amdx wrote:

I was in a hurry this morning and didn't ask my main question.
I think at this point I understand different frequencies travel at different
speeds.
Group Velocity vs Velocity Factor what is the difference?

Velocity factor is the ratio of the velocity of waves in the medium to
the velocity of the speed of light in a vacuum. That is, VF = v/c, where
v is the velocity in the medium, c is the speed of light in a vacuum,
and VF is the velocity factor. I'm more familiar with its use in
non-dispersive media, but there's no reason it couldn't also be used for
dispersive media. If it is, then group velocity and phase velocity would
each have a different velocity factor (and it would be greater than one
for the phase velocity in a hollow waveguide), and it would also be a
function of frequency.

If vg = c * sqrt(1 - (f/fc)^2) hmm, maybe I should tell what I think I know.
(I'm way over my head on this subject).

I made an error and reversed f and fc in the equations. I've posted a
correction.

If I generate a spark ( many frequencies) all these frequencies combine to
make a waveshape, as the wave travels down the waveguide the waveshape
changes because different frequencies are traveling at different speeds?
Correct me as needed.

Yes, that's correct. A dispersive medium distorts any waveshape except a
pure sine wave. People used to working in the frequency domain often
forget the vital importance of phase response in preserving waveshape
integrity. I learned the hard way that microstrip line is dispersive
when designing a delay line loss compensator for a high-speed sampling
oscilloscope used for TDR, where very good waveshape integrity is essential.

So is 'group velocity' the velocity the peak of the signal as it travels
down the waveguide?

Assuming that by "signal" you mean something other than a sine wave, it
becomes a matter of definition depending on the application. When
dealing with step functions, for example, the 50% point on the step is
commonly used.

Forgive my ignorance but the formula vg = c * sqrt(1 - (f/fc)^2) doesn't
work for me. (1-(f/fc)^2) is negative and I can't get the sqrt of a
negative. What did I miss?

Nothing, it was I that missed my error in reversing f and fc. My
apology. I've posted a correction.

Roy Lewallen, W7EL