Constant wavelength of light (the fundamental axiom of future, Einstein-free physics) emerges any time physicists ignore for a while Einstein's relativity and return to common sense:
University of Illinois at Urbana-Champaign: "Consider a falling object. Its speed increases as it is falling. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, we should observe the same effect for light. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction." https://courses.physics.illinois.edu/phys419/sp2011/lectures/Lecture13/L13r.html
Two principles implied in this particular scenario are actually valid in any scenario:
(1) Frequency and speed of light vary proportionally.
(2) The wavelength of light remains constant.
It is easy to see that (1) and (2) are equivalent, given the formula (frequency)=(speed of light)/(wavelength).
Another scenario where (1) and (2) are obviously true is Doppler (moving observer):
"Thus, the moving observer sees a wave possessing the same wavelength [...] but a different frequency [...] to that seen by the stationary observer." http://farside.ph.utexas.edu/teaching/315/Waveshtml/node41.html
"The wavelength is staying the same in this case." https://www.youtube.com/watch?v=MHepfIIsKcE
"Vo is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + Vo. [...] The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time." http://a-levelphysicstutor.com/wav-doppler.php
Physicists would not readily agree that the wavelength remains constant when the moving-emitter scenario is considered:
Stephen Hawking, "A Brief History of Time", Chapter 3: "Now imagine a source of light at a constant distance from us, such as a star, emitting waves of light at a constant wavelength. Obviously the wavelength of the waves we receive will be the same as the wavelength at which they are emitted (the gravitational field of the galaxy will not be large enough to have a significant effect). Suppose now that the source starts moving toward us. When the source emits the next wave crest it will be nearer to us, so the distance between wave crests will be smaller than when the star was stationary." http://www.fisica.net/relatividade/stephen_hawking_a_brief_history_of_time.pdf
Hawking is not alone - all physicists believe that the wavelength of light varies with the speed of the emitter. Here is an animation: https://youtu.be/3mJTRXCMU6o?t=77
Variable wavelength of light contradicts the principle of relativity. If the wavelength varied, the emitter could regularly measure the variations inside his spaceship - so he would know his spaceship's speed without looking outside. If, for instance, measurements inside the spaceship show that the wavelength has decreased, the emitter will conclude that his spaceship is now moving faster than before.
The wavelength of light depends only on the nature of the emitting substance and is constant otherwise. In future, Einstein-free physics the wavelength of light will be nothing more than an invariable coefficient in the formula
(speed of light) = (wavelength)(frequency)
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