The light propagation is described, in the simplest case, by Asin(k⃗ r⃗ −2πνt)$Asin(\overrightarrow{k}\overrightarrow{r}-2\pi \nu t)$. The quantity ϕ=k⃗ r⃗ −2πνt $\varphi =\overrightarrow{k}\overrightarrow{r}-2\pi \nu t\text{}$ is called phase, and a surface on which at a given time t$t$ the phase is constant, is called wave-front. The phase-velocity is the ratio of the distance between two neighbor wave-fronts of the same phase ϕ$\varphi $, and the time T$T$ needed for the light to travel a distance equal to the distance d$d$between two neighbor wave-fronts. This time is given by

T=1/ν.(i)$$\begin{array}{}\text{(i)}& T=1/\nu .\end{array}$$

Note that the light frequency ν$\nu $ doesn't change from medium to medium, therefore T$T$ doesn't change.

For illustration of the situation imagine the following scenario:

Consider a front wave of phase ϕ$\varphi $ (red line) touching at the time t0${t}_{0}$ the point A$A$ on the separation surface between the two media, then at a time t1${t}_{1}$ the point B$B$ of the front wave touches a point B1${B}_{1}$ of the separation surface, and at a time t2${t}_{2}$ the point C$C$ of the front wave touches the point C2${C}_{2}$ of the separation surface. Let the points A,B,C$A,B,C$ be chosen so as t1=t0+T${t}_{1}={t}_{0}+T$, and t2=t1+T${t}_{2}={t}_{1}+T$. As the velocity of the wave is smaller in M2${M}_{2}$,

v2v1=n1n2(ii)$$\begin{array}{}\text{(ii)}& \frac{{v}_{2}}{{v}_{1}}=\frac{{n}_{1}}{{n}_{2}}\end{array}$$

the distance that the wave can travel during the time T$T$ is smaller,

d1=v1T, d2=v2T==>d2<d1(iii)$$\begin{array}{}\text{(iii)}& {d}_{1}={v}_{1}T,\text{}\text{}\text{}{d}_{2}={v}_{2}T=={d}_{2}{d}_{1}\end{array}$$

(hence the distance between two consecutive wave-fronts is smaller.)

In consequence, the angle θ$\theta $ of the wave-front with the separation surface is smaller in M2${M}_{2}$ than in M1${M}_{1}$. Finally, note that the angle between the front wave in M2${M}_{2}$ and the separation surface is exactly equal to the refraction angle (i.e. between the normal to the separation surface and the normal to the wave-front).