The ratio of intensities of two waves are given by `4:1`. The ratio of the amplitudes of the two waves is

Options

A. `2:1`
B. `1:2`
C. `4:1`
D. `1:4`

Explanation

To solve the problem, we need to find the ratio of the amplitudes of two waves given that the ratio of their intensities is 4:1.

### 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:
\[ I \propto A^2 \]

2. **Express the ratio of intensities**:
Given the ratio of the intensities of the two waves is:
\[ \frac{I_1}{I_2} = \frac{4}{1} \]

3. **Relate the intensities to amplitudes**:
Since intensity is proportional to the square of the amplitude, we can write:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
Therefore, substituting the intensity ratio:
\[ \frac{A_1^2}{A_2^2} = \frac{4}{1} \]

4. **Take the square root to find the amplitude ratio**:
To find the ratio of the amplitudes, we take the square root of both sides:
\[ \frac{A_1}{A_2} = \sqrt{\frac{4}{1}} = \sqrt{4} = 2 \]

5. **Express the ratio in a standard form**:
Thus, the ratio of the amplitudes \( A_1 : A_2 \) is:
\[ A_1 : A_2 = 2 : 1 \]

### Final Answer:
The ratio of the amplitudes of the two waves is \( 2 : 1 \).

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