= Refraction
{wiki}
Refraction changes the direction of a transmitted <wave> when it crosses between media with different <phase velocities>. For <light> at a planar interface, <Snell's law> relates the angles measured from the <normal vector> through the <refractive indices>. At normal incidence the direction remains unchanged even though the <wavelength> and <phase velocity> can change. The worked example under <geometrical optics> distinguishes transmission from <total internal reflection>.
For <light>, <refraction> explains why a submerged object looks nearer the surface and why an <optical lens> can form an image. A <wavefront> crossing a boundary must retain a consistent <wave phase> along that boundary: its normal spacing changes with <phase velocity>, and oblique <light rays> change direction. <Normal incidence> separates these two effects especially clearly.
Take stationary, transparent, homogeneous media with <isotropy>, <linearity> of their optical response, and real positive <refractive indices>, at one chosen <frequency> $f$. With $c$ the <vacuum> <speed of light> and $\lambda_0=c/f$ the <vacuum> <wavelength>, medium $j$ has
$$
n_j=\frac{c}{v_{\mathrm{phase},j}},\qquad
v_{\mathrm{phase},j}=f\lambda_j,\qquad
\lambda_j=\frac{\lambda_0}{n_j},\qquad
k_j=\frac{2\pi}{\lambda_j}=\frac{n_j\omega}{c},\qquad \omega=2\pi f.
$$
The <frequency> is unchanged at a stationary boundary; the <phase velocity>, <wavelength> and <wavenumber> can change. Here $n_j$ is a phase <refractive index>, not the <group refractive index>. If there is <optical dispersion>, evaluate $n_j$ at this same <frequency> on both sides.<College Physics by Paul Peter Urone and others/27.1 The Wave Aspect of Light: Interference>[§27.1, paragraph preceding equation 27.2 and equation 27.2: unchanged frequency and wavelength in a medium]{ref}
For an original idealized air–<water> example, stipulate $n_{\mathrm a}=1$ and $n_{\mathrm w}=4/3$, with <vacuum> <wavelength> $600\,\mathrm{nm}$. Then $f=c/(600\,\mathrm{nm})\approx4.99654\times10^{14}\,\mathrm{Hz}$, the <water> <phase velocity> is $3c/4$, and its <wavelength> is $450\,\mathrm{nm}$. At an air <angle of incidence> $30^\circ$, <Snell's law> gives a <water> <angle of refraction> $\arcsin(3/8)\approx22.0243^\circ$. These stipulated values illustrate the relations; actual <refractive indices> depend on material conditions and <frequency>. By <reversibility of an optical ray>, the same path emerging from <water> has outgoing air <angle> $30^\circ$.
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