A fixed source of sound emitting a certain frequency appears as `f_(a)` when the observer is apporoaching the source with `upsilon_(0)` and `f_(r)` when the observer recedes from the source with the same speed. The frequency of source is

Options

A. (a) `(f_(r) + f_(a))/(2)`
B. (b) `(f_(r) - f_(a))/(2)`
C. (c ) `sqrt(f_(a) f_(r)`
D. (d) `(2f_(r)f_(a))/(f_(a) + f_(a))`

Explanation

To solve the problem, we will use the Doppler effect equations for sound waves. The apparent frequency when the observer approaches the source and when the observer recedes from the source can be expressed using the following equations:

1. **When the observer approaches the source:**
\[ f_A = f \frac{V + V_0}{V} \]
where:
- \( f_A \) is the apparent frequency when approaching,
- \( f \) is the actual frequency of the source,
- \( V \) is the speed of sound in the medium,
- \( V_0 \) is the speed of the observer.

2. **When the observer recedes from the source:**
\[ f_R = f \frac{V - V_0}{V} \]
where:
- \( f_R \) is the apparent frequency when receding.

### Step 1: Write the equations for both cases
From the above, we have:
\[ f_A = f \frac{V + V_0}{V} \quad \text{(1)} \]
\[ f_R = f \frac{V - V_0}{V} \quad \text{(2)} \]

### Step 2: Rearrange the equations to express \( f \)
From equation (1):
\[ f = f_A \frac{V}{V + V_0} \quad \text{(3)} \]
From equation (2):
\[ f = f_R \frac{V}{V - V_0} \quad \text{(4)} \]

### Step 3: Set equations (3) and (4) equal to each other
Since both equations equal \( f \):
\[ f_A \frac{V}{V + V_0} = f_R \frac{V}{V - V_0} \]

### Step 4: Cross-multiply to eliminate \( V \)
\[ f_A (V - V_0) = f_R (V + V_0) \]

### Step 5: Expand and rearrange the equation
Expanding both sides:
\[ f_A V - f_A V_0 = f_R V + f_R V_0 \]
Rearranging gives:
\[ f_A V - f_R V = f_A V_0 + f_R V_0 \]
Factoring out \( V \):
\[ V (f_A - f_R) = V_0 (f_A + f_R) \]

### Step 6: Solve for \( f \)
Now, we can find \( f \) by using the average of \( f_A \) and \( f_R \):
\[ f = \frac{f_A + f_R}{2} \]

### Final Answer
Thus, the frequency of the source is:
\[ f = \frac{f_A + f_R}{2} \]