Question

Four independent waves are expressed as \[ (i)\; y_1=A_1\sin\omega t, \] \[ (ii)\; y_2=A_2\sin 2\omega t, \] \[ (iii)\; y_3=A_3\cos\omega t, \] \[ (iv)\; y_4=A_4\sin\left(\omega t+\frac{\pi}{3}\right) \] 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

Hint:

For interference, remember the keyword: \[ \boxed{\text{Same frequency + Constant phase difference}} \] Different frequencies do not produce sustained interference.

Answer 1

The Correct Option is C

Concept: For sustained interference, the two waves must be coherent. The conditions for coherence are:
• Same frequency.
• Constant phase difference. Only waves having the same angular frequency can interfere.

Step 1: Examine the frequency of each wave.
Wave (i): \[ y_1=A_1\sin\omega t \] has angular frequency \(\omega\). Wave (ii): \[ y_2=A_2\sin2\omega t \] has angular frequency \(2\omega\). Wave (iii): \[ y_3=A_3\cos\omega t \] has angular frequency \(\omega\). Wave (iv): \[ y_4=A_4\sin\left(\omega t+\frac{\pi}{3}\right) \] has angular frequency \(\omega\).

Step 2: Identify the waves with identical frequencies.
Waves (i), (iii), and (iv) all have angular frequency \(\omega\). Also, \[ \cos\omega t=\sin\left(\omega t+\frac{\pi}{2}\right), \] which means wave (iii) differs from wave (i) only by a constant phase difference. Similarly, wave (iv) also has a constant phase difference. Hence, waves (i), (iii), and (iv) can interfere. Wave (ii) cannot interfere with the others because its frequency is different. Therefore, \[ \boxed{\text{(i), (iii) and (iv) only}} \]