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

Options

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

Answer

To solve the problem, we need to find the ratio of the amplitudes of two waves given the ratio of their intensities.

### 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:
\[ I \propto A^2 \]
This means that if we have two waves, the ratio of their intensities can be related to the ratio of their amplitudes.

2. **Set up the ratio of intensities**:
According to the problem, the ratio of the intensities of the two waves is given as:
\[ \frac{I_1}{I_2} = \frac{1}{9} \]

3. **Relate the intensities to the amplitudes**:
Using the relationship established earlier, we can write:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
Substituting the known ratio of intensities into this equation gives:
\[ \frac{A_1^2}{A_2^2} = \frac{1}{9} \]

4. **Take the square root to find the ratio of amplitudes**:
To find the ratio of the amplitudes, we take the square root of both sides:
\[ \frac{A_1}{A_2} = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

5. **Conclusion**:
Thus, the ratio of the amplitudes of the two waves is:
\[ \frac{A_1}{A_2} = \frac{1}{3} \]

### Final Answer:
The ratio of the amplitudes of the two waves is \( 1 : 3 \).