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

Options

A. `10:8`
B. `9:1`
C. `4:1`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of maximum to minimum intensity when two waves with a given intensity ratio superimpose. Here's a step-by-step solution:

### Step 1: Understand the relationship between intensity and amplitude
The intensity \( I \) of a wave is directly proportional to the square of its amplitude \( A \):
\[ I \propto A^2 \]

### Step 2: Set up the ratio of intensities
Given the ratio of intensities of two waves is \( I_1 : I_2 = 9 : 1 \), we can express this as:
\[ \frac{I_1}{I_2} = \frac{9}{1} \]

### Step 3: Find the ratio of amplitudes
Since intensity is proportional to the square of the amplitude, we can find the ratio of the amplitudes \( A_1 \) and \( A_2 \):
\[ \frac{A_1^2}{A_2^2} = \frac{I_1}{I_2} = \frac{9}{1} \]
Taking the square root of both sides:
\[ \frac{A_1}{A_2} = \sqrt{\frac{9}{1}} = \frac{3}{1} \]

### Step 4: Calculate maximum and minimum intensity
The maximum intensity \( I_{\text{max}} \) when the two waves superimpose is given by:
\[ I_{\text{max}} = (A_1 + A_2)^2 \]
The minimum intensity \( I_{\text{min}} \) is given by:
\[ I_{\text{min}} = (A_1 - A_2)^2 \]

### Step 5: Substitute the values of amplitudes
Let \( A_2 = A \). Then, \( A_1 = 3A \). Now substituting these into the equations for maximum and minimum intensity:
\[ I_{\text{max}} = (3A + A)^2 = (4A)^2 = 16A^2 \]
\[ I_{\text{min}} = (3A - A)^2 = (2A)^2 = 4A^2 \]

### Step 6: Find the ratio of maximum to minimum intensity
Now we can find the ratio:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{16A^2}{4A^2} = \frac{16}{4} = 4 \]

### Conclusion
Thus, the ratio of maximum to minimum intensity is:
\[ \text{Ratio} = 4 : 1 \]