If a vibrating fork is rapidly moved towards a wall, beats may beheard between the direct and reflected sounds. Calculate beat frequency if the frequency of fork is 512 Hz and approaches the wall with a velocity of 300 cm/s. The velocity of sound is 330 m/s. Consider observer is behind the fork.

Explanation

To solve the problem of calculating the beat frequency when a vibrating fork is rapidly moved towards a wall, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Frequency of the fork, \( f_0 = 512 \, \text{Hz} \)
- Velocity of the fork, \( v_f = 300 \, \text{cm/s} = 3 \, \text{m/s} \) (conversion from cm/s to m/s)
- Velocity of sound, \( v_s = 330 \, \text{m/s} \)

2. **Calculate the Apparent Frequency of Direct Sound:**
Since the observer is behind the fork and the fork is moving towards the wall, the apparent frequency of the direct sound (\( f_1 \)) can be calculated using the Doppler effect formula:
\[ f_1 = f_0 \left( \frac{v_s}{v_s - v_f} \right) \]
Substituting the values:
\[ f_1 = 512 \left( \frac{330}{330 - 3} \right) = 512 \left( \frac{330}{327} \right) \]

3. **Calculate the Apparent Frequency of Reflected Sound:**
The reflected sound will have a different apparent frequency (\( f_2 \)) since it travels back to the observer after reflecting off the wall. The formula for the reflected sound is:
\[ f_2 = f_0 \left( \frac{v_s}{v_s + v_f} \right) \]
Substituting the values:
\[ f_2 = 512 \left( \frac{330}{330 + 3} \right) = 512 \left( \frac{330}{333} \right) \]

4. **Calculate the Beat Frequency:**
The beat frequency (\( f_b \)) is the difference between the two frequencies:
\[ f_b = |f_2 - f_1| \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ f_b = \left| 512 \left( \frac{330}{333} \right) - 512 \left( \frac{330}{327} \right) \right| \]
Factor out \( 512 \):
\[ f_b = 512 \left| \frac{330}{333} - \frac{330}{327} \right| \]
To simplify:
\[ f_b = 512 \cdot 330 \left| \frac{1}{333} - \frac{1}{327} \right| \]

5. **Calculate the Difference:**
Finding a common denominator:
\[ \frac{1}{333} - \frac{1}{327} = \frac{327 - 333}{333 \cdot 327} = \frac{-6}{333 \cdot 327} \]
Thus:
\[ f_b = 512 \cdot 330 \cdot \left| \frac{-6}{333 \cdot 327} \right| \]

6. **Final Calculation:**
Calculate the numerical values:
\[ f_b = 512 \cdot 330 \cdot \frac{6}{333 \cdot 327} \]
After calculating, you will find:
\[ f_b \approx 9.3 \, \text{Hz} \]

### Final Answer:
The beat frequency is approximately \( 9.3 \, \text{Hz} \).