Four waves are expressed as
1. `y_1=a_1 sin omega t`
2. `y_2=a_2 sin2 omega t`
3. `y_3=a_3 cos omega t`
4. `y_4=a_4 sin (omega t+phi)`
The interference is possible between

Options

A. (i) and (iii)
B. (i) and (ii)
C. (ii) and (iv)
D. Not possible at all

Answer

To determine which waves can interfere with each other, we need to analyze the angular frequencies of the given waves. Interference is possible only when the waves have the same angular frequency.

### Step-by-Step Solution:

1. **Identify the Angular Frequencies**:
- For the waves given:
- \( y_1 = a_1 \sin(\omega t) \) has angular frequency \( \omega \).
- \( y_2 = a_2 \sin(2\omega t) \) has angular frequency \( 2\omega \).
- \( y_3 = a_3 \cos(\omega t) \) has angular frequency \( \omega \).
- \( y_4 = a_4 \sin(\omega t + \phi) \) has angular frequency \( \omega \).

2. **Compare Angular Frequencies**:
- Waves \( y_1 \), \( y_3 \), and \( y_4 \) all have the same angular frequency \( \omega \).
- Wave \( y_2 \) has a different angular frequency \( 2\omega \).

3. **Determine Possible Interference**:
- **Between \( y_1 \) and \( y_3 \)**:
- Both have angular frequency \( \omega \). They can interfere.
- **Between \( y_1 \) and \( y_2 \)**:
- \( y_1 \) has \( \omega \) and \( y_2 \) has \( 2\omega \). They cannot interfere.
- **Between \( y_2 \) and \( y_3 \)**:
- \( y_2 \) has \( 2\omega \) and \( y_3 \) has \( \omega \). They cannot interfere.
- **Between \( y_2 \) and \( y_4 \)**:
- \( y_2 \) has \( 2\omega \) and \( y_4 \) has \( \omega \). They cannot interfere.
- **Between \( y_3 \) and \( y_4 \)**:
- Both have angular frequency \( \omega \). They can interfere.

4. **Conclusion**:
- The waves that can interfere are:
- \( y_1 \) and \( y_3 \)
- \( y_1 \) and \( y_4 \)
- \( y_3 \) and \( y_4 \)

### Final Answer:
Interference is possible between:
- \( y_1 \) and \( y_3 \)
- \( y_1 \) and \( y_4 \)
- \( y_3 \) and \( y_4 \)