Four sound sources produce the following four waves
(i) `y_(1)=a sin (omega t+phi_(1))`
(ii) `y_(2)=a sin 2 omega t`
(iii) `y_(3)= a' sin (omega t+phi_(2))`
(iv) `y_(4)=a' sin (3 omega t+phi)`
Superposition of which two waves gives rise to interference?

Options

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

Answer

To determine which two waves from the given sources produce interference, we need to analyze the conditions for interference. The two main conditions for interference are:

1. The waves must travel in the same direction.
2. The sources must be coherent, meaning they should have the same frequency.

Given the four waves:

1. \( y_1 = a \sin(\omega t + \phi_1) \)
2. \( y_2 = a \sin(2\omega t) \)
3. \( y_3 = a' \sin(\omega t + \phi_2) \)
4. \( y_4 = a' \sin(3\omega t + \phi) \)

### Step 1: Identify the angular frequencies of each wave
- For wave \( y_1 \), the angular frequency is \( \omega \).
- For wave \( y_2 \), the angular frequency is \( 2\omega \).
- For wave \( y_3 \), the angular frequency is \( \omega \).
- For wave \( y_4 \), the angular frequency is \( 3\omega \).

### Step 2: Compare the angular frequencies
- Waves \( y_1 \) and \( y_3 \) both have an angular frequency of \( \omega \).
- Waves \( y_2 \) and \( y_4 \) have angular frequencies of \( 2\omega \) and \( 3\omega \) respectively, which are different from each other and from \( y_1 \) and \( y_3 \).
- Wave \( y_2 \) has an angular frequency of \( 2\omega \), which is different from \( y_1 \) and \( y_3 \).
- Wave \( y_4 \) has an angular frequency of \( 3\omega \), which is also different from \( y_1 \) and \( y_3 \).

### Step 3: Identify coherent pairs
- The only pair of waves that have the same angular frequency are \( y_1 \) and \( y_3 \) (both have frequency \( \omega \)).
- Therefore, \( y_1 \) and \( y_3 \) can produce interference.

### Conclusion
The superposition of waves \( y_1 \) and \( y_3 \) gives rise to interference.

### Final Answer
The waves that give rise to interference are \( y_1 \) and \( y_3 \).

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