Gene Fuller wrote:
... how does one learn of such a "hidden mathematical concept",
when it does not seem to be embodied in the formalism?

The standing wave function equation, cos(kz)*cos*wt), is different
in kind and function from the traveling wave function equation,
cos (kz ± wt).

When two traveling waves are moving along the same path in opposite
directions, their two phasors are rotating in opposite directions. It
is the sum of their phase angles that is a constant number of degrees.
It is that constant phase angle that has been measured and reported
here. Kraus shows a plot of the standing wave angle for a 1/2WL thin-
wire dipole. It is zero from tip to tip. Kraus has already told us that
its value is zero degrees. For a non-thin-wire, it deviates from zero
degrees, but not by much. There's no good reason to keep measuring it
over and over. A quantity whose phase is fixed at zero degrees cannot
tell us anything about the phase shift (delay) through a coil or even
through a wire.

Given: The phase shift in the standing wave current through 1/8WL of
wire in a 1/2WL thin-wire dipole is zero degrees.

What valid technical conclusions can be drawn from that statement?
That there is no phase shift in 45 degrees of wire in a 1/2WL dipole?

Suppose the standing wave is examined to perfection. Everything that can
be determined is measured without error. Now we take the superposition
in reverse; specifically we divide the standing wave into forward and
reverse traveling components. It would seem that we have a complete and
accurate definition for the two traveling wave components. The
interrelations, as you call them, between the variables and parameters
are fully defined by the basic math and the carefully measured standing
wave.

No argument. What some individuals seem to have missed are key
concepts involved in that process. In fact, that very process is
what I am presenting here.

What else is needed to describe the traveling waves? Additional
variables? Additional coefficients or parameters? Additional hidden
mathematical concepts?

What else is needed is already there but unrecognized by a number of
individuals. The equations for the forward and reflected waves are
different in kind and function from the equations for the standing
wave. Assuming equal magnitudes and phases for the forward and
reflected waves, the superposition of those two phasors yields a
result that is really not a bona fide phasor because it doesn't
rotate.

One cannot use a quantity whose phasor doesn't rotate to measure
phase shifts (delays) through coils or through wires. Pardon me
for having to state the obvious.

Picture one end of the 1/2WL thin-wire dipole and set the reference
phase of the forward current at 90 degrees. This is for reference
only to make the math easy. When the forward current hits the end
of the dipole, it undergoes a 180 degree phase shift and starts
traveling in the opposite direction as the reflected current. For
ease of math, let's assume the magnitude of the forward current and
reflected current at the end of the dipole is one amp.

Here's what the standing wave current will be at points along the
dipole wire looking back toward the center. The first column is
the number of degrees back toward the center from the end of
the dipole, i.e. the end of the dipole is the zero degree reference
for 'z'. The center of the dipole is obviously 90 degrees away
from the end.

Back forward current reflected current standing wave current
0 deg 1 at 90 deg 1 at -90 deg zero
15 deg 1 at 75 deg 1 at -75 deg 0.52 at 0 deg
30 deg 1 at 60 deg 1 at -60 deg 1.00 at 0 deg
45 deg 1 at 45 deg 1 at -45 deg 1.41 at 0 deg
60 deg 1 at 30 deg 1 at -30 deg 1.73 at 0 deg
75 deg 1 at 15 deg 1 at -15 deg 1.93 at 0 deg
90 deg 1 at 0 deg 1 at 0 deg 2.00 at 0 deg

Seven points on the standing wave current curve have been produced
by superposing the forward current and reflected current. One can
observe the phase rotation of the forward and reflected waves.
Please note the phase of the standing wave current is fixed
at zero degrees. Measuring it in the real world will produce
a measurement close to zero degrees. Its phase is already known.
Measuring it multiple times over multiple years continues to
yield the same close-to-zero value. Except for proving something
already known, those measurements were a waste of time.

The above magnitudes and phases of the standing wave current are
reproduced in a graph by Kraus, "Antennas for All Applications",
3rd edition, Figure 14-2, page 464.

There seems to be a lack of understanding and appreciation for what the
concepts of "linear" and "superposition" really mean. These are not just
mathematical concepts. When they apply it means that the system under
study is fully and completely described by ** either ** the individual
functional subcomponents ** or ** the full superimposed functional
component. It is not necessary to use both formats, and there is no
added information by doing so.

No argument there. But the individual doing the superposition needs to
understand exactly what he is doing or else he may make some conceptual
mental blunders. Trying to measure the phase shift of a quantity that
doesn't shift phases is one of those mental blunders.

Take a look at any of your favorite antenna references with an eye
toward the treatment of standing wave antennas. I believe you will find
only passing discussion of traveling waves. There will be some mention
of the equivalence between the two types of waves, but little else. It
is unlikely that you will find anything that says you will get more
information if you take the time and trouble to analyze traveling waves.

My only bona fide antenna references are Kraus and Balanis. Quoting:

Kraus: "A sinusoidal current distribution may be regarded as the standing
wave produced by two uniform (unattenuated) traveling waves of equal
amplitude moving in opposite directions along the antenna."

Balanis: "The sinusoidal current distribution of long open-ended linear
antennas is a standing wave constructed by two waves of equal amplitude
and 180 degree phase difference at the open-end traveling in opposite
directions along its length."

Balanis: "The current and voltage distributions on open-ended wire
antennas are similar to the standing wave patterns on open-ended
transmission lines."

Balanis: "Standing wave antennas, such as the dipole, can be analyzed
as traveling wave antennas with waves propagating in opposite directions
(forward and backward) and represented by traveling wave currents ..."
--
73, Cecil http://www.qsl.net/w5dxp