Q.(a) Define coherent sources.

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).

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 …

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 …

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 …

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:

1. The same frequency (and hence the same wavelength), and
2. A constant phase difference between them that does not change randomly with time (ideally zero or a fixed value). …

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 …

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

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 …

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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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.

…

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) …

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: …