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.
Answer
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 its 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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