The phase factors of all three waves must be equal on the actual boundary itself, hence:
, the three
's must lie in a plane. The angles of
, 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.