The intensity ratio of two waves is `1 : 16`. The ratio of their amplitudes is

Answer

To solve the problem of finding the ratio of the amplitudes of two waves given their intensity ratio of 1:16, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Intensity and Amplitude**:
The intensity \( I \) of a wave is related to the amplitude \( A \) by the formula:
\[ I \propto A^2 \]
This means that the intensity is proportional to the square of the amplitude.

2. **Set Up the Intensity Ratio**:
Given the intensity ratio of the two waves as:
\[ \frac{I_1}{I_2} = \frac{1}{16} \]
We can express this in terms of their amplitudes:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]

3. **Substitute the Intensity Values**:
From the intensity ratio, we substitute:
\[ \frac{A_1^2}{A_2^2} = \frac{1}{16} \]

4. **Cross-Multiply to Relate Amplitudes**:
Rearranging gives us:
\[ A_1^2 = \frac{1}{16} A_2^2 \]

5. **Take the Square Root**:
To find the ratio of the amplitudes, we take the square root of both sides:
\[ \frac{A_1}{A_2} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

6. **Write the Final Ratio**:
Thus, the ratio of the amplitudes \( A_1 : A_2 \) is:
\[ A_1 : A_2 = 1 : 4 \]

### Final Answer:
The ratio of their amplitudes is \( 1 : 4 \).

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