The displacements of four light waves are given below. If the amplitudes \( a_1 \) and \( a_2 \) are nearly equal, then the interference fringes may be observed due to superposition of
(A) \( y = a_1\sin\omega t \)
(B) \( y = a_2\sin(\omega t + \phi) \)
(C) \( y = a_1\sin 2\omega t \)
(D) \( y = a_2\sin 3(\omega t + \phi) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
Steady interference fringes need coherent waves: the two waves must have the same frequency and a constant phase difference. Nearly equal amplitudes then give fringes of good contrast.
(A) \( y = a_1\sin\omega t \) has angular frequency \( \omega \).
(B) \( y = a_2\sin(\omega t + \phi) \) also has angular frequency \( \omega \); it differs from (A) only by a constant phase \( \phi \), so (A) and (B) are coherent.
(C) \( y = a_1\sin 2\omega t \) has angular frequency \( 2\omega \), different from (A) and (B).
(D) \( y = a_2\sin 3(\omega t + \phi) \) has angular frequency \( 3\omega \), different from (A) and (B).
Only (A) and (B) can give steady fringes, which is Option (1).
NCERT (2026-27):
Coherent and Incoherent Addition of Waves – “… 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.”