Two sinusoidal plane waves of same frequency having intensities `I_(0)` and `4I_(0)` are travelling in the same direction. The resultant intensity at a point at which waves meet with a phase difference of zero radian is

Options

A. `I_(0)`
B. `5I_(0)`
C. `9I_(0)`
D. `3I_(0)`

Answer

To solve the problem of finding the resultant intensity of two sinusoidal plane waves with intensities \( I_1 = I_0 \) and \( I_2 = 4I_0 \) that are traveling in the same direction with a phase difference of zero radians, we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \):
\[ I \propto A^2 \]
This means that if we know the intensities of the waves, we can find their amplitudes.

### Step 2: Calculate the amplitudes of the two waves
Let the amplitudes of the two waves be \( A_1 \) and \( A_2 \). From the relationship between intensity and amplitude, we have:
\[ I_1 = k A_1^2 \quad \text{and} \quad I_2 = k A_2^2 \]
where \( k \) is a constant. Therefore, we can express the amplitudes as:
\[ A_1 = \sqrt{\frac{I_1}{k}} = \sqrt{\frac{I_0}{k}} \quad \text{and} \quad A_2 = \sqrt{\frac{I_2}{k}} = \sqrt{\frac{4I_0}{k}} = 2\sqrt{\frac{I_0}{k}} \]

### Step 3: Use the formula for resultant amplitude
When two waves interfere, the resultant amplitude \( A_{net} \) when they meet with a phase difference \( \phi = 0 \) is given by:
\[ A_{net} = A_1 + A_2 \]
Substituting the values of \( A_1 \) and \( A_2 \):
\[ A_{net} = \sqrt{\frac{I_0}{k}} + 2\sqrt{\frac{I_0}{k}} = 3\sqrt{\frac{I_0}{k}} \]

### Step 4: Calculate the resultant intensity
Now, the resultant intensity \( I_{net} \) can be calculated using the resultant amplitude:
\[ I_{net} = k A_{net}^2 \]
Substituting \( A_{net} \):
\[ I_{net} = k \left(3\sqrt{\frac{I_0}{k}}\right)^2 = k \cdot 9 \cdot \frac{I_0}{k} = 9I_0 \]

### Final Result
Thus, the resultant intensity at the point where the two waves meet is:
\[ \boxed{9I_0} \]