If A is the amplitude of the wave coming from a line source at a distance r, then :

Options

A. `A prop r^(- 1/2)`
B. `A prop r^(-1)`
C. `A prop r^(2)`
D. `A prop r^(1)`

Answer

To solve the problem, we need to establish the relationship between the amplitude \( A \) of a wave coming from a line source and the distance \( r \) from that source.

### Step-by-Step Solution:

1. **Understanding the Source Types**:
- We differentiate between a point source and a line source. A point source radiates in all directions, while a line source emits waves along its length.

2. **Amplitude and Intensity Relationship**:
- For a point source, the intensity \( I \) is inversely proportional to the square of the distance \( r \):
\[ I \propto \frac{1}{r^2} \]
- The amplitude \( A \) is related to intensity by:
\[ A \propto \sqrt{I} \]
- Therefore, for a point source:
\[ A \propto \sqrt{\frac{1}{r^2}} \Rightarrow A \propto \frac{1}{r} \]

3. **For a Line Source**:
- For a line source, the intensity \( I \) is inversely proportional to the distance \( r \):
\[ I \propto \frac{1}{r} \]
- Again, using the relationship between amplitude and intensity:
\[ A \propto \sqrt{I} \]
- Thus, for a line source:
\[ A \propto \sqrt{\frac{1}{r}} \Rightarrow A \propto \frac{1}{\sqrt{r}} \Rightarrow A \propto r^{-1/2} \]

4. **Conclusion**:
- Therefore, the amplitude \( A \) of the wave coming from a line source at a distance \( r \) can be expressed as:
\[ A \propto r^{-1/2} \]
- This means that the correct answer is that the amplitude varies as \( r^{-1/2} \).

### Final Answer:
The amplitude \( A \) varies as \( r^{-1/2} \).