Waves from two different sources overlap near a particular point. The amplitude and the frequency of the two waves are same. The ratio of the intensity when the two waves arrive in phase to that when they arrive `90^(@)` out phase is

Options

A. `1:1`
B. `sqrt2` : 1
C. `2:1`
D. `4:1`

Answer

To solve the problem, we need to find the ratio of the intensity of two waves when they arrive in phase to that when they arrive 90 degrees out of phase. Let's break down the solution step by step.

### Step 1: Understand the Intensity of Waves
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \). For two waves with the same amplitude and frequency, we can denote their intensities as \( I_1 = I \) and \( I_2 = I \).

### Step 2: Resultant Intensity When Waves are In Phase
When two waves arrive in phase, the phase difference \( \phi \) is \( 0 \) degrees. The resultant intensity \( I_R \) can be calculated using the formula:
\[ I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
Substituting \( I_1 = I \), \( I_2 = I \), and \( \phi = 0 \):
\[ I_R = I + I + 2\sqrt{I \cdot I} \cos(0) \]
\[ I_R = 2I + 2I = 4I \]

### Step 3: Resultant Intensity When Waves are 90 Degrees Out of Phase
When the two waves arrive 90 degrees out of phase, the phase difference \( \phi \) is \( 90 \) degrees. Using the same formula:
\[ I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
Substituting \( I_1 = I \), \( I_2 = I \), and \( \phi = 90 \):
\[ I_R = I + I + 2\sqrt{I \cdot I} \cos(90) \]
Since \( \cos(90) = 0 \):
\[ I_R = 2I + 0 = 2I \]

### Step 4: Calculate the Ratio of Intensities
Now we can find the ratio of the intensity when the waves are in phase to when they are 90 degrees out of phase:
\[ \text{Ratio} = \frac{I_{R \text{ (in phase)}}}{I_{R \text{ (90 degrees out of phase)}}} = \frac{4I}{2I} = 2 \]

### Final Answer
The ratio of the intensity when the two waves arrive in phase to that when they arrive 90 degrees out of phase is \( 2:1 \).

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