Question

Two sound waves each of intensity I are superimposed. If the phase difference between the waves is \( \frac{\pi}{2} \), then the intensity of the resultant wave is

• A. 2 I
• B. 3 I
• C. 4 I
• D. I

Hint:

For equal intensities \( I_0 \), the resultant intensity is \( 4I_0 \cos^2(\phi/2) \). Here \( \phi = 90^\circ \), so \( 4I_0 (\frac{1}{\sqrt{2}})^2 = 2I_0 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:

When two coherent waves interfere, the resultant intensity depends on their individual intensities and the phase difference between them.
Step 2: Key Formula or Approach:

Resultant Intensity Formula: \[ I_{res} = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi \]
Step 3: Detailed Explanation:

Given: \( I_1 = I \), \( I_2 = I \) Phase difference \( \phi = \frac{\pi}{2} \) Substitute into the formula: \[ I_{res} = I + I + 2\sqrt{I \cdot I} \cos\left(\frac{\pi}{2}\right) \] Since \( \cos(\frac{\pi}{2}) = 0 \): \[ I_{res} = I + I + 0 = 2I \]
Step 4: Final Answer:

The resultant intensity is 2I.