Roger wrote:

I think we are in sync here, but something is missing. When I think of
a traveling sine wave, it must have a beginning as a point of beginning
discussion. I pick a point which is the zero voltage point between wave
halves. It follows that the maximum voltage point will be 90 degrees
later. I think you are doing the same thing, but maybe not.

Next I imagine the whole wave moving down the transmission line as an
intact physical object, with the peak always 90 degrees behind the
leading edge. In our example 1/2 wave line, the leading edge would
reach the open end 180 degrees in time after entering the example. We
can see then, that the current peak will be at the center of the
transmission line when the leading edge reaches the end.

Yes, you're describing some of the properties of a sinusoidal traveling
wave. I generally describe them mathematically.

Again, we seem to be in complete agreement except for the statement "In
those situations, infinite currents or voltages occur during runup". For
many years I thought that "initial current into a transmission line at
startup" would be very high, limited only by the inductive
characteristics of the line. With this understanding, I thought that
voltage would lead current at runup. It was not until I saw the formula
Zo = 1/cC that I realized that a transmission line presents a true
resistive load at startup. Current and voltage are always in phase at
startup.

They are provided that Z0 is purely resistive. That follows from the
simplifying assumption that loss is zero or in the special case of a
distortionless line, and it's often a reasonable approximation. But it's
generally not strictly true.

But that doesn't have anything to do with my statement, which deals with
theoretical cases where neither end of the line has loss. For example,
look at a half wavelength short circuited line driven by a voltage
source. Everything is fine until the initial traveling wave reaches the
end and returns to the source end.

If we agree that voltage and current are always in phase in the
traveling wave, then we should find that in our example, the system
comes to complete stability after one whole wave (two half cycles) is
applied to the system.

Assuming you're talking about the half wavelength open circuited line
driven by a voltage source -- please do the math and show the magnitude
and phase of the initial forward wave, the reflected wave, the wave
re-reflected from the source, and so forth for a few cycles, to show
that what you say is true. My calculations show it is not. I'd do it,
but I find that the effort of showing anything mathematically is pretty
much a waste of effort here, since it's generally ignored. It appears
that the general reader isn't comfortable with high school level
trigonometry and basic complex arithmetic, which is a good explanation
of why this is such fertile ground for pseudo-science. But I promise
I'll read your mathematical analysis of the transmission line run-up.

Roy Lewallen, W7EL