A point source of sound is placed in a non-absorbing medium two points A and B are at the distance of 1 m and 2 m, respectively, from the source. The ratio of amplitudes of waves at A to B is

Options

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

Explanation

To solve the problem step by step, we need to find the ratio of the amplitudes of sound waves at points A and B, which are at distances of 1 m and 2 m from a point source of sound, respectively.

### 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. **Define the Distances:**
Let:
- Distance from the source to point A, \( r_A = 1 \, \text{m} \)
- Distance from the source to point B, \( r_B = 2 \, \text{m} \)

3. **Intensity Ratio Calculation:**
Since intensity is inversely proportional to the square of the distance from the source, we can express the intensity at points A and B as follows:
\[ I_A \propto \frac{P}{r_A^2} \quad \text{and} \quad I_B \propto \frac{P}{r_B^2} \]
where \( P \) is the power of the source.

Therefore, the ratio of intensities at points A and B is:
\[ \frac{I_A}{I_B} = \frac{P/r_A^2}{P/r_B^2} = \frac{r_B^2}{r_A^2} \]

4. **Substituting the Distances:**
Now substituting the distances:
\[ \frac{I_A}{I_B} = \frac{(2)^2}{(1)^2} = \frac{4}{1} \]

5. **Finding the Amplitude Ratio:**
Since \( I \propto A^2 \), we can write:
\[ \frac{A_A^2}{A_B^2} = \frac{I_A}{I_B} = 4 \]
Taking the square root to find the ratio of amplitudes:
\[ \frac{A_A}{A_B} = \sqrt{4} = 2 \]

6. **Final Ratio:**
Thus, the ratio of the amplitudes of waves at points A and B is:
\[ \frac{A_A}{A_B} = 2:1 \]

### Conclusion:
The ratio of amplitudes of waves at points A to B is \( 2:1 \).

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