The relationship between the speed of light in a medium and the refractive index is the following:
Therefore it can be understood that for a medium of higher refractive index, the speed of light in that medium will be slower. Light will not achieve a speed higher than c or 2.99 x 10^8 m/s. When light is traveling at this speed the refractive index of the medium is 1.00.
Now, what about the wavelength? Interestingly, one might begin to understand that the wavelength is the determining factor for color. In fact, this is not the case. Frequency is what defines the color of the light, which can vary from an invisible infrared range to the visible range to the invisible ultraviolet range. In a monochromatic system, the frequency of light (and therefore color) will stay the same. The velocity and wavelength will change with the refractive index.
As the above picture suggests, we might beleive that wavelength and frequency are forever tied together. The above example would in fact be incomplete at best, were we to consider that light can travel at more than one speed. However, let us review the relationship between wavelength and frequency. The following formula is normally presented for wavelength:
Now, here is the question: does c in this equation correspond to the speed of light in a vacuum, or does it correspond to the speed of the travelling light wave? Let’s consider, what does the speed of light in a vacuum have to say about the speed of light in water? It really doesn’t have much to say, does it? Which is why we can use instead, v to denote the speed of light.
Note that I’ve written the wavelength as a function of the speed of light in the medium. Taking this to it’s conclusions, we would understand that actually, the wavelength is not exclusively dependent on frequency and that multiple wavelengths may exist for one frequency. The determining factor in such a case is the refractive index, given that frequency is constant.
Given the wavelength, frequency and refractive index, the speed of the light wave may also be calculated.
Physically, one may picture that the frequency is the rate at which the peak of a wave passes by a point. A longer wavelength wave will need to move faster to keep at the same frequency.
The applications and implications of this physical relationship will be explored next.