An observer rides with a moving sound source directly toward a large vertical wall. The vehicle has a constant speed of 10m/s and the source has a frequency of 150Hz. What beat frequency is heard by the observer as a result of the combination of the direct and reflected sounds?
Received frequency shift (df):
df/f = 2*(v/a) = 2*10/340 = 0.059
df = 150*.059 = 8.8 Hz
Received frequency = 150 + 8.8
= 158.8 Hz
Beat frequency = 158.8 - 150 = 8.8 Hz
To find the beat frequency heard by the observer, we need to consider the Doppler effect and the reflection of sound waves.
The beat frequency is caused by the interference between the direct sound waves from the moving source and the reflected sound waves from the stationary wall.
First, let's consider the Doppler effect. When the sound source is moving towards the observer, the frequency of the sound waves appears higher. The formula for the apparent frequency due to the Doppler effect is:
f' = (v + v₀) / (v + vs) * f₀
where:
- f' is the apparent frequency heard by the observer
- v is the speed of sound in air (approximately 343 m/s at room temperature)
- v₀ is the speed of the observer
- vs is the speed of the source
- f₀ is the actual frequency of the source
In this case, the sound source is moving towards the observer with a speed of 10 m/s (vs = 10 m/s). The actual frequency of the source is 150 Hz (f₀ = 150 Hz). The speed of sound in air is 343 m/s (v = 343 m/s).
Now, let's consider the reflection of sound waves. When the sound waves hit a stationary wall, they are reflected back towards the observer. The apparent frequency of the reflected sound waves is the same as the actual frequency of the source (f₀).
The beat frequency (fbeat) is the difference between the apparent frequency due to the Doppler effect and the actual frequency (fbeat = f' - f₀).
Putting it all together, we can calculate the beat frequency:
f' = (v + v₀) / (v + vs) * f₀
fbeat = f' - f₀
fbeat = [(v + v₀) / (v + vs) * f₀] - f₀
To find the beat frequency, we just need to substitute the given values into the equation.
f' = (343 + 0) / (343 + 10) * 150
f' = 343 / 353 * 150
f' ≈ 145.03 Hz
fbeat = 145.03 Hz - 150 Hz
fbeat ≈ -4.97 Hz
Therefore, the beat frequency heard by the observer as a result of the combination of the direct and reflected sounds is approximately -4.97 Hz.