Derivation of Snell's law

(a) Refraction of a water wave. The water in the upper left part of the tank is shallower, so the speed of the waves is slower there, and their wavelength is shorter. The reflected part of the wave is also very faintly visible. Retouched from an uncopyrighted PSSC College Physics photograph. (b) A close-up view of what happens at the interface between the deeper medium and the shallower medium. The dashed lines are normals to the interface. The two marked angles on the right side are both equal to θ1, and the two on the left equal θ2.

However intuitively appealing the mechanical model may be, light is a wave, and we should be using wave models to describe refraction. In fact Snell's law can be derived quite simply from wave concepts. In figure (b), trigonometry gives

Eliminating h by dividing the equations, we find

The frequencies of the two waves must be equal or else they would get out of step, so by v=fλ we know that their wavelengths are proportional to their velocities. Combining λ v with v 1/n gives λ 1/n, so we find

which is one form of Snell's law.

Ocean waves near and far from shore