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Old Jun 20th 2016, 03:18 PM   #1
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SR Doppler Shift of EM Waves

Plugging away on my undergrad textbook. Into the more advanced problems, stuck in SR again (maybe a song title there?).

Arrived at an answer per the given hint in the problem. Hoping for a qualitative explanation as to why this is (or is not) correct.

Problem given as: "A spaceship moving at constant speed u relative to us broadcasts a radio signal at constant frequency f-naught. As the spaceship approaches us, we receive a higher frequency f. After it has passed, we receive a lower frequency. As the spaceship passes by, so that it is instantaneously neither moving toward or away from us, show that the frequency we receive is not f-naught and derive an expression for the frequency we do receive. Is the frequency we receive larger or smaller than f-naught? Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus f equals 1/T. Use the time dilation formula to relate the periods in the stationary and moving frames."

I took the not-so-subtle hint and dilated the period T to (gamma T). Thus f = 1/(gamma T) and the frequency is lower (red-shifted).

I'm trying to reconcile this with my image of what's going on. In the emitter's frame of reference, I see the EM waves as concentric circles propagating outward. Then I have the emitter moving at speed u in the +x direction in relation to the observer's stationary frame, and those once concentric circles are now more densely spaced on the leading edge (+x), while less densely spaced on the trailing edge (-x). The emitter is traveling left-to-right, and is instantaneously "above" the observer in my visualization.

So the observer is looking upward at all this, and relative velocity is instantaneously perpendicular. Why would I dilate the wave period with gamma? I'm not sure how to approach this. Should I attack this geometrically with my once-concentric but now smushed circles? Is difficult for me to envision the propagation of the waves in motion, relative to the stationary observer, and particularly at the moment of truth when the emitter strafes overhead and hits x=0 where my observer resides. More confusion from the fact that the emitter is moving at speed u, implied to be an appreciable percentage of c, such that the picture is changing rapidly as the emitter approaches x=0.
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Old Jun 22nd 2016, 01:07 AM   #2
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I found info on Transverse Doppler Shift, and see that the equation does indeed simplify to f = f-naught / gamma when the angle of approach is 90 degrees. I believe the intent of the textbook's authors was to simply have students time-dilate the wave period and produce the desired answer. A deeper understanding or derivation of Transverse Doppler Shift was not intended at this level of instruction. Was getting in over my head.
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