The ratio of intensities of two waves is `9 : 1` When they superimpose, the ratio of maximum to minimum intensity will become :-

Options

A. `4 : 1`
B. `3 : 1`
C. `2 : 1`
D. `1 : 1`

Explanation

To solve the problem of finding the ratio of maximum to minimum intensity when two waves with a given intensity ratio superimpose, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Intensities**:
Let the intensities of the two waves be \( I_1 \) and \( I_2 \). According to the problem, the ratio of the intensities is given as:
\[ I_1 : I_2 = 9 : 1 \]
This can be expressed as:
\[ I_1 = 9k \quad \text{and} \quad I_2 = k \]
where \( k \) is a constant.

2. **Calculate the Maximum and Minimum Intensities**:
When two waves superimpose, the maximum intensity \( I_{max} \) and minimum intensity \( I_{min} \) can be calculated using the following formulas:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

3. **Calculate \( I_{max} \)**:
Substitute \( I_1 \) and \( I_2 \) into the formula for maximum intensity:
\[ I_{max} = (\sqrt{9k} + \sqrt{k})^2 = (3\sqrt{k} + \sqrt{k})^2 = (4\sqrt{k})^2 = 16k \]

4. **Calculate \( I_{min} \)**:
Now substitute \( I_1 \) and \( I_2 \) into the formula for minimum intensity:
\[ I_{min} = (\sqrt{9k} - \sqrt{k})^2 = (3\sqrt{k} - \sqrt{k})^2 = (2\sqrt{k})^2 = 4k \]

5. **Find the Ratio of Maximum to Minimum Intensity**:
Now we can find the ratio of maximum to minimum intensity:
\[ \frac{I_{max}}{I_{min}} = \frac{16k}{4k} = 4 \]

6. **Express the Ratio**:
Therefore, the ratio of maximum to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = 4 : 1 \]

### Final Answer:
The ratio of maximum to minimum intensity when the two waves superimpose is **4 : 1**.