Consider the three waves `z_(1), z_(2) "and" z_(3)` as
`z_(1) = A "sin"(kx - omega t)`
`z_(2) = A "sin"(kx + omega t)`
`z_(3) = A "sin"(ky - omega t)`
Which of the following represents a standing wave?

Options

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

Answer

To determine which of the given waves represents a standing wave, we need to analyze the equations of the waves provided.

1. **Identify the waves**:
- The first wave is given by \( z_1 = A \sin(kx - \omega t) \).
- The second wave is given by \( z_2 = A \sin(kx + \omega t) \).
- The third wave is given by \( z_3 = A \sin(ky - \omega t) \).

2. **Understand the concept of standing waves**:
- Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions. This can be observed in the case of waves that have the same wave number \( k \) and angular frequency \( \omega \).

3. **Analyze the first two waves**:
- The wave \( z_1 \) travels in the positive x-direction (since it has the term \( -\omega t \)).
- The wave \( z_2 \) travels in the negative x-direction (since it has the term \( +\omega t \)).
- Since \( z_1 \) and \( z_2 \) are traveling in opposite directions, they can interfere to form a standing wave.

4. **Check the third wave**:
- The wave \( z_3 = A \sin(ky - \omega t) \) involves a different variable \( y \) and does not interact with \( z_1 \) and \( z_2 \) in the same manner. Thus, it cannot form a standing wave with the other two.

5. **Conclusion**:
- The combination of \( z_1 \) and \( z_2 \) will produce a standing wave, while \( z_3 \) does not contribute to this formation. Therefore, the correct answer is that the waves \( z_1 \) and \( z_2 \) represent a standing wave.

### Final Answer:
The waves \( z_1 \) and \( z_2 \) represent a standing wave.