QUESTION:

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

MY ATTEMPT:

Using Doppler's Effect:

f ' = (c/c-v)f

where c = speed of sound in the medium = 330 m/s

v= velocity of source = 5m/s

f= 256 Hz

On solving the above equation.. u get f' as approximately 260Hz..

Hence beats per second = 260-256 = 4Hz..

But the answer says 8Hz... How???

To solve the problem of how many beats per second will be heard when a sound source is moving towards a wall, we need to apply the Doppler Effect correctly. Let's break it down step by step to clarify where the discrepancy might be coming from.

Understanding the Doppler Effect

The Doppler Effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave. In this case, we have a sound source moving towards a wall, which acts as a stationary observer. The frequency heard by the wall will be higher than the original frequency due to the motion of the source.

Calculating the Frequency Heard by the Wall

We can use the formula for the Doppler Effect when the source is moving towards a stationary observer:

• f' = f (c / (c - v))

Where:

• f' = observed frequency
• f = source frequency (256 Hz)
• c = speed of sound in air (330 m/s)
• v = speed of the source (5 m/s)

Plugging in the values:

• f' = 256 Hz * (330 m/s / (330 m/s - 5 m/s))
• f' = 256 Hz * (330 / 325)
• f' ≈ 256 Hz * 1.01538
• f' ≈ 260 Hz

Reflection of Sound

Now, the sound wave reflects off the wall, and the wall acts as a new source of sound for the observer (you). The frequency of the sound wave that reflects back to the moving source will also be affected by the Doppler Effect. The source is now moving towards the wall, and the wall is effectively a stationary source emitting the frequency f' that we just calculated.

Calculating the Frequency of the Reflected Sound

Now we need to find the frequency of the sound that the moving source hears after the reflection:

• f'' = f' (c + v) / c

Where:

• f'' = frequency heard by the moving source after reflection

Substituting the values:

• f'' = 260 Hz * (330 m/s + 5 m/s) / 330 m/s
• f'' = 260 Hz * (335 / 330)
• f'' ≈ 260 Hz * 1.01515
• f'' ≈ 264 Hz

Calculating Beats Per Second

Finally, to find the number of beats per second, we subtract the original frequency from the frequency of the reflected sound:

• Beats per second = f'' - f = 264 Hz - 256 Hz = 8 Hz

This explains why the answer is 8 Hz. The key point is that the sound reflects off the wall, and the moving source then hears this reflected sound, which has a higher frequency due to the Doppler Effect acting twice—once when the sound travels to the wall and again when it reflects back. This double application of the Doppler Effect is what leads to the increased beat frequency.