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 analyze the conditions required for stationary waves to form. Stationary waves, also known as standing waves, occur when two waves of the same frequency and amplitude travel 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. **Understand the Conditions for Stationary Waves**:
- For stationary waves to form, we need two waves with the same frequency and amplitude traveling in opposite directions. This means one wave should have a positive wave vector (kx) and the other should have a negative wave vector (-kx).

3. **Analyze Each Equation**:
- **Equation (i)**: \( y = A \sin(\omega t - kx) \) represents a wave traveling in the positive x-direction.
- **Equation (ii)**: \( y = A \cos(\omega t - kx) \) also represents a wave traveling in the positive x-direction.
- **Equation (iii)**: \( y = A \sin(\omega t + kx) \) represents a wave traveling in the negative x-direction.
- **Equation (iv)**: \( y = A \cos(\omega t + kx) \) represents a wave traveling in the negative x-direction.

4. **Form Possible Pairs**:
- To form stationary waves, we can pair:
- (i) and (iii): One travels in the positive direction and the other in the negative direction.
- (ii) and (iv): One travels in the positive direction and the other in the negative direction.
- The pairs (i) with (ii) and (iii) with (iv) cannot form stationary waves since they both travel in the same direction.

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

### Final Answer:
The equations that can form stationary waves are:
- (i) and (iii)
- (ii) and (iv)