Five different waves travelling in the same direction are represented by:
(A) \( y_1=a_1\sin\omega t \)
(B) \( y_2=a_2\sin2\omega t \)
(C) \( y_3=a_3\sin3\omega t \)
(D) \( y_4=a_4\cos\omega t \)
(E) \( y_5=a_5\sin(\omega t+\pi/3) \)
Choose the correct answer from the options given below for interference to take place:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (E), (A) and (D), (D) and (E)
Sustained interference needs coherent waves: the waves must have the same frequency and a constant phase difference between them. Read the angular frequency of each wave.
(A) \( y_1=a_1\sin\omega t \) — frequency \( \omega \).
(B) \( y_2=a_2\sin2\omega t \) — frequency \( 2\omega \).
(C) \( y_3=a_3\sin3\omega t \) — frequency \( 3\omega \).
(D) \( y_4=a_4\cos\omega t = a_4\sin(\omega t+\pi/2) \) — frequency \( \omega \).
(E) \( y_5=a_5\sin(\omega t+\pi/3) \) — frequency \( \omega \).
Only (A), (D) and (E) share the frequency \( \omega \), and their phase differences (\( \pi/2 \), \( \pi/3 \) and \( \pi/6 \)) are constant. So the pairs that can interfere are (A) and (E), (A) and (D), and (D) and (E). Waves (B) and (C) have different frequencies from everything else, so they cannot form a steady interference pattern with any other wave.