Four independent waves are expressed as (i) $y_1 = A_1 \sin \omega t$ (ii) $y_2 = A_2 \sin 2\omega t$ (iii) $y_3 = A_3 \cos \omega t$ (iv) $y_4 = A_4 \sin (\omega t + \pi/3)$. The interference between two of these waves is possible in

This tests the conditions for sustained (observable) interference: equal frequency and a time-independent phase relationship.

Concept: Two waves produce a stable interference pattern only if:

• they have the same frequency (equal $\omega$), and
• they maintain a constant phase difference (are coherent).

Applying it:

• (i) $y_1 = A_1 \sin \omega t$ — frequency $\omega$.
• (iii) $y_3 = A_3 \cos \omega t = A_3 \sin(\omega t + \pi/2)$ — frequency $\omega$, constant phase lead of $\pi/2$.
• (iv) $y_4 = A_4 \sin(\omega t + \pi/3)$ — frequency $\omega$, constant phase lead of $\pi/3$.
• (ii) $y_2 = A_2 \sin 2\omega t$ — frequency $2\omega$, different from the rest.

Waves (i), (iii) and (iv) share the same $\omega$ and fixed relative phases, so any two of them interfere. Wave (ii) has double the frequency, giving a continuously changing phase difference with the others — no steady pattern.

Why (C) is correct: exactly the three $\omega$-waves qualify.

Why the others are wrong:

• (A) and (B) are too restrictive — they omit valid $\omega$-pairs.
• (D) wrongly includes (ii), whose frequency $2\omega$ breaks the coherence condition.