Longitudinal Doppler Effect

In summary: Y (1-v/c)) = YT (1+v/c) / (1-v^2/c^2) = YT (1+v/c) / (1-v/c)(1+v/c) = YT (1+v/c)^2 / (1-v^2/c^2) = YT (1+v/c)^2 / (1-(v/c)^2) = YT (1+v/c)^2 / (1-Y^2 v^2/c^2) = YT (1+v/c)^2 / (1-(1/Y)^2 (v/c)^2) = YT (1+v/c)^2 / (1-(1/Y)^2 (v/c)^2) = YT (1+v/c)^2
  • #1
greendog77
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A and B leave from a common point and travel in opposite directions with
relative speed v. When B’s clock shows that a time T has elapsed, he (B)
sends out a light signal. When A receives the signal, what time does his (A’s)
clock show? Answer this question by doing the calculation entirely in (a) A’s
frame, and then (b) B’s frame.

(Y = gamma)

a)

In A's frame, when A's clock reads YT, B's clock reads T. This means B is at a distance YTv from A. When B emits the photon, the photon takes time YTv/c to reach A in A's frame. Thus, the total time for A is YT(1 + v/c).

b)

In B's frame, when his clock reads T, A is at a distance Tv away. Then, B emits a photon which travels at a speed of (c-v) relative to A in B's reference frame. Thus, the time taken for the photon to reach A is Tv/(c-v). Thus the total time this takes according to B is T + Tv/(c-v) = T(1 + v/(c-v)). By time dilation, in A the total time is YT(1 + v/(c-v)).

I get these two different answers. Does anyone know what I'm doing wrong?
 
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  • #2
greendog77 said:
A and B leave from a common point and travel in opposite directions with
relative speed v. When B’s clock shows that a time T has elapsed, he (B)
sends out a light signal. When A receives the signal, what time does his (A’s)
clock show? Answer this question by doing the calculation entirely in (a) A’s
frame, and then (b) B’s frame.

(Y = gamma)

a)

In A's frame, when A's clock reads YT, B's clock reads T. This means B is at a distance YTv from A. When B emits the photon, the photon takes time YTv/c to reach A in A's frame. Thus, the total time for A is YT(1 + v/c).

b)

In B's frame, when his clock reads T, A is at a distance Tv away. Then, B emits a photon which travels at a speed of (c-v) relative to A in B's reference frame. Thus, the time taken for the photon to reach A is Tv/(c-v). Thus the total time this takes according to B is T + Tv/(c-v) = T(1 + v/(c-v)). By time dilation, in A the total time is YT(1 + v/(c-v)).

I get these two different answers. Does anyone know what I'm doing wrong?

Your calculation in B's frame is wrong. You got it correct, that the time, according to B's frame, for the photon to reach A is T_arrive = T(1+v/(c-v)) = T/(1 - v/c). But in B's frame, A's clock is running slower, so the elapsed time on A's clock is T'_arrive = T_arrive/Y = T/(Y (1-v/c)).

That's the same as your calculation in A's frame, since

T/(Y (1-v/c)) = YT (1+v/c)
 
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1. What is the Longitudinal Doppler Effect?

The Longitudinal Doppler Effect is a phenomenon that occurs when there is a relative motion between a source of sound and an observer. It describes the perceived change in frequency of a sound wave as the source and observer move towards or away from each other.

2. How is the Longitudinal Doppler Effect different from the Transverse Doppler Effect?

The Longitudinal Doppler Effect involves the perceived change in frequency of a sound wave, while the Transverse Doppler Effect involves the perceived change in frequency of a light wave. The Longitudinal Doppler Effect also takes into account the speed of sound in a medium, while the Transverse Doppler Effect does not.

3. What factors affect the magnitude of the Longitudinal Doppler Effect?

The magnitude of the Longitudinal Doppler Effect is affected by the speed of the source and observer, the speed of sound in the medium, and the angle of motion between the source and observer.

4. How is the Longitudinal Doppler Effect used in real-life applications?

The Longitudinal Doppler Effect is used in various real-life applications such as speed radar guns, medical ultrasound imaging, and weather radar systems. It is also used in the study of celestial objects and their motion, such as stars and galaxies.

5. Can the Longitudinal Doppler Effect be observed in other types of waves besides sound waves?

Yes, the Longitudinal Doppler Effect can also be observed in other types of waves such as water waves and seismic waves. It is a phenomenon that occurs in any type of wave that has a frequency and wavelength, and can be observed when there is relative motion between a source and observer.

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