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 an intensity ratio of 9:1 superimpose, we can follow these steps:

### Step 1: Define the Intensities
Let the intensities of the two waves be:
- \( I_1 = 9I \)
- \( I_2 = I \)

### Step 2: Calculate Maximum Intensity
The formula for maximum intensity \( I_{\text{max}} \) when two waves superimpose is given by:
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{\text{max}} = 9I + I + 2\sqrt{9I \cdot I} \]
\[ I_{\text{max}} = 10I + 2\sqrt{9I^2} \]
\[ I_{\text{max}} = 10I + 6I = 16I \]

### Step 3: Calculate Minimum Intensity
The formula for minimum intensity \( I_{\text{min}} \) is given by:
\[ I_{\text{min}} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{\text{min}} = 9I + I - 2\sqrt{9I \cdot I} \]
\[ I_{\text{min}} = 10I - 6I = 4I \]

### Step 4: Calculate the Ratio of Maximum to Minimum Intensity
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{16I}{4I} = 4 \]
Thus, the ratio of maximum to minimum intensity is:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = 4:1 \]

### Conclusion
The final answer is that the ratio of maximum to minimum intensity when the two waves superimpose is \( 4:1 \).

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