The amplitude ratio of two superposing waves 3:1. what is the ratio of the maximum and minimum intensities?

Options

A. `1:1`
B. `9:1`
C. `4:1`
D. `3:1`

Explanation

To solve the problem of finding the ratio of maximum and minimum intensities of two superimposing waves with an amplitude ratio of 3:1, we can follow these steps:

### Step 1: Define the Amplitudes
Let the amplitudes of the two waves be:
- \( A_1 = 3 \) (for the first wave)
- \( A_2 = 1 \) (for the second wave)

### Step 2: Write the Formulas for Maximum and Minimum Intensities
The formulas for the maximum and minimum intensities when two waves interfere are given by:
- Maximum Intensity (\( I_{max} \)):
\[ I_{max} = K (A_1 + A_2)^2 \]
- Minimum Intensity (\( I_{min} \)):
\[ I_{min} = K (A_1 - A_2)^2 \]
Where \( K \) is a constant.

### Step 3: Calculate the Maximum Intensity
Substituting the values of \( A_1 \) and \( A_2 \) into the formula for maximum intensity:
\[ I_{max} = K (3 + 1)^2 = K (4)^2 = K \cdot 16 \]

### Step 4: Calculate the Minimum Intensity
Now, substituting the values into the formula for minimum intensity:
\[ I_{min} = K (3 - 1)^2 = K (2)^2 = K \cdot 4 \]

### Step 5: Find the Ratio of Maximum to Minimum Intensities
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{max}}{I_{min}} = \frac{K \cdot 16}{K \cdot 4} \]
The \( K \) cancels out:
\[ \frac{I_{max}}{I_{min}} = \frac{16}{4} = 4 \]

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