Out of given four waves (1),(2),(3) and (4)
`y=asin(kx+omegat)` .(1)
`y=asin(omegat-kx)` ..(2)
`y=acos(kx+omegat)` ..(3)
`y=acos(omegat-kx)` .(4)
emitted by four different source `S_1`,`S_2`,`S_3` and `S_4` respectively, interference phenomena would be observed in space under appropriate conditions when

Options

A. (a)source `S_(1)` emits wave (1) and `S_(4)` emits wave (2)
B. (b)source `S_(3)` emits wave (3) and `S_(4)` emits wave (4)
C. (c)source `S_(2)` emits wave (2) and `S_(4)` emits wave (4)
D. (d)`S_(4)` smits waves (4) and `S_(3)` emits waves (3)

Answer

To solve the problem regarding the interference of the given waves, we need to analyze the equations of the waves and determine the conditions under which interference occurs. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Wave Equations
The four waves given are:
1. \( y = a \sin(kx + \omega t) \) (Wave 1)
2. \( y = a \sin(\omega t - kx) \) (Wave 2)
3. \( y = a \cos(kx + \omega t) \) (Wave 3)
4. \( y = a \cos(\omega t - kx) \) (Wave 4)

### Step 2: Understand the Condition for Interference
For interference to occur between two waves, they must have a constant phase relationship. This means that the difference in their phase should be constant over time.

### Step 3: Analyze the Phase Relationships
- **Wave 1 and Wave 3**:
- Wave 1: \( \sin(kx + \omega t) \)
- Wave 3: \( \cos(kx + \omega t) \)

We can express Wave 3 in terms of sine:
\[ \cos(kx + \omega t) = \sin\left(kx + \omega t + \frac{\pi}{2}\right) \]
This indicates a phase difference of \( \frac{\pi}{2} \) (90 degrees).

- **Wave 2 and Wave 4**:
- Wave 2: \( \sin(\omega t - kx) \)
- Wave 4: \( \cos(\omega t - kx) \)

Similarly, we can express Wave 4 in terms of sine:
\[ \cos(\omega t - kx) = \sin\left(\omega t - kx + \frac{\pi}{2}\right) \]
This also indicates a phase difference of \( \frac{\pi}{2} \).

### Step 4: Determine Which Waves Can Interfere
From the analysis:
- Waves 1 and 3 can interfere due to their phase difference of \( \frac{\pi}{2} \).
- Waves 2 and 4 can also interfere due to their phase difference of \( \frac{\pi}{2} \).

### Step 5: Identify the Sources
- Wave 1 is emitted by source \( S_1 \).
- Wave 2 is emitted by source \( S_2 \).
- Wave 3 is emitted by source \( S_3 \).
- Wave 4 is emitted by source \( S_4 \).

### Conclusion
The pairs of waves that can interfere are:
- \( S_1 \) (Wave 1) and \( S_3 \) (Wave 3)
- \( S_2 \) (Wave 2) and \( S_4 \) (Wave 4)

Thus, the correct answer is that interference phenomena would be observed when:
- \( S_2 \) emits Wave 2 and \( S_4 \) emits Wave 4.

### Final Answer
The correct option is **C**.

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