If galilean relativity transformations do not apply to waves, then the speed of light can only be invariant

The reason why Galilean velocity additions cannot apply to light is because light is a WAVE, and waves do not obey galilean relativity because they have a PREFERRED FRAME- the rest frame of their medium. In Galilean relativity there is NO PREFERRED FRAME because it treats all inertial frames as equivalent, therefore it is incompatible with waves. So you cannot simply change reference frames and add the speed of the train to the speed of light.

This has nothing to do with Einstein's relativity, it is a principle of classical wave physics. The only reason why Einstein's theory of relativity was developed is because all physicists were incorrectly applying galilean transformations to waves as if they were particles, and wrongly predicted a variable speed of light based on this fundamental error- which was experimentally infirmed.

If they correctly calculated the observed wave speed for moving observers, using the WAVE EQUATION and multiplying OBSERVED FREQUENCY to OBSERVED WAVELENGTH via the DOPPLER-FIZEAU effect, they would have realized that the speed of light is INVARIANT for ALL OBSERVERS: *f'lambda'=c.*

Because the observed frequency f' changes inversely proportionally to the observed wavelength lambda', i.e. the observed frequency increases by a factor of (c+vo)/c+vs) and the observed wavelength decreases by the inverse amount (c+vs)/(c+vo) (signs can also be negative depending on direction). And these Doppler shifts are canceled when multiplying f' to lambda', leaving simply flambda=c.  It is THAT simple, no time dilation or length contraction are required because classical wave physics ALREADY PREDICTS INVARIANT WAVE SPEED if you apply it correctly

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