Objection, your Honor! Answer is unresponsive to the question.

Sustained.

8-)

Gee Cecil, how does one learn of such a "hidden mathematical concept",
when it does not seem to be embodied in the formalism?

Let's try again.

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.

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

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.

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.

73,
Gene
W4SZ


Cecil Moore wrote:
Gene Fuller wrote:

As you have stated, including references from Hecht, it is customary
to mathematically show traveling waves in the form: cos (kz +/- wt)
Through straightforward addition and simple trigonometry is is seen
that the standing wave corresponding to the sum of equal magnitude
forward and reverse traveling waves has the form: cos (kz) * cos (wt)


I see I made a typo and typed a '+' sign in my previous equation.
Of course, it should have been a '*' sign for multiply.

Are there some hidden variables that have not been considered?


Not a hidden variable, but there seems to be a hidden
mathematical concept, at least hidden from some individuals.

In case some might not know, 'z' is the position up and down
the wire, omega (w) is our old friend 2*pi*f, and 't' is, of
course, time.

In the traveling wave equation, cos(kz +/- wt), the position on
the wire and omega*time are added or subtracted *before* the cosine
function is taken. That means that the position on the wire and the
phase velocity are inter-related. One cannot have one without the
other. And that is indeed a characteristic of a traveling
wave. Physical position, frequency, and time all go into
making a traveling wave. It is modeled as a rotating phasor.

However, in the equation, cos(kz) * cos(wt), the physical position,
'z' is disconnected from the phase velocity, 'wt'. The standing wave
is no longer moving in the 'z' dimension. If you pick a 'z' and hold
it constant, i.e. choose a single point on the wire, the standing
wave becomes simply some constant times cos(wt). Thus at any fixed
point on the line, the standing wave is not moving - it is just
oscillating at the 'wt' rate and a current probe will certainly pick
up the H-field signal. The phase of the standing wave current is
everywhere, up and down the 1/2WL thin-wire, equal to zero. The sum
of the forward phasor and reflected phasor doesn't rotate. Its phase
doesn't change with position. Only its magnitude changes with position
and if the forward wave magnitude equals the reflected wave magnitude,
it is not flowing in the real sense that current flows. It is a
standing wave and it is just standing there.

The main thing to realize is that the standing wave equation
divorces the position of the standing wave from its phase
velocity such that the phase velocity is not active in the
'z' dimension, i.e. up and down the wire. The standing wave
current "pseudo phasor" is not rotating. The standing wave
is not going anywhere. It is not flowing along a wire or
through a coil. Measuring its phase is meaningless because
the phase is already known to be constant and unchanging from
tip to tip in a 1/2WL dipole or across a loading coil in a
mobile antenna.

Thinking that standing wave current flows from the middle of a
dipole to the ends is just a misconception. The equation for
a standing wave indicates that it doesn't flow. What is flowing
are the forward and reflected waves.