Past exam of the physics 2 paper of the University of Oxford / 2014 / 6

Huygens' principle treats each point of a wavefront as the source of a secondary wavelet. The later wavefront is the common forward envelope of those wavelets. Consider a plane front reaching a flat interface between media with speeds $v_1$ and $v_2$. In the same interval $\Delta t$, the incident front advances $v_1\Delta t$ while the first transmitted wavelet grows to radius $v_2\Delta t$.

The two right triangles sharing the interface segment of length $d$ give $d\sin i=v_1\Delta t$ and $d\sin r=v_2\Delta t$, where $i$ and $r$ are ray angles to the normal. HenceThis is Snell's law. The frequency is unchanged at a stationary interface, while wavelength changes with the propagation speed.