The `(x, y)` co-ordinates of the corners of a square plate are `(0, 0)`,`(L, 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. `a cos(pi r//2 L) cos(pi y//2 L)`
B. `a sin(pi x//L) sin(pi y//L)`
C. `a sin (pi x//L) sin (2 pi y//L)`
D. `a cos(2 pi x//L) sin (pi y//L)`

Explanation

To solve the problem of finding the possible expressions for the displacement \( u(x, y) \) of a square plate with clamped edges, we need to analyze the boundary conditions and the nature of standing waves.

### Step-by-Step Solution:

1. **Understand the Boundary Conditions**:
- The edges of the square plate are clamped. This means that the displacement \( u(x, y) \) must be zero at the edges. The corners of the square plate are given as:
- \( (0, 0) \)
- \( (L, 0) \)
- \( (L, L) \)
- \( (0, L) \)

2. **Identify the Edges**:
- The edges of the square plate can be represented as:
- Bottom edge: \( y = 0 \) for \( 0 \leq x \leq L \)
- Right edge: \( x = L \) for \( 0 \leq y \leq L \)
- Top edge: \( y = L \) for \( 0 \leq x \leq L \)
- Left edge: \( x = 0 \) for \( 0 \leq y \leq L \)

3. **Formulate the Displacement Function**:
- The displacement function \( u(x, y) \) must satisfy the boundary conditions \( u(x, 0) = 0 \), \( u(x, L) = 0 \), \( u(0, y) = 0 \), and \( u(L, y) = 0 \).

4. **Examine Possible Forms of \( u(x, y) \)**:
- A general form for the displacement in terms of sine functions is:
\[ u(x, y) = A \sin\left(\frac{n \pi x}{L}\right) \sin\left(\frac{m \pi y}{L}\right) \]
- Here, \( n \) and \( m \) are positive integers, which ensure that the displacement is zero at the boundaries.

5. **Evaluate the Given Options**:
- **Option A**: \( u(x, y) = A \cos\left(\frac{\pi x}{2L}\right) \cos\left(\frac{\pi y}{2L}\right) \)
- At \( x = 0 \) and \( y = 0 \), \( u(0, 0) = A \) (not zero). This option is not valid.
- **Option B**: \( u(x, y) = A \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right) \)
- At \( x = 0 \), \( u(0, y) = 0 \) and at \( y = 0 \), \( u(x, 0) = 0 \). This option is valid.
- **Option C**: \( u(x, y) = A \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{2\pi y}{L}\right) \)
- At \( x = 0 \), \( u(0, y) = 0 \) and at \( y = 0 \), \( u(x, 0) = 0 \). This option is valid.
- **Option D**: \( u(x, y) = A \cos\left(\frac{2\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right) \)
- At \( x = 0 \), \( u(0, y) = A \) (not zero). This option is not valid.

6. **Conclusion**:
- The valid expressions for the displacement \( u(x, y) \) are:
- Option B: \( A \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right) \)
- Option C: \( A \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{2\pi y}{L}\right) \)

### Final Answer:
The possible expressions for \( u(x, y) \) are:
- \( u(x, y) = A \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right) \)
- \( u(x, y) = A \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{2\pi y}{L}\right) \)