When a train approaches a stationary observer, the apparent frequency of the whistle is `n'` and when the same train recedes away from the observer, the apparent frequency is `n''`. Then the apperent frquency `n` when the observer sitting in the train is :

Options

A. (a)`n=(n'+n)/(2)`
B. (b)`n=sqrt(n'n'')`
C. (c)`n=(2n'n'')/(n'+n'')`
D. (d)`n=(2n'n'')/(n'-n'')`

Answer

To solve the problem, we need to analyze the situation using the Doppler effect for sound. We have two cases to consider: when the train approaches the observer and when it recedes from the observer.

### Step-by-Step Solution:

1. **Understanding the Doppler Effect**:
- When a source of sound moves towards a stationary observer, the apparent frequency increases. Conversely, when the source moves away, the apparent frequency decreases.

2. **Case 1 - Train Approaching the Observer**:
- Let the natural frequency of the whistle be \( n \).
- The apparent frequency when the train approaches the observer is given by:
\[ n' = \frac{v}{v - v_s} \]
- Here, \( v \) is the speed of sound, and \( v_s \) is the speed of the train (source).

3. **Case 2 - Train Receding from the Observer**:
- The apparent frequency when the train recedes is given by:
\[ n'' = \frac{v}{v + v_s} \]

4. **Finding the Relationship Between Frequencies**:
- From the first case, we can rearrange the equation to express \( n \):
\[ n = n' \left( \frac{v - v_s}{v} \right) \]
- From the second case, we can express \( n \) as:
\[ n = n'' \left( \frac{v + v_s}{v} \right) \]

5. **Setting Up the Equations**:
- We have two equations for \( n \):
\[ n = \frac{n' (v - v_s)}{v} \quad \text{(1)} \]
\[ n = \frac{n'' (v + v_s)}{v} \quad \text{(2)} \]

6. **Equating the Two Expressions for \( n \)**:
- Set equation (1) equal to equation (2):
\[ \frac{n' (v - v_s)}{v} = \frac{n'' (v + v_s)}{v} \]

7. **Cross-Multiplying**:
- Cross-multiply to eliminate \( v \):
\[ n' (v - v_s) = n'' (v + v_s) \]

8. **Expanding and Rearranging**:
- Expand both sides:
\[ n' v - n' v_s = n'' v + n'' v_s \]
- Rearranging gives:
\[ (n' - n'') v = (n' + n'') v_s \]

9. **Solving for \( v_s \)**:
- From the above equation, we can express \( v_s \):
\[ v_s = \frac{(n' - n'') v}{(n' + n'')} \]

10. **Finding the Apparent Frequency \( n \)**:
- Substitute \( v_s \) back into either expression for \( n \):
\[ n = \frac{n' (v - v_s)}{v} \]
- After simplification, we find:
\[ n = \frac{2 n' n''}{n' + n''} \]

### Final Answer:
The apparent frequency \( n \) when the observer is sitting in the train is given by:
\[ n = \frac{2 n' n''}{n' + n''} \]