Case: Read the passage and answer the question.
A wave is represented by a variable y which gives the displacement of the particles of the medium in which the wave is travelling at any given instant of time. In the same context, consider two waves y1 and y2 which superpose such that y gives the resultant of the two waves. By the resultant of the two waves we mean that y gives the vector sum of the displacements due to each of the individual waves.
If x is the path difference between two waves of wavelength λ that interfere constructively, what is x?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( n\lambda,\ n=0,1,2,3,\ldots \)
Two waves interfere constructively where they arrive in phase, that is, where their phase difference is \( 0, 2\pi, 4\pi, \ldots \). A path difference of one wavelength corresponds to a phase difference of \( 2\pi \), so a path difference \( x \) gives a phase difference \( 2\pi x/\lambda \).
For the waves to be in phase:
\( \displaystyle \frac{2\pi}{\lambda}\,x=2n\pi \)
\( \Rightarrow\; x=n\lambda \)
Option (1) \( \lambda \) — Incorrect. A path difference of one wavelength does give constructive interference. But it is only the case \( n=1 \). Points with path difference \( 0, 2\lambda, 3\lambda, \ldots \) are also bright.
Option (2) \( n\lambda,\ n=0,1,2,3,\ldots \) — Correct. The condition holds for every whole number of wavelengths. The central point (\( n=0 \)) is included. At all these points the resultant intensity is maximum.
Option (3) \( \lambda/2 \) — Incorrect. A path difference of half a wavelength gives a phase difference of \( \pi \). The waves then arrive in opposite phase and interfere destructively.
Option (4) \( n\lambda/2,\ n=0,1,2,3,\ldots \) — Incorrect. For odd \( n \) this gives \( \lambda/2, 3\lambda/2, \ldots \). At these path differences the waves arrive in opposite phase and interfere destructively.
NCERT (2026-27):
Coherent and Incoherent Addition of Waves – “To summarise: If we have two coherent sources S1 and S2 vibrating in phase, then for an arbitrary point P whenever the path difference,
\( S_1P\sim S_2P=n\lambda\quad(n=0,1,2,3,\ldots) \)\( \qquad (10.9) \)
we will have constructive interference and the resultant intensity will be 4I0; the sign ~ between S1P and S2P represents the difference between S1P and S2P.”