A source of sonic oscillations with frequency and a receiver are located on the same normal to the wall. Both the source and the receiver are stationary and the wall recedes from source with velocity u. Find the beat frequency registered by the receiver. The velocity of sound is equal to v.

Answer

To solve the problem of finding the beat frequency registered by the receiver when a wall recedes from a stationary source of sonic oscillations, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Frequency of the source, \( f_0 \)
- Velocity of sound, \( v \)
- Velocity of the wall receding from the source, \( u \)

2. **Determine the Frequency Heard by the Wall:**
- The wall acts as a moving observer for the sound waves emitted by the source. Since the wall is moving away from the source, the frequency \( f_w \) heard by the wall can be calculated using the formula for the Doppler effect:
\[ f_w = f_0 \left( \frac{v - u}{v} \right) \]
Here, \( v - u \) is used because the wall is moving away from the source.

3. **Determine the Frequency Reflected Back to the Receiver:**
- The wall now acts as a new source emitting the frequency \( f_w \). The receiver, which is stationary, will hear this frequency. The frequency \( f_r \) heard by the receiver from the wall is given by:
\[ f_r = f_w \left( \frac{v}{v + u} \right) \]
Substituting \( f_w \) into this equation gives:
\[ f_r = \left( f_0 \left( \frac{v - u}{v} \right) \right) \left( \frac{v}{v + u} \right) \]
Simplifying this, we get:
\[ f_r = f_0 \left( \frac{(v - u)v}{v(v + u)} \right) = f_0 \left( \frac{v - u}{v + u} \right) \]

4. **Calculate the Beat Frequency:**
- The beat frequency \( f_b \) is the difference between the frequency heard by the receiver and the original frequency emitted by the source:
\[ f_b = |f_0 - f_r| \]
Substituting \( f_r \) into this equation gives:
\[ f_b = \left| f_0 - f_0 \left( \frac{v - u}{v + u} \right) \right| \]
Factoring out \( f_0 \):
\[ f_b = f_0 \left| 1 - \frac{v - u}{v + u} \right| \]
Simplifying the expression inside the absolute value:
\[ f_b = f_0 \left| \frac{(v + u) - (v - u)}{v + u} \right| = f_0 \left| \frac{2u}{v + u} \right| \]

5. **Final Expression for Beat Frequency:**
- Thus, the beat frequency registered by the receiver is:
\[ f_b = \frac{2u f_0}{v + u} \]