Four independent waves are expressed as (i) y₁ = A₁ sinω t (ii) y₂ = A₂ sin 2ω t (iii) y₃ = A₃ cosω t (iv) y₄

Question

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\left(\omega t + \dfrac{\pi}{3}\right)

The interference between two of these waves is possible in (A)

(i) and

(iii) only (B)

(iii) and

(iv) only (C) (i),

(iii) and

(iv) only (D) All of them

CBSECBSE Class XII Board 2026MCQ· 1mImportance★★★★★

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Interference requires waves to have the same frequency and a constant phase difference. Only waves (i), (iii), and (iv) share the same angular frequency $\omega$ , so interference is possible among them. The correct option is (C).

Concept and Intuition

Interference is the superposition of two or more waves that produces a resultant wave of greater, lower, or the same amplitude. For interference to be observable (i.e., to produce a stable pattern), the waves must satisfy two conditions:

Same frequency — so the phase difference between them does not change with time.
Constant phase difference — so the interference pattern is stationary.

If two waves have different frequencies, their phase difference keeps changing randomly, and the interference pattern averages out to no observable effect. This is why light from two different bulbs (different frequencies and random phases) never shows interference.

Here, we have four waves. Let's check which pairs meet the frequency condition.

Watch out

A common mistake is to think that any two sine/cosine waves can interfere. But if their frequencies differ (like $\omega$ vs $2\omega$ ), the phase difference is not constant — no stable interference.

Step-by-Step Solution

Identify the angular frequency of each wave
- (i) $y_1 = A_1 \sin\omega t$ → angular frequency = $\omega$
- (ii) $y_2 = A_2 \sin 2\omega t$ → angular frequency = $2\omega$
- (iii) $y_3 = A_3 \cos\omega t$ → angular frequency = $\omega$ (since $\cos\omega t = \sin(\omega t + \pi/2)$ )
- (iv) $y_4 = A_4 \sin\left(\omega t + \dfrac{\pi}{3}\right)$ → angular frequency = $\omega$
Group waves by frequency
- Frequency $\omega$ : (i), (iii), (iv)
- Frequency $2\omega$ : (ii) alone
Check interference possibility
- Waves (i) and (iii): both have frequency $\omega$ . Their phase difference is $\pi/2$ (since $\cos\omega t = \sin(\omega t + \pi/2)$ ), which is constant. Interference possible. …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set 55/1/11 markMCQ
$Q. Four independent waves are expressed as$
(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) $y_4 = A_4 \sin\left(\omega t + \dfrac{\pi}{3}\right)$ The interference between two of these waves is possible in (A)
(i) and
(iii) only (B)
(iii) and
(iv) only (C) (i),
(iii) and
(iv) only (D) All of them
›Reveal solutionSolution
Interference requires waves to have the same frequency and a constant phase difference. Only waves (i), (iii), and (iv) share the same angular frequency $\omega$ , so interference is possible among them. The correct option is (C).

Concept and Intuition

Interference is the superposition of two or more waves that produces a resultant wave of greater, lower, or the same amplitude. For interference to be observable (i.e., to produce a stable pattern), the waves must satisfy two conditions:

Same frequency — so the phase difference between them does not change with time.

Constant phase difference — so the interference pattern is stationary.

If two waves have different frequencies, their phase difference keeps changing randomly, and the interference pattern averages out to no observable effect. This is why light from two different bulbs (different frequencies and random phases) never shows interference.

Here, we have four waves. Let's check which pairs meet the frequency condition.

Watch out

A common mistake is to think that any two sine/cosine waves can interfere. But if their frequencies differ (like $\omega$ vs $2\omega$ ), the phase difference is not constant — no stable interference.

Step-by-Step Solution

Identify the angular frequency of each wave

(i) $y_1 = A_1 \sin\omega t$ → angular frequency = $\omega$

(ii) $y_2 = A_2 \sin 2\omega t$ → angular frequency = $2\omega$

(iii) $y_3 = A_3 \cos\omega t$ → angular frequency = $\omega$ (since $\cos\omega t = \sin(\omega t + \pi/2)$ )

(iv) $y_4 = A_4 \sin\left(\omega t + \dfrac{\pi}{3}\right)$ → angular frequency = $\omega$

Group waves by frequency

Frequency $\omega$ : (i), (iii), (iv)

Frequency $2\omega$ : (ii) alone

Check interference possibility

Waves (i) and (iii): both have frequency $\omega$ . Their phase difference is $\pi/2$ (since $\cos\omega t = \sin(\omega t + \pi/2)$ ), which is constant. Interference possible. …
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CBSE 2026Set A1 markMCQ
$Q. Coherent sources are those sources for which (A) phase difference remains constant (B) amplitude remains constant (C) frequency remains constant (D) both phase difference and frequency remain constant$
›Reveal solutionSolution

Two sources are coherent when they emit waves of the same frequency with a constant (time-independent) phase difference.

For a steady, observable interference pattern the relative phase of the two waves at any point must not change with time. This requires a constant phase difference, which in turn is only possible if the two sources have the same frequency. If the phase difference varied randomly, the bright and dark fringes wou …

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CBSE 2024Set IMPROVEMENT1 markMCQ
$Q. Assertion (A): In a stable interference pattern the positions of maxima and minima do not change with time. Reason (R): For a sustained interference the phase difference must be constant between waves emitting from two sources.$
(a) Both A and R are correct and R is the correct explanation of A.
(b) Both A and R are correct but R is not the correct explanation of A.
(c) A is correct but R is incorrect.
(d) Both A and R are incorrect.
›Reveal solutionSolution

A constant phase difference between the two sources is exactly what keeps the maxima/minima positions fixed in a stable interference pattern.

Assertion (A) is correct: in a stable (sustained) interference pattern, the positions of bright (maxima) and dark (minima) fringes do not shift with time. Reason (R) is also correct: this stability is only possible if the phase difference between the waves from the two sources remains constant in time (i.e. the sources are coherent). R cor …

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CBSE 2024Set A1 markMCQ
$Q. Two sources of monochromatic light is coherent, when their (A) intensities are equal (B) amplitudes are equal (C) phases are equal (D) none of these$
›Reveal solutionSolution

Two monochromatic sources are coherent when they have a constant/equal phase relationship → option (C).

Two sources are coherent if they emit waves of the same frequency and maintain a constant phase difference with time (in the ideal case their phases are equal, i.e. zero constant phase difference). Only then can a steady, observable interference pattern be produce …

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CBSE 2024Set ANNUAL1 mark
$Q. Define Coherent source.$
›Reveal solutionSolution
Sustained interference needs coherent sources: same frequency and a phase relationship that does not drift randomly with time.

Two sources are called coherent if they emit light waves of:

The same frequency (and hence the same wavelength), and

A constant phase difference between them that does not change randomly with time (ideally zero or a fixed value). …
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CBSE 2024Set ANNUAL1 mark
$Q. What type of source is used in Young's double slit experiment?$
›Reveal solutionSolution

YDSE requires a coherent, monochromatic source so that the two slits derived from it maintain a constant phase relationship needed for a stable interference pattern.

In Young's double slit experiment, a single monochromatic (one wavelength/colour, e.g. sodium light or a laser) source illuminates a narrow single slit S; the light diverging from S then falls on the two closely-spaced slits $S_1$ and $S_2$ . Because $S_1$ and $S_2$ are both derived from the same original source, they act as coherent sources - always maintaining a constant phase difference. This coherence and monochromaticity a …

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CBSE 2022Set M1 mark
$Q. What are coherent sources of light?$
›Reveal solutionSolution

Coherent sources: same frequency and a constant (time-independent) phase difference.

Coherent sources of light are two (or more) sources which emit light waves having the same frequency (or wavelength) and a constant phase difference between them that remains unchanged with time. …
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CBSE 2022Set TERM21 markMCQ
$Q. Study the following paragraph and answer Question Nos. 10 and 11 based on it. Consider the needles S1 and S2 moving periodically up and down in an identical fashion in a trough of water. They produce two water waves and at a particular point, the phase difference between the displacements produced by each of the waves does not change with time; when this happens the two sources are said to be coherent. Now, if the two sources are coherent (i.e., if the two needles are going up and down regularly), then the phase difference φ at any point will not change with time and we will have a stable interference pattern; i.e., the positions of maxima and minima will not change with time. When the phase difference between the two vibrating sources changes rapidly with time, we say that the two sources are incoherent and when this happens the intensities just add up. To demonstrate the Phenomena of interference we require two sources which emit radiation:$
(a) of different wavelength
(b) of the same frequency
(c) of nearly the same frequency
(d) of the same frequency and having a definite phase relationship
›Reveal solutionSolution

Sustained interference needs coherent sources: same frequency AND a time-independent phase difference.

As the passage states, a stable interference pattern (fixed positions of maxima and minima) requires the phase difference between the two waves at any point to stay constant in time. Two sources of merely the "same frequency" is not, by itself, enough if their initial phases drift relative to each other; the sources must additionally maintain a **definite (fixed) phase rel …

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CBSE 2022Set ANNUAL1 markMCQ
$Q. For sustained interference of light, we must have$
(a) two real monochromatic sources of same wavelength
(b) two real white sources
(c) two virtual sources obtained from a single monochromatic source
(d) two virtual white sources
›Reveal solutionSolution

Sustained (time-independent, observable) interference needs two coherent sources — sources with a constant phase difference. Two independent real sources can never stay coherent, so coherent sources are always obtained as two virtual images of a single source.

Why coherence is essential

For interference fringes to be sustained (not flickering/washing out), the phase difference $\delta$ between the two overlapping waves at any point must stay constant in time, so that the resultant intensity

$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\delta$

does not itself fluctuate faster than the eye/detector can respond.

Why two independent real sources fail

Ordinary light sources emit light as millions of independent wave-trains from different atoms, each lasting only about $10^{-8}\,\text{s}$ with a random, rapidly-changing phase. Even if two real sources have the same wavelength, their initial phases are uncorrelated and change randomly and independently every $\sim10^{-8}$ s — so $\delta$ randomizes, the cosine term averages to zero over any observable time, and no stationary fringe pattern can be seen.

Why virtual sources from one source work

…

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CBSE 2021Set TERM21 markMCQ
$Q. The coherence of two light sources means that the light waves emitted have:$
(a) same frequency
(b) same intensity
(c) constant phase difference
(d) same velocity
›Reveal solutionSolution

Two sources are coherent when the phase difference between the waves they emit stays constant in time — this is what makes a stable interference pattern possible.

If the relative phase fluctuated randomly with time, the positions of constructive/destructive interference (bright/dark fringes) would shift continuously and wash out to uniform illumination, so no stable fringe pattern (as in Young's double slit) would ever be seen.

…

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CBSE 2020Set NC1 markMCQ
$Q. To demonstrate the phenomenon of interference, we require$
(a) two sources which emit radiations of same frequency
(b) two sources which emit radiations of different wavelengths
(c) two sources which emit radiations of nearly the same frequency
(d) two sources which emit radiations of the same frequency and have a definite phase relationship
›Reveal solutionSolution

A steady interference pattern needs the phase difference between the two waves to stay fixed in time — same frequency alone is not enough without a locked phase relationship.

Reasoning

Two independent ordinary sources of light, even of exactly the same frequency, generally have randomly fluctuating relative phase (since each emits light in independent, uncorrelated wave trains). For a stable interference pattern (fixed positions of maxima and minima), the two sources must be coherent: same frequency and a constant phase difference. Options (a)–(c) …

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CBSE 2016Set ANNUAL1 mark
$Q. What are coherent sources of light?$
›Reveal solutionSolution

Coherent sources have the same frequency and a constant phase difference, so their superposition gives a stable, observable interference pattern.

Definition.

Two sources of light are said to be coherent if they emit waves of exactly the same frequency (and hence wavelength) and maintain a constant phase difference between them at all times.

Why coherence matters.

If two independent sources have random, rapidly-fluctuating phase differences, the positions of bright and dark fringes shift millions of times per second (since typical light sources change phase within nanoseconds), and the eye/detector only sees a time-averaged uniform illumination -- no interference pattern is observed. Only when the phase difference is constant does a stable pattern of maxima and minima persist long enough to be seen.

How coherent sources are produced in practice.

Since two independent, physically separate sources (e.g. two separate bulbs) can never be perfectly coherent, coherent sources are created by splitting the light from a single source into two parts, which then travel different paths and are recombined: …

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Q.Four independent waves are expressed as

Concept understanding — Conditions for Sustained Interference

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Q.Four independent waves are expressed as

Concept and Intuition

Step-by-Step Solution

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