A source of sound of frequency 256 Hz moves rapidly towards a wall with a velocity of `5 ms^(-1).` How many beats per second will be heard if sound travels at a speed of 330 m/s?

Answer

To solve the problem of how many beats per second will be heard when a source of sound moves towards a wall, we can use the Doppler effect. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Frequency of the source, \( f = 256 \, \text{Hz} \)
- Velocity of sound, \( v = 330 \, \text{m/s} \)
- Velocity of the source, \( v_s = 5 \, \text{m/s} \)
- Velocity of the observer, \( v_0 = 0 \, \text{m/s} \) (since the observer is at rest)

### Step 2: Calculate the frequency heard by the wall (reflected frequency)
Using the Doppler effect formula for the frequency heard by the wall:
\[ f' = f \frac{v - v_0}{v - v_s} \]
Substituting the values:
\[ f' = 256 \, \text{Hz} \cdot \frac{330 - 0}{330 - 5} \]
\[ f' = 256 \, \text{Hz} \cdot \frac{330}{325} \]
Calculating this gives:
\[ f' \approx 252.18 \, \text{Hz} \]

### Step 3: Calculate the frequency of the sound reflected back to the observer
Now, we need to calculate the frequency of the sound that is reflected back to the observer. In this case, the source is moving towards the observer after reflecting off the wall, so:
\[ f'' = f' \frac{v + v_0}{v - v_s} \]
Substituting the values:
\[ f'' = 252.18 \, \text{Hz} \cdot \frac{330 + 0}{330 - 5} \]
\[ f'' = 252.18 \, \text{Hz} \cdot \frac{330}{325} \]
Calculating this gives:
\[ f'' \approx 259.94 \, \text{Hz} \]

### Step 4: Calculate the beat frequency
The beat frequency \( f_b \) is given by the difference between the two frequencies:
\[ f_b = f'' - f' \]
Substituting the calculated frequencies:
\[ f_b = 259.94 \, \text{Hz} - 252.18 \, \text{Hz} \]
Calculating this gives:
\[ f_b \approx 7.76 \, \text{Hz} \]

### Final Answer
The number of beats per second heard is approximately \( 7.76 \, \text{Hz} \).

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