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

(9.54) 
as a kinematic constraint for the wave to be consistent. That is,
this has nothing to do with ``physics'' per se, it is just a
mathematical requirement for the wave description to work.
Consequently it is generally covered even in kiddyphysics classes,
where one can derive Snell's law just from pictures of incident waves
and triangles and a knowledge of the wavelength shift associated with
the speed shift with a fixed frequency wave.
At , the three
's must lie in a plane and we obtain:
which is both Snell's Law and the Law of Reflection, where
we use
to put it in terms of the index
of refraction, defined by . Note that we cancel ,
using the fact that the frequency is the same in both media.
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Robert G. Brown
20071228