A sound wave of frequency `f` propagating through air with a velocity `C`, is reflected from a surface which is moving away from the source with a constant speed `V`. Find the frequency of the reflected wave, measured by the observer at the position of the source.

Options

A. `( f( c + 2v))/(c + v ) `
B. `(f ( c + v))/( c-v)`
C. `(f(c-v))/( c+v)`
D. `(f ( c-v))/( c- 2 v)`

Answer

To solve the problem of finding the frequency of the reflected wave measured by the observer at the position of the source, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Information:**
- Frequency of the source sound wave: \( f \)
- Velocity of sound in air: \( C \)
- Velocity of the reflecting surface moving away from the source: \( V \)

2. **Determine the Incident Frequency:**
- When the sound wave travels from the source to the moving surface, we can use the Doppler effect formula to find the frequency of the sound wave as it reaches the moving surface.
- The formula for the frequency observed by a moving observer is given by:
\[ f_i = \frac{C - V}{C} \cdot f \]
- Here, \( C \) is the speed of sound, \( V \) is the speed of the moving surface (which is moving away from the source), and \( f \) is the original frequency.

3. **Calculate the Reflected Frequency:**
- Now, the surface acts as a new source of sound with frequency \( f_i \) and reflects the sound back towards the observer (the original source).
- The frequency of the reflected wave \( f_r \) can be calculated using the Doppler effect again:
\[ f_r = \frac{C}{C + V} \cdot f_i \]
- Substituting \( f_i \) from the previous step:
\[ f_r = \frac{C}{C + V} \cdot \left(\frac{C - V}{C} \cdot f\right) \]

4. **Simplify the Expression:**
- Simplifying the expression for \( f_r \):
\[ f_r = \frac{C(C - V)}{C(C + V)} \cdot f \]
- This simplifies to:
\[ f_r = \frac{C - V}{C + V} \cdot f \]

5. **Final Result:**
- The frequency of the reflected wave measured by the observer at the position of the source is:
\[ f_r = \frac{C - V}{C + V} \cdot f \]