Re: Total Internal Reflection

[John Denker <jsd@AV8N.COM>]

Quoting SSHS KPHOX <kphox@CHERRYCREEKSCHOOLS.ORG>:
> A wave is traveling from a slow to a faster material. At the critical
> angle the wave is totally reflected. I know the math. I know the argument
> that the exiting wave has a limited finite speed and can play Huygens'
> principle to get all the way to the critical angle. One would think a HS
> teacher should be happy. We know it happens and we can solve problems
> about it happening but I cannot answer the very curious student (me
> included) who wants a deeper understanding, a mechanism for this
> happening.

1) Anybody who wants to go to the next level of understanding
is going in the right direction IMHO.

2) The quoted passage is pretty much a RAY-based approach.  I
suspect it would be helpful to write down the WAVE functions
in each region.
  a) For simplicity, I
would start by working with a scalar wave in one dimension,
such as a wave on a string made of two pieces (dissimilar
but each uniform and nondispersive).  Match the wavefunctions
at the boundary to get the T and R amplitude-and-phase
relative to the I wave.
  b) Generalize to a dispersive medium such as a waveguide.
(Knowing about this is useful for a hundred reasons unrelated
to the question at hand, but I digress....)
See what happens to the T wave when the "far" side of the
boundary is beyond cutoff (but the near side isn't).   You
get an evanescent wave, that is, an omega for which there
is no (real) k.
  c) Then work up to a scalar wave in two dimensions,
perhaps modelling sound at the air/water interface.  Again
there is an evanescent wave, that is, an (omega,kx) for
which kz is not real.
  d) Eventually work up to full-blown vector equations
(electromagnetism).  A reasonable treatment of this is
Feynman volume II chapter 33.

I don't know of any good high-school-level references for
items (a) through (c), sorry.  But I suspect you could do
pretty well by taking any of the standard references and
throwing away the derivations (which involve div grad and
curl) and keeping the solutions i.e. wavefunctions (which
are mostly just algebraic).  As Bob L said, Feynman's _QED_
is a good read ... but it might be a long way to go just
for reflection and refraction.

Huyghens' construction (wavelets) appeals to some folks and
not to others.  The simple version is not quantitative and
the quantitative version is not simple ... but it gives a
nice intuitive picture in any case, and serves to reinforce
the other two approaches (rays and wave equations).