Mike Monett wrote:
Roy Lewallen wrote:
. . .
Again my apology. You do indeed have it right. Incidentally, it's
not possible for a medium to have a purely reactive (imaginary) Z0
at any non-zero frequency.

Roy Lewallen, W7EL

Thanks very much, Roy. It was probably my mistake, using the word
"Orthogonal" when quadrature would probably have worked better.

Can you explain your last sentence? Why does this happen?

The "non-zero" was unnecessary, and a result of a too-quick evaluation
of an equation, although the meaning of Z0 at DC isn't clear anyway.

There are at least two related ways to show that a medium can't have a
purely imaginary Z0 (more correctly, intrinsic impedance). One is to use
the telegrapher's equation for a transmission line immersed in the medium:

Z0 = sqrt((R + jwL)/(G + jwC)) (w = omega, the rotational frequency)

For Z0 to be purely imaginary, the quantity under the radical has to be
purely real and negative. A little algebraic manipulation shows that
this requires that RG + w^2LC 0. All the quantities are positive, so
it can't happen.

You can also use

Zc = sqrt(mu/ceps) where ceps = the complex permittivity, mu = the
permeability of the medium, and Zc the intrinsic impedance.

The complex permittivity ceps = eps - j*sigma/w

where eps = the real (DC) permittivity
sigma = the conductivity of the material

You end up with the same situation, where for Zc to be purely imaginary,
the quantity under the radical has to be purely real and negative, which
requires that mu * eps 0. Remember that mu and eps here are the actual
permeability and permittivity, not the relative values we often use.

A little further research reveals that there are some fairly recently
created man-made materials which have a negative permeability. Those
could presumably have a purely imaginary intrinsic impedance, provided
that they have a positive permittivity. So there might be an exception
to my statement, although it isn't something you're likely to encounter
for some time to come.

I have been following these threads with some interest, and I very
much appreciate your analysis, as it adds greatly to my
understanding. Thank you very much for taking the time to write so
clearly.

There is one point I still have trouble with. The concept of power
flowing in standing waves where the superposition goes to zero, and
yet the energy flow is unaffected and continues in opposite
directions on either side of the null point.

Anyway, I have googled until my fingers get sore, and I haven't
found a good explanation of why this happens. Everyone says it is
well understood from basic undergraduate theory, but the only
references I can find are from graduate studies in Quantum
Electrodynamics. This is not much help.

Please exclude me from the "everyone" in "everyone says". I don't say
that power flows, period. We've seen the serious traps people have
fallen into by making this assumption and trying to build from it.
That's why you won't find it in texts.

So I have to form some image in my mind of why these waves do not
interact. Here is a partial pictu

1. Electromagnetic waves travel at the speed of light in whatever
medium they are in. For them to interact, there must be some advance
information they are about to collide. But that would require
transferring information faster than the speed of light, which is
forbidden.

You don't need a reason for them to not interact, you need a reason for
them to do so. In a linear medium, there is none.

2. The fields in electromagnetic waves are at right angles to the
direction of propagation. There is no longitudinal component, and
therefore the waves have no advance warning they are about to
collide. There is no vector component that is common to both that
would allow any interaction, so there is no way this can happen.

3. Photons carry no charge. They are not deflected by electrostatic
or electromagnetic fields, and do not interact with other photons.
Electromagnetic waves are made up of photons. Since photons do not
interact, EM waves also do not interact with each other.

The above concepts seem to make sense, and allow me to get some
sleep at night. Can you tell me if they are valid, and if there are
other ways of explaining this phenomenon?

I'm glad they work for you. I'll have to leave it to others to comment
on their validity, since I don't buy into the notion of flowing power in
the first place.

Roy Lewallen, W7EL