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When light refracts from a medium to a second one, its frequency stays the same, and its wavelength changes. If this is true, why we see the refracted light ray's colour is the same as the incident light ray in the second medium? The colors should not be the same. If the wavelength changes, colour should change too.

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marked as duplicate by DumpsterDoofus, Kyle Kanos, jinawee, Brandon Enright, Chris White Mar 18 at 6:30

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When seeing, isn't the last medium the light passes through always the stuff in your eyeballs? –  Glen The Udderboat Mar 17 at 20:15
    
Frequency does not change moving through another medium, so a (better?) different question to ask is whether color is dependent on wavelength or frequency. –  Joshua Mar 18 at 5:26
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The color will not change. What you're not taking into account is the speed of light in the medium. It's not the same $ c $ en vacuo. The frequency stays the same. What changes is that speed of light in the refracting medium and as a result wavelength. This difference for speed is the exact reason we have refractive effects, and I believe was the observation that led to Snell's Law. In symbols $$ \lambda = \frac {c} {\nu} $$ where $\lambda$ is the wavelength and $ \nu $ is the frequency.

The speed of light changes because the photons have to have it's energy ( and therefore it's presence ) propagated across very long molecular chains. Electrons have to absorb the incident photons, re-emit, and repeat this in a longitudinal direction. Depending on what the material is made of, this will take variable time in variable media. That notion manifests itself in the different indices of refraction that different objects have.

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Doryan pretty much explained it, I'll just add some intuitive points. The wavelength changes BECAUSE the speed of light changes and the ratio of these remains constant which is the frequency of the light wave. Colour is a direct result of the frequency of a wave, not it's wavelength. Wavelength is merely the distance between two consecutive crests/troughs of a wave. If a travelling wave changes it's speed, this distance must surely also change so that their ratio remains the same. Try looking at a wave graph to get the idea.

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You seem to be mixing up a few concepts. The wavelength does indeed effectively change inside the medium (let's just say glass, like a glass window, to be simple). However, when it comes out of the glass, it turns back into its initial wavelength.

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