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

(11.55) |

as a

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:
k (&thetas#theta;_i) & = & k'(&thetas#theta;_r) = k (&thetas#theta;_l)

n (&thetas#theta;_i) & = & n'(&thetas#theta;_r) = n (&thetas#theta;_l)
which is both *Snell's Law* and the *Law of Reflection*,
obtained in one fell swoop.

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.