Q10

1 markMCQSection A

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

Q: 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

**Correct option: (C) (i), (iii) and (iv) only** Sustained interference needs the **same frequency** and a **constant phase difference**. - (i) $\sin \omega t$, (iii) $\cos \omega t = \sin(\omega t + \pi/2)$ and (iv) $\sin(\omega t + \pi/3)$ all have angular frequency **$\omega$** — mutually coherent, so any pair among them can interfere. - (ii) $\sin 2\omega t$ has frequency **$2\omega$**, so it can never form a steady interference pattern with the others.

Wave Optics

Conditions for Interference — Coherence

Options

(A)(i) and (iii) only

(B)(iii) and (iv) only

(C)(i), (iii) and (iv) only

(D)All of them

Official Answer

Correct option: (C) (i), (iii) and (iv) only

Sustained interference needs the same frequency and a constant phase difference.

(i) $\sin \omega t$ , (iii) $\cos \omega t = \sin(\omega t + \pi/2)$ and (iv) $\sin(\omega t + \pi/3)$ all have angular frequency $\omega$ — mutually coherent, so any pair among them can interfere.
(ii) $\sin 2\omega t$ has frequency $2\omega$ , so it can never form a steady interference pattern with the others.

conditions for interferencesame frequencyconstant phase differencecoherent wavessustained interferencecos ωt equals sin(ωt+π/2)wave opticssuperposition

Marking Scheme

11 mark: correct option (C) (i), (iii) and (iv) only.
2Key reasoning: same-frequency ( $\omega$ ) waves are coherent; the $2\omega$ wave (ii) cannot interfere steadily.

Hint

Interference is sustained only between waves of the same frequency — spot the odd wave with $2\omega$ .

Quick Oral Answer

Only waves of the same frequency with a constant phase difference can interfere steadily; here waves (i), (iii) and (iv) all have frequency $\omega$ , while wave (ii) has $2\omega$ and is excluded.

Analysis & Explanation

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.

Common Mistakes

1Thinking only waves with the same phase can interfere — a constant (non-zero) phase difference like $\pi/3$ or $\pi/2$ is perfectly acceptable.
2Including the $2\omega$ wave because it 'looks similar', ignoring that different frequency destroys sustained interference.
3Not recognising $\cos \omega t$ as $\sin(\omega t + \pi/2)$ , and hence wrongly excluding wave (iii).

Interesting Facts

This is why interference is easy to see with laser light (a single frequency and phase) but not with two ordinary bulbs, whose light contains many uncorrelated frequencies and random phases.

Coherence over even a tiny frequency spread is why white-light interference shows only a few coloured fringes near the centre before they wash out.

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Frequently Asked Questions

Can two waves with a phase difference of $\pi/3$ still interfere?

Yes. Interference requires the phase difference to be constant in time, not zero. A fixed phase difference such as $\pi/3$ or $\pi/2$ still gives a stable pattern; only a continuously changing phase difference (from different frequencies) destroys it.

Why can't the $2\omega$ wave interfere with the ω waves?

Because its frequency is different, the phase difference between the $2\omega$ wave and any ω wave changes continuously with time. The resulting intensity averages out, so no steady interference pattern forms.