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 of finding the ratio of amplitudes of waves at points A and B, we can follow these steps:

### Step 1: Understand the relationship between intensity and distance
The intensity (I) of sound from a point source is inversely proportional to the square of the distance (R) from the source. This can be expressed mathematically as:
\[ I \propto \frac{1}{R^2} \]

### Step 2: Define the distances for points A and B
Given:
- Distance from the source to point A, \( R_1 = 1 \, \text{m} \)
- Distance from the source to point B, \( R_2 = 2 \, \text{m} \)

### Step 3: Calculate the intensity ratio
Using the relationship from Step 1, we can find the ratio of intensities at points A and B:
\[ \frac{I_1}{I_2} = \frac{R_2^2}{R_1^2} \]
Substituting the values:
\[ \frac{I_1}{I_2} = \frac{(2)^2}{(1)^2} = \frac{4}{1} \]
Thus, the intensity ratio \( \frac{I_1}{I_2} = 4:1 \).

### Step 4: Relate intensity to amplitude
The intensity of a wave is directly proportional to the square of its amplitude (A). This can be expressed as:
\[ I \propto A^2 \]
Thus, we can write the ratio of intensities in terms of amplitudes:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]

### Step 5: Calculate the amplitude ratio
From the intensity ratio calculated in Step 3, we can express the amplitude ratio:
\[ \frac{A_1^2}{A_2^2} = \frac{4}{1} \]
Taking the square root of both sides gives:
\[ \frac{A_1}{A_2} = \sqrt{\frac{4}{1}} = 2 \]
Thus, the ratio of amplitudes \( A_1 : A_2 = 2 : 1 \).

### Final Answer
The ratio of amplitudes of waves at points A to B is:
\[ \boxed{2:1} \]

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