Four light waves are represented by
`i. y=a_1sin omegat`
ii. `y=a_2sin(omegat+epsilon)`
iii. `y=a_1sin 2omegat`
iv.` y=a_2sin2(omegat+epsilon)` Inteference fringes may be observed due to superposition of

Options

A. (i) and (ii)
B. (i) and (iii)
C. (ii) and (iv)
D. (iii) and (iv).

Answer

To solve the problem of identifying which pairs of light waves can produce interference fringes due to superposition, we need to analyze the given wave equations based on two criteria: the frequencies of the waves must be the same, and the initial phase difference must be constant.

### Step-by-Step Solution:

1. **Identify the Given Waves**:
- Wave 1: \( y_1 = A_1 \sin(\omega t) \)
- Wave 2: \( y_2 = A_2 \sin(\omega t + \epsilon) \)
- Wave 3: \( y_3 = A_1 \sin(2\omega t) \)
- Wave 4: \( y_4 = A_2 \sin(2\omega t + \epsilon) \)

2. **Check Frequencies of the Waves**:
- For Wave 1 and Wave 2, the angular frequency is \( \omega \).
- For Wave 3 and Wave 4, the angular frequency is \( 2\omega \).
- Waves 1 and 2 have a frequency of \( \frac{\omega}{2\pi} \).
- Waves 3 and 4 have a frequency of \( \frac{2\omega}{2\pi} \).

3. **Pair Analysis for Interference**:
- **Pair 1 (Wave 1 and Wave 2)**:
- Frequencies: Both have frequency \( \frac{\omega}{2\pi} \) (same).
- Phase difference: \( 0 \) (for Wave 1) and \( \epsilon \) (for Wave 2) → constant.
- **Conclusion**: Pair 1 can produce interference fringes.

- **Pair 2 (Wave 3 and Wave 4)**:
- Frequencies: Both have frequency \( \frac{2\omega}{2\pi} \) (same).
- Phase difference: \( 0 \) (for Wave 3) and \( \epsilon \) (for Wave 4) → constant.
- **Conclusion**: Pair 2 can also produce interference fringes.

- **Pair 3 (Wave 1 and Wave 3)**:
- Frequencies: \( \frac{\omega}{2\pi} \) (for Wave 1) and \( \frac{2\omega}{2\pi} \) (for Wave 3) → different.
- **Conclusion**: Pair 3 cannot produce interference fringes.

- **Pair 4 (Wave 2 and Wave 4)**:
- Frequencies: \( \frac{\omega}{2\pi} \) (for Wave 2) and \( \frac{2\omega}{2\pi} \) (for Wave 4) → different.
- **Conclusion**: Pair 4 cannot produce interference fringes.

4. **Final Conclusion**:
- The pairs that can produce interference fringes are:
- Pair 1: Waves 1 and 2
- Pair 2: Waves 3 and 4

### Answer:
The pairs of light waves that can produce interference fringes are:
- Pair 1: Waves 1 and 2
- Pair 2: Waves 3 and 4