Richard
An analogy; You shrink, and sit at rest in the NEW medium frame (K' at n=1), which is doing v through the background frame (K at n=1). You are at the refractive plane with a tape measure and a stopwatch. Your bird at rest (K) in the approach medium has told you the approaching waves are doing c and are 10 metres apart in her frame.
As a wave hits the refractive plane (at relative c+v until the collision) you hook the end of your tape measure to it and start your stopwatch as it continues at c in the new medium frame (K'). Now when the next wave hits the refractive plane you look at the tape measure. Because you are approaching the oncoming waves you find a SHORTER distance than 10 metres! Only THEN can you check your watch and calculate the frequency (actually your mate 'Brains' who does that is well behind you up the optic nerve). The trouble with Brains is that he can calculate 'f 'ok, but he can't see the big picture, so he complacently assumes his simple maths are good enough to describe nature.
Back at the BIG scale, what we've all been missing is the important detail, and we've just been making that dumb simplistic assumption. That wrong assumption is what has maintained the current paradigm and paradoxes. There can be no detection without a lens medium, all lenses are made of dense matter, and all matter re-scatters absorbed energy at c. Local c. NOT some 'absolute' c.
There are three elements to your formula; f, c, and L. If f and L change inversely c is conserved. All precisely as always found. There never was a need for the assumption of "no background frames" implicit in the STR, background frames are always LOCAL or 'discrete', not absolute, so fully equivalent to the almost infinite 'compound proposition' structure of logic. The POSTULATES of SR are then identified as not the problem, and are logically produced by the quantum mechanism of Raman scattering, Unifying physics at last.
Note, there are a number of 'cases', which are the cause of confusion, of both observer frame and signal transitions at frame boundaries. The Cartesian system must be completely abandoned. It uses geometric 'vector' space and motion is not a valid concept in geometry. Inertial frames are simply 'states of motion', so apply to ALL matter in relative motion, and are separated simply by an acceleration. Length contraction is then simply what happens in a car crash! Nature is far simpler than old physics, just unfamiliar at first, as Feynman predicted. And that is the ontological construction of the 'discrete field model' (DFM).
Do re-read the essay with that new light pouring in, ask any questions, then mark your own papers and pass me the scores.
(I just posted the above in reply to Pentcho on my string but wanted to ensure you saw it.) There is of course a stack of physics behind it, but I think the above is enough to handle in one go. Please confirm you agree, or comment, and advise if you can see any better way to present it to overcome it's big issue of initial unfamiliarity.
Many thanks
Peter