Step 1 (Governing Law & Condition for Interference): [0.25 Mark]
For two waves to produce a stable interference pattern, they must be coherent. The fundamental condition for coherence is that the waves must have the same frequency (or same angular frequency, $\omega$).
Given waves:
- (i) $y_1 = A_1 \sin \omega t$ (Frequency: $\omega$)
- (ii) $y_2 = A_2 \sin 2\omega t$ (Frequency: $2\omega$)
- (iii) $y_3 = A_3 \cos \omega t = A_3 \sin(\omega t + \pi/2)$ (Frequency: $\omega$)
- (iv) $y_4 = A_4 \sin(\omega t + \pi/3)$ (Frequency: $\omega$)
Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
We compare the angular frequencies of the given waves:
- Wave (i), (iii), and (iv) all possess the same angular frequency, $\omega$.
- Wave (ii) has an angular frequency of $2\omega$, which is different from the others.
Since interference requires the same frequency, waves (i), (iii), and (iv) can interfere with each other because they satisfy the condition:
$$\omega_1 = \omega_3 = \omega_4 = \omega$$Wave (ii) cannot interfere with any of the others because $2\omega \neq \omega$.
Step 3 (Final Concluded Answer): [0.25 Mark]
The waves that can interfere are (i), (iii), and (iv). Therefore, the correct option is (C).
💡 EL10 Concept Insight:
Think of interference like two people trying to dance in sync; if one person is dancing at a different speed (frequency) than the other, they can never stay in step to create a steady pattern. Only waves moving at the exact same speed can "lock in" to create a consistent interference pattern!