Four monochromatic light waves are represented as follows:
`P : E = E_0 sin omega t`
`Q : E = 2 E_0 sin(omega t + delta)`
`R : E = E_0 sin 2 omegat`
`S : E = 2E_0 sin (2 omega t + delta)`
Sustained interference pattern is obtained due to superposition of

Options

A. P and Q
B. R and S
C. P and S
D. Q and R

Explanation

To determine which pairs of light waves will produce a sustained interference pattern due to superposition, we need to analyze the given equations for the four monochromatic light waves.

### Step-by-Step Solution:

1. **Identify the Given Waves**:
- Wave P: \( E_P = E_0 \sin(\omega t) \)
- Wave Q: \( E_Q = 2E_0 \sin(\omega t + \delta) \)
- Wave R: \( E_R = E_0 \sin(2\omega t) \)
- Wave S: \( E_S = 2E_0 \sin(2\omega t + \delta) \)

2. **Determine the Angular Frequencies**:
- Waves P and Q both have an angular frequency of \( \omega \).
- Waves R and S both have an angular frequency of \( 2\omega \).

3. **Check for Constant Phase Difference**:
- For sustained interference, the phase difference between the two waves must be constant over time.
- For waves P and Q:
- The phase of P is \( \omega t \).
- The phase of Q is \( \omega t + \delta \).
- The phase difference \( \Delta \phi_{PQ} = (\omega t + \delta) - \omega t = \delta \) (constant).
- For waves R and S:
- The phase of R is \( 2\omega t \).
- The phase of S is \( 2\omega t + \delta \).
- The phase difference \( \Delta \phi_{RS} = (2\omega t + \delta) - 2\omega t = \delta \) (constant).

4. **Analyze Other Combinations**:
- For waves P and R:
- The phase difference \( \Delta \phi_{PR} = 2\omega t - \omega t = \omega t \) (not constant).
- For waves P and S:
- The phase difference \( \Delta \phi_{PS} = (2\omega t + \delta) - \omega t = \omega t + \delta \) (not constant).
- For waves Q and R:
- The phase difference \( \Delta \phi_{QR} = 2\omega t - (\omega t + \delta) = \omega t - \delta \) (not constant).
- For waves Q and S:
- The phase difference \( \Delta \phi_{QS} = (2\omega t + \delta) - (\omega t + \delta) = \omega t \) (not constant).

5. **Conclusion**:
- The pairs that show sustained interference patterns are:
- Waves P and Q (constant phase difference \( \delta \)).
- Waves R and S (constant phase difference \( \delta \)).
- Therefore, the correct pairs are:
- **P and Q**
- **R and S**

### Final Answer:
The sustained interference patterns are obtained due to the superposition of:
- Waves P and Q
- Waves R and S