Which of the following equations can form stationary waves?
(i)`y= A sin (omegat - kx)` (ii) `y= A cos (omegat - kx)`
(iii) `y= A sin (omegat + kx)` (iv) `y= A cos (omegat - kx)` .

Options

A. (i) and (ii)
B. (i) and (iii)
C. (iii) and (iv)
D. (ii) and (iv)

Explanation

To determine which of the given equations can form stationary waves, we need to understand the conditions required for stationary waves to exist. Stationary waves are formed by the superposition of two waves traveling in opposite directions.

### Step-by-step Solution:

1. **Identify the Equations**:
The equations given are:
- (i) \( y = A \sin(\omega t - kx) \)
- (ii) \( y = A \cos(\omega t - kx) \)
- (iii) \( y = A \sin(\omega t + kx) \)
- (iv) \( y = A \cos(\omega t + kx) \)

2. **Understanding Wave Direction**:
- The term \( \sin(\omega t - kx) \) represents a wave traveling in the positive x-direction.
- The term \( \sin(\omega t + kx) \) represents a wave traveling in the negative x-direction.
- Similarly, \( \cos(\omega t - kx) \) represents a wave traveling in the positive x-direction, while \( \cos(\omega t + kx) \) represents a wave traveling in the negative x-direction.

3. **Pairing the Waves**:
- For stationary waves to form, we need pairs of waves traveling in opposite directions:
- Pair 1: \( y = A \sin(\omega t - kx) \) (positive direction) and \( y = A \sin(\omega t + kx) \) (negative direction).
- Pair 2: \( y = A \cos(\omega t - kx) \) (positive direction) and \( y = A \cos(\omega t + kx) \) (negative direction).

4. **Identifying Valid Equations**:
- From the pairs identified:
- Equations (i) and (iii) can form stationary waves.
- Equations (ii) and (iv) can also form stationary waves.
- Therefore, equations (ii) and (iv) can also form stationary waves because they are both cosine functions traveling in opposite directions.

5. **Conclusion**:
The equations that can form stationary waves are:
- (ii) \( y = A \cos(\omega t - kx) \)
- (iv) \( y = A \cos(\omega t + kx) \)

### Final Answer:
The correct options that can form stationary waves are (ii) and (iv).