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. `5.8 I_(0)`
B. `0.2 I_(0)`
C. `3 I_(0)`
D. `I_(0)`

Answer

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 follows:
- 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 Waves 2 and 3
To find the resultant of \( I_2 \) and \( I_3 \):
\[ I_2 + I_3 = I_0 \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) + I_0 \left( \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right) \]
The imaginary parts cancel out:
\[ I_2 + I_3 = I_0 \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \right) = I_0 \sqrt{2} \]

### Step 3: Resultant of All Three Waves
Now we need to find the resultant of \( I_1 \) and the resultant of \( I_2 + I_3 \):
\[ I_{\text{resultant}} = I_1 + (I_2 + I_3) = I_0 + I_0 \sqrt{2} \]
This can be expressed as:
\[ I_{\text{resultant}} = I_0 (1 + \sqrt{2}) \]

### Step 4: Calculate the Intensity
The intensity \( I \) is related to the amplitude \( A \) by the formula:
\[ I \propto A^2 \]
Thus, if we denote the resultant amplitude as \( A_{\text{resultant}} \):
\[ A_{\text{resultant}} = I_0 (1 + \sqrt{2}) \]
The resultant intensity \( I_{\text{resultant}} \) will be:
\[ I_{\text{resultant}} = \left( A_{\text{resultant}} \right)^2 = \left( I_0 (1 + \sqrt{2}) \right)^2 = I_0^2 (1 + \sqrt{2})^2 \]

### Step 5: Expand and Simplify
Now, we expand \( (1 + \sqrt{2})^2 \):
\[ (1 + \sqrt{2})^2 = 1 + 2 + 2\sqrt{2} = 3 + 2\sqrt{2} \]
Thus, the resultant intensity is:
\[ I_{\text{resultant}} = I_0^2 (3 + 2\sqrt{2}) \]

### Step 6: Final Result
The final expression for the resultant intensity is:
\[ I_{\text{resultant}} = (3 + 2\sqrt{2}) I_0 \]