Two periodic waves of intensities $I_1$ and $I_2$ pass through a region at the same time in the same direction

Question

Two periodic waves of intensities $I_1$ and $I_2$ pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is:

Accepted Answer

1. **Identify the formulas for maximum and minimum intensity:** When two waves of intensities $I_1$ and $I_2$ interfere, the resultant intensity is given by $I = I_1 + I_2 + 2\sqrt{I_1I_2} \cos(\Delta\phi)$ . The maximum intensity (constructive interference) is $I_{\max} = (\sqrt{I_1} + \sqrt{I_2})^2 = I_1 + I_2 + 2\sqrt{I_1I_2}$ . The minimum intensity (destructive interference) is $I_{\min} = (\sqrt{I_1} - \sqrt{I_2})^2 = I_1 + I_2 - 2\sqrt{I_1I_2}$ .
2. **Calculate the sum:** Add the expressions for maximum and minimum intensities.
   $$I_{\max} + I_{\min} = (I_1 + I_2 + 2\sqrt{I_1I_2}) + (I_1 + I_2 - 2\sqrt{I_1I_2})$$
   $$I_{\max} + I_{\min} = 2I_1 + 2I_2 = 2(I_1 + I_2)$$

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