Four different independent waves are represented by
`y_1=a_1 sin omegat`
`y_2=a_2 sin 2 omega t`
`y_3=a_3 cos omega t`
`y_4=a_4 sin( omega t+lamda/3)`
With which of two waves interference is possible

Options

A. (i) and (ii)
B. (i) and (iv)
C. (iii) and (iv)
D. Not possible with any combination

Answer

To determine which pairs of waves can interfere with each other, we need to analyze the equations of the waves provided and check for coherence. Coherence means that the waves must have a constant phase difference over time.

The waves given are:

1. \( y_1 = a_1 \sin(\omega t) \)
2. \( y_2 = a_2 \sin(2\omega t) \)
3. \( y_3 = a_3 \cos(\omega t) \)
4. \( y_4 = a_4 \sin(\omega t + \frac{\lambda}{3}) \)

### Step-by-Step Solution:

**Step 1: Identify the frequency and phase of each wave.**

- For \( y_1 = a_1 \sin(\omega t) \):
- Frequency: \( \omega \)
- Phase: \( 0 \)

- For \( y_2 = a_2 \sin(2\omega t) \):
- Frequency: \( 2\omega \)
- Phase: \( 0 \)

- For \( y_3 = a_3 \cos(\omega t) \):
- Frequency: \( \omega \)
- Phase: \( \frac{\pi}{2} \) (since \( \cos(\theta) = \sin(\theta + \frac{\pi}{2}) \))

- For \( y_4 = a_4 \sin(\omega t + \frac{\lambda}{3}) \):
- Frequency: \( \omega \)
- Phase: \( \frac{\lambda}{3} \)

**Step 2: Check for coherence between pairs of waves.**

- **Pair \( (y_1, y_2) \)**:
- Frequencies are different (\( \omega \) and \( 2\omega \)).
- **Not coherent.**

- **Pair \( (y_1, y_3) \)**:
- Both have the same frequency (\( \omega \)).
- Phase difference: \( \frac{\pi}{2} - 0 = \frac{\pi}{2} \) (constant).
- **Coherent.**

- **Pair \( (y_1, y_4) \)**:
- Both have the same frequency (\( \omega \)).
- Phase difference: \( \frac{\lambda}{3} - 0 = \frac{\lambda}{3} \) (constant).
- **Coherent.**

- **Pair \( (y_2, y_3) \)**:
- Frequencies are different (\( 2\omega \) and \( \omega \)).
- **Not coherent.**

- **Pair \( (y_2, y_4) \)**:
- Frequencies are different (\( 2\omega \) and \( \omega \)).
- **Not coherent.**

- **Pair \( (y_3, y_4) \)**:
- Both have the same frequency (\( \omega \)).
- Phase difference: \( \frac{\lambda}{3} - \frac{\pi}{2} \) (constant).
- **Coherent.**

**Step 3: Conclusion on interference possibilities.**

The pairs of waves that can interfere with each other are:
- \( y_1 \) and \( y_3 \)
- \( y_1 \) and \( y_4 \)
- \( y_3 \) and \( y_4 \)

Thus, interference is possible between:
- **Waves 1 and 3**
- **Waves 1 and 4**
- **Waves 3 and 4**