A source of sound of frequency `256Hz` is moving rapidly towards wall with a velocity of `5m//sec`. How many beats per second will be heard if sound travels at a speed of `330m//sec`.

Explanation

To solve the problem of how many beats per second will be heard when a source of sound of frequency \(256 \, \text{Hz}\) is moving towards a wall at a velocity of \(5 \, \text{m/s}\), we can follow these steps:

### Step 1: Understand the Doppler Effect
When a sound source moves towards a stationary observer (or wall in this case), the frequency of the sound heard by the observer changes due to the Doppler effect. The formula for the observed frequency \(f'\) when the source is moving towards a stationary observer is given by:

\[ f' = \frac{v}{v - v_s} f \]

where:
- \(f'\) = observed frequency
- \(v\) = speed of sound in air (\(330 \, \text{m/s}\))
- \(v_s\) = speed of the source (\(5 \, \text{m/s}\))
- \(f\) = original frequency of the source (\(256 \, \text{Hz}\))

### Step 2: Calculate the Frequency Reflected from the Wall
The sound reflects off the wall and acts as if the wall is a new source of sound. The frequency of the sound that returns to the observer (the original source) is given by:

\[ f'' = \frac{v}{v + v_s} f' \]

### Step 3: Substitute \(f'\) into the Equation for \(f''\)
Substituting the expression for \(f'\) into the equation for \(f''\):

\[ f'' = \frac{v}{v + v_s} \left(\frac{v}{v - v_s} f\right) \]

### Step 4: Simplify the Expression
Now we can simplify this expression:

\[ f'' = \frac{v^2}{(v + v_s)(v - v_s)} f \]

### Step 5: Calculate Beat Frequency
The beat frequency \(f_b\) is the absolute difference between the original frequency and the frequency of the sound after it reflects off the wall:

\[ f_b = |f'' - f| \]

### Step 6: Substitute Values
Now, we substitute the values into the equation:

- \(v = 330 \, \text{m/s}\)
- \(v_s = 5 \, \text{m/s}\)
- \(f = 256 \, \text{Hz}\)

Calculating \(f''\):

\[ f'' = \frac{330^2}{(330 + 5)(330 - 5)} \cdot 256 \]

Calculating the denominator:

\[ (330 + 5)(330 - 5) = 335 \cdot 325 = 109375 \]

Calculating \(f''\):

\[ f'' = \frac{108900}{109375} \cdot 256 \approx 254.4 \, \text{Hz} \]

### Step 7: Calculate Beat Frequency
Now, we calculate the beat frequency:

\[ f_b = |254.4 - 256| \approx 1.6 \, \text{Hz} \]

### Final Result
Thus, the number of beats per second heard is approximately \(1.6 \, \text{Hz}\).

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