Hello everyone,
Here is something I've discussed with some colleagues recently and that generated a lot of disagreement, so I wanted to bring the topic to Physics Forums to ask what is your understanding of the situations and wheter you agree with the "solution" I provide below or not.
Imagine that observers [itex]A[/itex] and [itex]B[/itex] are in a motionless media. Observer [itex]A[/itex] moves to the right with velocity [itex]v_{a}[/itex] and observer [itex]B[/itex] moves to the left with velocity [itex]v_{b}[/itex]. At the same time, [itex]A[/itex] emits sound waves of frequency [itex]f_{o}[/itex] in the direction of [itex]B[/itex]. These sound waves are reflected at [itex]B[/itex] and return in the direction of [itex]A[/itex]. The question is: what is the expression that gives the frequency of the reflected wave as measured by [itex]A[/itex].
I argue the following, and I would like to really appreciate to hear more opinions on this matter to reach a final answer:
Since both observers are moving with respect to each other, the frequency [itex]f^{'}[/itex] measured by [itex]B[/itex] is given by the Doppler relation (considering that the speed of sound in that media is [itex]u[/itex]):
[itex]f^{'} = (\frac{u + v_{b}}{u - v_{a}})f_{o}[/itex]
Now, the wave reflected at [itex]B[/itex] has just the same frequency as that measured by [itex]B[/itex], and therefore has exactly the value [itex]f^{'}[/itex]. However, [itex]A[/itex] observes this reflected wave not as a wave emitted by a moving source, but as if it was being emitted by a source that stationary with respect to the media.
This is the point where there is a lot of disagreement, so I would like to make it clear beforehand that the reasoning above does not state that [itex]B[/itex] is stationary (this would be a contradition, since we start with a different assumption), it only states that the reflected wave as measured by [itex]A[/itex] appears to be emitted by a source which is stationaty with respect to the media.
Thus, since [itex]A[/itex] is moving into the reflected wave with velocity [itex]v_{a}[/itex], another Doppler shift piles up on top of the previous one, giving the following frequency [itex]f^{''}[/itex] measured by [itex]A[/itex]:
[itex]f^{''} = (\frac{u + v_{a}}{u})f^{'}= (\frac{u + v_{a}}{u})(\frac{u + v_{b}}{u - v_{a}})f_{o}[/itex]
So, this is the story. I hope to hear some comments soon!
Best regards,
Zag
Here is something I've discussed with some colleagues recently and that generated a lot of disagreement, so I wanted to bring the topic to Physics Forums to ask what is your understanding of the situations and wheter you agree with the "solution" I provide below or not.
Imagine that observers [itex]A[/itex] and [itex]B[/itex] are in a motionless media. Observer [itex]A[/itex] moves to the right with velocity [itex]v_{a}[/itex] and observer [itex]B[/itex] moves to the left with velocity [itex]v_{b}[/itex]. At the same time, [itex]A[/itex] emits sound waves of frequency [itex]f_{o}[/itex] in the direction of [itex]B[/itex]. These sound waves are reflected at [itex]B[/itex] and return in the direction of [itex]A[/itex]. The question is: what is the expression that gives the frequency of the reflected wave as measured by [itex]A[/itex].
I argue the following, and I would like to really appreciate to hear more opinions on this matter to reach a final answer:
Since both observers are moving with respect to each other, the frequency [itex]f^{'}[/itex] measured by [itex]B[/itex] is given by the Doppler relation (considering that the speed of sound in that media is [itex]u[/itex]):
[itex]f^{'} = (\frac{u + v_{b}}{u - v_{a}})f_{o}[/itex]
Now, the wave reflected at [itex]B[/itex] has just the same frequency as that measured by [itex]B[/itex], and therefore has exactly the value [itex]f^{'}[/itex]. However, [itex]A[/itex] observes this reflected wave not as a wave emitted by a moving source, but as if it was being emitted by a source that stationary with respect to the media.
This is the point where there is a lot of disagreement, so I would like to make it clear beforehand that the reasoning above does not state that [itex]B[/itex] is stationary (this would be a contradition, since we start with a different assumption), it only states that the reflected wave as measured by [itex]A[/itex] appears to be emitted by a source which is stationaty with respect to the media.
Thus, since [itex]A[/itex] is moving into the reflected wave with velocity [itex]v_{a}[/itex], another Doppler shift piles up on top of the previous one, giving the following frequency [itex]f^{''}[/itex] measured by [itex]A[/itex]:
[itex]f^{''} = (\frac{u + v_{a}}{u})f^{'}= (\frac{u + v_{a}}{u})(\frac{u + v_{b}}{u - v_{a}})f_{o}[/itex]
So, this is the story. I hope to hear some comments soon!
Best regards,
Zag