The following equations represent progressive transverse waves
`z_(1) = A cos ( omega t - kx)`
`z_(2) = A cos ( omega t + kx)`
`z_(3) = A cos ( omega t + ky)`
`z_(4) = A cos (2 omega t - 2 ky)`
A stationary wave will be formed by superposing

Options

A. `z_(1) and z_(2)`
B. `z_(1) and z_(4)`
C. `z_(2) and z_(3)`
D. `z_(3) and z_(4)`

Explanation

To determine which two waves can be superimposed to create a stationary wave, we need to analyze the given wave equations.

1. **Identify the wave equations:**
- \( z_1 = A \cos(\omega t - kx) \)
- \( z_2 = A \cos(\omega t + kx) \)
- \( z_3 = A \cos(\omega t + ky) \)
- \( z_4 = A \cos(2\omega t - 2ky) \)

2. **Check the conditions for forming a stationary wave:**
- For two waves to form a stationary wave, they must have the same frequency and travel in opposite directions. The general form of a stationary wave can be expressed as the superposition of two waves traveling in opposite directions.

3. **Examine pairs of waves:**
- **Pair \( z_1 \) and \( z_2 \):**
- Both have the same angular frequency \( \omega \).
- \( z_1 \) travels in the positive x-direction and \( z_2 \) travels in the negative x-direction.
- Since they meet the criteria (same frequency and opposite directions), they can form a stationary wave.

- **Pair \( z_1 \) and \( z_4 \):**
- \( z_1 \) has frequency \( \omega \) and \( z_4 \) has frequency \( 2\omega \).
- Since the frequencies are different, they cannot form a stationary wave.

- **Pair \( z_2 \) and \( z_3 \):**
- \( z_2 \) travels in the x-direction and \( z_3 \) travels in the y-direction.
- Since they are in different directions, they cannot form a stationary wave.

- **Pair \( z_3 \) and \( z_4 \):**
- \( z_3 \) has frequency \( \omega \) and \( z_4 \) has frequency \( 2\omega \).
- Since the frequencies are different, they cannot form a stationary wave.

4. **Conclusion:**
- The only pair of waves that can be superimposed to create a stationary wave is \( z_1 \) and \( z_2 \).

### Final Answer:
The stationary wave will be formed by superposing \( z_1 \) and \( z_2 \).