(For context, I originally thought of this question in the context of electromagnetic Doppler shift, but I'm also curious if the same logic applies for acoustic Doppler shift.)
Assume you are watching an object approaching you at relativistic speeds, for example fast enough that the measured frequency of its emissions is shifted by $10\%$. The object is not on a collision course, but the point of closest approach is a reasonably short, i.e. non-relativistic distance away. If the object emits a continous wave radio signal, over what timescale does the measured frequency of that signal change as it passes the point of closest approach?
I cannot intuitively accept that it changes $20\%$ of a potentially very large number (e.g. $1\text{ GHz}$) instantly, because classical mechanics really dislikes discontinuities. But the transition between moving towards and moving away is in some sense instantaneous, given the boundary between the two is infinitesimal. What actually happens then?
Best Answer
The instantaneous change occurs when you consider the Doppler shift in only one dimension. In three dimensions you can consider the correction when the velocity vector and the separation vector are not parallel. Usually such corrections go like $\cos\theta$, where $\theta$ is the angle between the two vectors, but more complicated things are possible.
Years ago I sat down and computed the speeds for which acoustic Doppler shifts correspond to musical intervals. That gave me the superpower of being able to stand on a sidewalk, listen to the WEEE-ooom as a car drove past, and say to myself “a major third? They're speeding!” But because of the $\cos\theta$ dependence, the trick gets harder as you get further from the road.