The wavelength of light in air and a medium is ${\mathrm{\lambda }}_{\mathrm{a}}$ and ${\mathrm{\lambda }}_{\mathrm{m}}$ respectively. In terms of wavelengths, the refractive index of a medium with respect to air can be represented as follows:

The refractive index ($\mathrm{\mu }$) of a medium is defined as the ratio of the speed of light in air to the speed of light in that medium.

$\mathrm{\mu }=\frac{\mathrm{Speed} \mathrm{of} \mathrm{light} \mathrm{in} \mathrm{air}\left({\mathrm{v}}_a\right)}{\mathrm{Speed} \mathrm{of} \mathrm{light} \mathrm{in} \mathrm{medium} \left({\mathrm{v}}_m\right)}$    ......(1)

The speed of light in a medium can be expressed as the product of its wavelength and frequency in the medium.

${\mathrm{v}}_a={\mathrm{\lambda }}_a\mathrm{f}$        ....(2)

When light enters from one medium to another, the frequency of the light does not change. Therefore, it will remain constant.

$\therefore$${\mathrm{v}}_m={\mathrm{\lambda }}_m\mathrm{f}$                .....(3)

Substituting equations (2) and (3) in equation (1), we get,

$\mathrm{\mu }=\frac{{\mathrm{\lambda }}_{\mathrm{a}}}{{\mathrm{\lambda }}_{\mathrm{m}}}$

Therefore, the refractive index of a medium with respect to air can be expressed in terms of wavelength as: $\frac{{\lambda }_a}{{\lambda }_m}$