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 equations represent waves that can form a stationary wave through superposition, 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. **Understand the conditions for stationary waves**:
- For two waves to form a stationary wave, they must have the same frequency and opposite directions of propagation. This means one wave should be traveling in the positive direction and the other in the negative direction.

3. **Analyze the first two equations**:
- \( z_1 = A \cos(\omega t - kx) \) represents a wave traveling in the positive x-direction.
- \( z_2 = A \cos(\omega t + kx) \) represents a wave traveling in the negative x-direction.
- Both waves have the same angular frequency \( \omega \).

4. **Check the other equations**:
- \( z_3 = A \cos(\omega t + ky) \) represents a wave traveling in the negative y-direction. It cannot form a stationary wave with the others since it involves a different variable (y).
- \( z_4 = A \cos(2\omega t - 2ky) \) has a different frequency (2ω) and also travels in the negative y-direction. It cannot form a stationary wave with the others.

5. **Conclusion**:
- The only pair of waves that can form a stationary wave is \( z_1 \) and \( z_2 \) since they have the same frequency and are moving in opposite directions.

Thus, the correct answer is that a stationary wave will be formed by superposing \( z_1 \) and \( z_2 \).