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

Options

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

Answer

To solve the problem, we need to determine the ratio of the intensities of two waves when they arrive in phase and when they arrive 90 degrees out of phase.

### Step-by-Step Solution:

1. **Understanding Wave Parameters**:
We have two waves with the same amplitude \( A \) and frequency. The equations of the waves can be represented as:
- Wave 1: \( Y_1 = A \sin(\omega t - kx) \)
- Wave 2: \( Y_2 = A \sin(\omega t - kx + \phi) \)
where \( \phi \) is the phase difference between the two waves.

2. **Case 1: Waves in Phase** (\( \phi = 0^\circ \)):
When the two waves are in phase, the resultant amplitude \( A_R \) can be calculated using the formula:
\[ A_R = A_1 + A_2 + 2A_1A_2 \cos(0^\circ) \]
Since \( A_1 = A_2 = A \):
\[ A_R = A + A + 2A \cdot A \cdot 1 = 2A + 2A = 4A \]

3. **Calculating Intensity for In-Phase Waves**:
The intensity \( I_1 \) is proportional to the square of the amplitude:
\[ I_1 \propto (A_R)^2 = (4A)^2 = 16A^2 \]

4. **Case 2: Waves 90 Degrees Out of Phase** (\( \phi = 90^\circ \)):
When the two waves are 90 degrees out of phase, the resultant amplitude \( A_R \) is given by:
\[ A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(90^\circ)} \]
Since \( \cos(90^\circ) = 0 \):
\[ A_R = \sqrt{A^2 + A^2} = \sqrt{2A^2} = \sqrt{2}A \]

5. **Calculating Intensity for 90 Degrees Out of Phase**:
The intensity \( I_2 \) is:
\[ I_2 \propto (A_R)^2 = (\sqrt{2}A)^2 = 2A^2 \]

6. **Finding the Ratio of Intensities**:
Now, we can find the ratio of the intensities when the waves are in phase to when they are 90 degrees out of phase:
\[ \text{Ratio} = \frac{I_1}{I_2} = \frac{16A^2}{2A^2} = \frac{16}{2} = 8 \]

### 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 **8**.

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