The phase factors of all three waves must be *equal* on the actual
boundary itself, hence:

(9.59) |

At , the three
's must lie in a plane. The angles of
incidence , reflection , and refraction
add to the angles in the dot product to make , so the cosine in
the dot product becomes the sine of these angles and we obtain:

(9.60) |

which is both

Note well that we used
and
to put it in terms of the *index of
refraction*, defined by and . Then we cancel
, using the fact that the frequency is the *same* in both
media.

Snell's Law and the Law of Reflection are thus seen to be kinematic relations that are the result of the requirement of phase continuity on the plane interface - a ``wavefront'' of the (or ) wave must be the same as the wavefront of the wave.