The maximum intensities produced by two coherent waves of intensity `I_(1)` and `I_(2)` will be:

Options

A. `I_(1) + I_(2)`
B. `I_(1)^(2) + I_(2)^(2)`
C. `I_(1) + I_(2) + 2sqrt(I_(1)I_(2))`
D. zero

Answer

To find the maximum intensity produced by two coherent waves with intensities \( I_1 \) and \( I_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Coherent Waves**:
Coherent waves are waves that have a constant phase difference and the same frequency. The intensity of a wave is related to the square of its amplitude.

2. **Relating Intensity to Amplitude**:
The intensity \( I \) of a wave is given by the formula:
\[ I = A^2 \]
where \( A \) is the amplitude of the wave. Therefore, if the intensities of the two waves are \( I_1 \) and \( I_2 \), their amplitudes can be expressed as:
\[ A_1 = \sqrt{I_1} \quad \text{and} \quad A_2 = \sqrt{I_2} \]

3. **Net Amplitude Calculation**:
When two coherent waves interfere, the resultant amplitude \( A \) can be calculated using the formula:
\[ A = A_1 + A_2 + 2 \sqrt{A_1 A_2} \cos(\theta) \]
where \( \theta \) is the phase difference between the two waves.

4. **Finding Maximum Intensity**:
To find the maximum intensity, we need to maximize the amplitude. The maximum value of \( \cos(\theta) \) is 1 (which occurs during constructive interference). Thus, we set \( \cos(\theta) = 1 \):
\[ A_{\text{max}} = A_1 + A_2 + 2 \sqrt{A_1 A_2} \]
Substituting the expressions for \( A_1 \) and \( A_2 \):
\[ A_{\text{max}} = \sqrt{I_1} + \sqrt{I_2} + 2 \sqrt{\sqrt{I_1} \sqrt{I_2}} = \sqrt{I_1} + \sqrt{I_2} + 2 \sqrt{I_1 I_2} \]

5. **Calculating Maximum Intensity**:
The maximum intensity \( I_{\text{max}} \) is given by:
\[ I_{\text{max}} = A_{\text{max}}^2 \]
Therefore:
\[ I_{\text{max}} = \left( \sqrt{I_1} + \sqrt{I_2} + 2 \sqrt{I_1 I_2} \right)^2 \]

6. **Final Expression**:
Expanding the square gives:
\[ I_{\text{max}} = I_1 + I_2 + 4 \sqrt{I_1 I_2} \]

### Conclusion:
The maximum intensity produced by two coherent waves of intensity \( I_1 \) and \( I_2 \) is:
\[ I_{\text{max}} = I_1 + I_2 + 4 \sqrt{I_1 I_2} \]