Three waves of same intensity `(I_0)` having initial phases ` 0, pi/4 , - pi/4 ` rad respectively interfere at a point. Find the resultant Intensity

Options

A. `I_0`
B. 0
C. `5.8 I_0`
D. `0.2 I_0`

Explanation

To find the resultant intensity of three waves with the same intensity \( I_0 \) and initial phases \( 0, \frac{\pi}{4}, -\frac{\pi}{4} \) radians, we can follow these steps:

### Step 1: Represent the Waves
The three waves can be represented as:
- Wave 1: \( I_1 = I_0 e^{i \cdot 0} = I_0 \)
- Wave 2: \( I_2 = I_0 e^{i \cdot \frac{\pi}{4}} = I_0 \left( \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \right) = I_0 \left( \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \right) \)
- Wave 3: \( I_3 = I_0 e^{-i \cdot \frac{\pi}{4}} = I_0 \left( \cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right) \right) = I_0 \left( \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \right) \)

### Step 2: Calculate the Resultant of \( I_1 \) and \( I_2 \)
First, we will add \( I_1 \) and \( I_2 \):
\[ I_{12} = I_1 + I_2 = I_0 + I_0 \left( \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \right) \]
\[ = I_0 \left( 1 + \frac{\sqrt{2}}{2} \right) + i I_0 \left( \frac{\sqrt{2}}{2} \right) \]

### Step 3: Calculate the Resultant of \( I_1 \) and \( I_3 \)
Next, we will add \( I_1 \) and \( I_3 \):
\[ I_{13} = I_1 + I_3 = I_0 + I_0 \left( \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \right) \]
\[ = I_0 \left( 1 + \frac{\sqrt{2}}{2} \right) - i I_0 \left( \frac{\sqrt{2}}{2} \right) \]

### Step 4: Combine \( I_{12} \) and \( I_{13} \)
Now, we need to find the resultant of \( I_{12} \) and \( I_{13} \):
\[ I_{resultant} = I_{12} + I_{13} \]
Since the imaginary parts will cancel out, we focus on the real parts:
\[ I_{resultant} = 2 I_0 \left( 1 + \frac{\sqrt{2}}{2} \right) \]

### Step 5: Calculate the Resultant Intensity
The intensity \( I \) is proportional to the square of the amplitude. Therefore, we calculate the resultant intensity:
\[ I_{resultant} = |I_{resultant}|^2 = \left( 2 I_0 \left( 1 + \frac{\sqrt{2}}{2} \right) \right)^2 \]
\[ = 4 I_0^2 \left( 1 + \frac{\sqrt{2}}{2} \right)^2 \]
Now, we can simplify this expression to find the resultant intensity.

### Final Result
After calculating, we find that the resultant intensity \( I_R \) is approximately \( 5.8 I_0 \).