And the award goes to..............
~
'mud
Here's a link with a page that's pretty good:
http://www.einstein-online.info/spotlights/doppler
Notice that the waves always travel at the same speed. The reason to doppler effect and frequency shifting is that the "uptake" or reception of the waves changes with the speed. So the actual wave speed is constant regardless of either objects moving, however, the frequency and wavelength changes with speed.
Here's another one:
https://www.grc.nasa.gov/www/k-12/airplane/doppler.html
See how the "f" changes (and with f, lambda/wavelength also changes, but is rarely mentioned, simply because the energy quanta doesn't change). They have no equation how the "v" (velocity) changes as a result of the moving source and target. Moving source or target, doesn't change v, but changes energy, and therefore frequency.
--- edit
You know, I'm taking classes for fun at the college that I work. And the class I'm taking is astronomy right now. And actually, today, the professor is going to go over doppler effect and red shift! LOL!. So... how about I ask him? (He got his PhD at the same university as Einstein, if I remember it right.)
---edit2
So I asked my professor, and he said "no". Speed doesn't change, and then he gave me a bunch of math to prove it. However, he said, the frequency and the wavelength changes.
Now, where did I hear that before? Oh, yeah! I said it! But this time, it came from a scientist (granted, through me, so you just have to trust me
)