Question

Following are expressions for four plane simple harmonic waves $$ y_1 = A \cos 2 \pi \left( n_1 t + \frac{x}{\lambda_1} \right), \quad y_2 = A \cos 2 \pi \left( n_1 t + \frac{x}{\lambda_1} + \pi \right), \quad y_3 = A \cos 2 \pi \left( n_2 t + \frac{x}{\lambda_2} \right), \quad y_4 = A \cos 2 \pi \left( n_2 t - \frac{x}{\lambda_2} \right) $$ The pairs of waves which will produce destructive interference and stationary waves respectively in a medium, are 

• A. (iii), (iv), (i), (ii)
• B. (i), (iii), (ii), (iv)
• C. (i), (iv), (ii), (iii)
• D. (i), (ii), (iii), (iv)

Hint:

For stationary waves, the waveforms need to have a fixed phase difference of either 0 or \( \pi \). For destructive interference, the phase difference between the waves should be \( \pi \).

Answer 1

The Correct Option is D

Step 1: Destructive interference occurs when two waves of the same frequency and amplitude are exactly out of phase. Here, the phase difference between waves (i) and (ii) is \( \pi \), which will result in destructive interference. 
Step 2: The waves with phase differences of \( 0 \) or \( \pi \) will result in stationary waves. The waves (i) and (ii) will combine to form stationary waves due to their phase relationship. Similarly, waves (iii) and (iv) will also form stationary waves. 
Step 3: Therefore, the correct pairs are (i), (ii), (iii), and (iv) for destructive interference and stationary waves.