to \( L \). Thus, **Expression A is not valid**.

4. **Checking Expression B**:
- \( u(x, y) = a \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right) \)
- For \( y = 0 \):
\[ u(x, 0) = a \sin\left(\frac{\pi x}{L}\right) \sin(0) = 0 \]
- For \( x = L \):
\[ u(L, y) = a \sin(\pi) \sin\left(\frac{\pi y}{L}\right) = 0 \]
- For \( y = L \):
\[ u(x, L) = a \sin\left(\frac{\pi x}{L}\right) \sin(\pi) = 0 \]
- For \( x = 0 \):
\[ u(0, y) = a \sin(0) \sin\left(\frac{\pi y}{L}\right) = 0 \]
- **Expression B is valid**.

5. **Checking Expression C**:
- \( u(x, y) = a \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{2\pi y}{L}\right) \)
- For \( y = 0 \):
\[ u(x, 0) = a \sin\left(\frac{\pi x}{L}\right) \sin(0) = 0 \]
- For \( x = L \):
\[ u(L, y) = a \sin(\pi) \sin\left(\frac{2\pi y}{L}\right) = 0 \]
- For \( y = L \):
\[ u(x, L) = a \sin\left(\frac{\pi x}{L}\right) \sin(2\pi) = 0 \]
- For \( x = 0 \):
\[ u(0, y) = a \sin(0) \sin\left(\frac{2\pi y}{L}\right) = 0 \]
- **Expression C is also valid**.

6. **Checking Expression D**:
- \( u(x, y) = a \cos\left(\frac{2\pi x}{L}\right) \sin\left(\frac{2\pi y}{L}\right) \)
- For \( y = 0 \):
\[ u(x, 0) = a \cos\left(\frac{2\pi x}{L}\right) \sin(0) = 0 \]
- For \( x = L \):
\[ u(L, y) = a \cos(2\pi) \sin\left(\frac{2\pi y}{L}\right) = a \sin\left(\frac{2\pi y}{L}\right) \]
- This is not zero for \( y \) varying from \( 0 \) to \( L \). Thus, **Expression D is not valid**.

### Conclusion:
The valid expressions for \( u(x, y) \) that satisfy the boundary conditions are **B and C**.