The ratio of the intensities two waves is 1 : 9 The ratio of their amplitudes is

Options

A. (a)`3:1`
B. (b)`1:3`
C. (c)`1:9`
D. (d)`9:1`

Answer

To solve the problem of finding the ratio of the amplitudes of two waves given the ratio of their intensities, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Intensity and Amplitude**:
The intensity \( I \) of a wave is directly proportional to the square of its amplitude \( A \). This can be expressed mathematically as:
\[ I \propto A^2 \]
Therefore, we can write:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]

2. **Use the Given Ratio of Intensities**:
We are given that the ratio of the intensities of the two waves is:
\[ \frac{I_1}{I_2} = \frac{1}{9} \]

3. **Set Up the Equation for Amplitudes**:
From the relationship established, we can substitute the intensity ratio into the equation:
\[ \frac{1}{9} = \frac{A_1^2}{A_2^2} \]

4. **Cross-Multiply to Solve for Amplitude Ratio**:
Rearranging the equation gives us:
\[ A_1^2 = \frac{1}{9} A_2^2 \]

5. **Take the Square Root of Both Sides**:
Taking the square root of both sides to find the amplitude ratio:
\[ A_1 = \frac{1}{3} A_2 \]

6. **Express the Ratio of Amplitudes**:
Therefore, the ratio of the amplitudes \( A_1 \) to \( A_2 \) can be expressed as:
\[ \frac{A_1}{A_2} = \frac{1}{3} \]

### Final Answer:
The ratio of the amplitudes of the two waves is:
\[ \frac{A_1}{A_2} = \frac{1}{3} \]