# Relative motion between a light source and a stationary observer

I've been taught that for a stationary observer moving at constant relative velocity away from a light source, this light source will decrease in frequency as the distance between the two increases. I don't quite understand this, I would've thought the frequency would be lower but unchanging, and that only the magnitude of the velocity influences the frequency of light. Can someone explain this phenomenon in depth please?

If the source is moving directly away from the receiver (i.e. on the line connecting the source and receiver), then you are right. The observed frequency at the receiver is constant as long as the frequency and velocity of the source is constant in that case.

The issue is when the source moves along a line that does not contain the receiver, e.g. like a train that passes by while you stand to the side on the platform. In that case, only the component of the (constant!) velocity of the source that along the line to the receiver contributes to the (nonrelativistic) Doppler effect. (In the relativistic Doppler effect, there is also a shift due to the time dilation of the source, which depends on its total velocity, but again this is constant.) But the source is moving, so the line connecting the source to the receiver is changing. Therefore, though the source has constant velocity, the component of that velocity that is "away from the receiver" changes as the source moves. So the received frequency changes with time.

Another way to see it: for a source moving along the line connecting it to the receiver, we receive a higher frequency while it's moving towards the receiver and a lower one when it's moving away. There is a discontinuous jump from the high frequency to the low when the source passes "through" the receiver. But when we remove the "physical weirdness" of the setup by making the source pass the receiver instead of through it, we expect the discontinuity to go away. So now we expect a smooth transition of the frequency from high to low. But once the source has passed the receiver, there is no natural cutoff where the frequency should go from smoothly decreasing to being constant. It decreases slower and slower while approaching the constant value you'd expect in the "passing through" case, but never reaches it.

If you want a formula, at any point in the source's motion you can find the angle $$\theta$$ between the direction "away from the receiver" and direction of the source's velocity. Then you should use $$v_\text{away}=v_\text{total}\cos\theta$$ instead of $$v_\text{total}$$ to calculate the (nonrelativistic portion of) the Doppler effect.