In an experiment, four identical coherent waves of intensity `I_0` are interfered. Calculated the maximum intensity in the interference pattern.

Answer

To solve the problem of finding the maximum intensity in the interference pattern created by four identical coherent waves of intensity \( I_0 \), we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity \( I \) of a wave is directly proportional to the square of its amplitude \( A \):
\[ I \propto A^2 \]

### Step 2: Determine the amplitude of the waves
Since all four waves are identical and coherent, we can denote the amplitude of each wave as \( A \). Therefore, the intensity of each wave can be expressed as:
\[ I_0 \propto A^2 \]

### Step 3: Calculate the resultant amplitude when the waves interfere constructively
When four waves interfere constructively (i.e., they are in phase), the resultant amplitude \( A_{\text{net}} \) is the sum of the individual amplitudes:
\[ A_{\text{net}} = A + A + A + A = 4A \]

### Step 4: Relate the resultant intensity to the resultant amplitude
Using the relationship between intensity and amplitude, we can express the maximum intensity \( I_{\text{max}} \) in terms of the resultant amplitude:
\[ I_{\text{max}} \propto (A_{\text{net}})^2 = (4A)^2 = 16A^2 \]

### Step 5: Substitute the expression for amplitude in terms of intensity
Since \( A^2 \) can be replaced using the intensity of one wave:
\[ A^2 \propto I_0 \]
Thus,
\[ I_{\text{max}} \propto 16A^2 = 16 \cdot I_0 \]

### Step 6: Write the final expression for maximum intensity
Therefore, the maximum intensity \( I_{\text{max}} \) in the interference pattern is:
\[ I_{\text{max}} = 16 I_0 \]

### Final Answer
The maximum intensity in the interference pattern created by four identical coherent waves of intensity \( I_0 \) is:
\[ \boxed{16 I_0} \]