Two Train whistles, A and B each have a frequency of 408Hz. A is stationary and B is moving toward the right (away from A) at a speed of 35.0m/s. A listener is between the two whistles and is moving toward the right at a speed of 15.0m/s. What is the frequency from A and B as heard by the listener?
From an ancient reply:
fr = fs[(v-vd)/(v-vs)]
fs is the frequency of the source, relative to the medium carrying the waves. We assume that is also the speed with respect to the ground (no wind)
fr is the received frequency.
v is the speed of sound in the air.
vd is the speed of the detector (listener)away from the source
vs is the speed of the source towards the detector.
Two Train whistles, A and B each have a frequency of 408Hz. A is stationary and B is moving toward the right (away from A) at a speed of 35.0m/s. A listener is between the two whistles and is moving toward the right at a speed of 15.0m/s. What is the frequency from A and B as heard by the listener?
Using 340 m/s for v
First due to A
d moving away from A at 15 m/s
fr = 408 (340-15)/(340-0) = 390 Hz
Now dues to B
Vd = -15
Vs = -35
fr = 408(340+15)/(340+35) = 386 Hz
To determine the frequency heard by the listener, we need to consider the Doppler effect. The Doppler effect is the change in frequency or wavelength of a wave as perceived by an observer moving relative to the source of the wave.
The formula to calculate the observed frequency is:
f' = f * (v + v₀) / (v - vᵥ)
where:
- f' is the observed frequency
- f is the original frequency emitted by the source
- v is the speed of sound
- v₀ is the speed of the listener (positive if moving towards the source, negative if moving away from the source)
- vᵥ is the speed of the source (positive if moving towards the listener, negative if moving away from the listener)
In this case:
- The original frequency of both A and B is 408 Hz.
- The speed of sound is typically 343 m/s in air.
- The speed of the listener, v₀, is +15.0 m/s (positive because the listener is moving towards the source).
- The speed of source B relative to the listener, vᵥ, is 35.0 m/s (positive because B is moving towards the listener).
Using the formula, we can calculate the observed frequency for each whistle:
For whistle A:
f'A = f * (v + v₀) / (v - vᵥ)
= 408 Hz * (343 m/s + 15.0 m/s) / (343 m/s - 0 m/s)
= 408 Hz * (358 m/s) / (343 m/s)
= 424.83 Hz
For whistle B:
f'B = f * (v + v₀) / (v - vᵥ)
= 408 Hz * (343 m/s + 15.0 m/s) / (343 m/s - (-35.0 m/s))
= 408 Hz * (358 m/s) / (378 m/s)
= 386.03 Hz
Therefore, the frequency heard by the listener from A is approximately 424.83 Hz, and the frequency heard from B is approximately 386.03 Hz.