Roy Lewallen wrote:
It turns out that we're saved -- For the forward traveling pulse, Ip1 =
Vp1/Z0. For the reverse traveling pulse, Ip2 = -Vp2/Z0. So when the
appropriate substitutions are made, we find that 2*Vp1*Vp2 +
2*Z0*Ip1*Ip2 = 0, so the energy in the sum of the pulses is equal to the
sum of the energies of the pulses. And this is true regardless of the
values of Vp1, Vp2, Ip1, and Ip2. That is, it's true for any two pulses,
for any overlap length. _Provided they're traveling in opposite
directions._

Yes, signals traveling in opposite directions don't interfere.

What happens when one pulse is the inverse of the other, that is, one is
positive and the other negative? Don't they cancel?

No, they don't. In the overlap region, the voltage is indeed zero. But
the current is twice that of each original pulse. The energy is simply
all stored in the magnetic field (line inductance) during the overlap.
The above equations still hold.

Yes, signals traveling in opposite directions don't interfere.

The conclusion I reach is that yes, a specific amount of energy
accompanies a pulse on a transmission line having purely real Z0, and is
confined to the pulse width. Although it can swap between E and H
fields, the energy in the confines of the pulse stays constant in value,
and simply adding when pulses overlap.

This is simply not true for coherent, collinear waves traveling
in the same direction. "Optics", by Hecht has an entire chapter
on "Interference". He says: "Briefly then, interference
corresponds to the interaction of two or more lightwaves yielding
a resultant irradiance that deviates from the sum of the component
irradiances." Irradiance is the power density of a lightwave, i.e.
watts per unit-area. Paraphrasing Hecht: Interference corresponds
to the interaction of two RF waves in a transmission line yielding
a resultant total power that deviates from the sum of the component
powers. If the total power is less than the sum of the component
powers, destructive interference has taken place (normally toward
the source). If the total power is greater than the sum of the
component powers, constructive interference has taken place
(normally toward the load). It is the goal of amateur radio
operators to cause *total destructive interference* toward the
source and *total constructive interference* toward the antenna.
These terms are defined in "Optics", by Hecht, 4th edition on
page 388. Quoting Hecht:

"In the case of *total constructive interference*, the phase
difference between the two waves is an integer multiple of
2*pi and the disturbances are in-phase."

When the phase angle is an odd multiple of of pi, "it is
referred to as *total destructive interference*.

If anyone works out the phase angles between the voltages, one
will discover that they match Hecht's definitions above.

Every text on EM wave interference that you can find will explain
how the bright interference rings are four times the intensity of
the dark interference rings so the average intensity is two times
the intensity of each equal-magnitude wave. Of course, that outcome
honors the conservation of energy principle. Using 'P' for power
density, the equation that governs such interference phenomena
in EM waves is:

Ptot = P1 + P2 + 2*SQRT(P1*P2)cos(A)

where 'A' is the angle between the two electric fields. Every
textbook on optical physics contains that irradiance equation.
If Ptot is ever zero while P1 and P2 are not zero, one can be
absolutely certain that the "lost" energy has headed in the
opposite direction in a transmission line because there is
no other possibility. Energy is *never* lost.

RF waves in a transmission line obey the same laws of physics as
do light waves in free space. Coherent, collinear waves traveling
in the same direction do indeed interfere with each other.
Sometimes the interference is permanent as it is at an ideal
1/4WL anti-reflective thin-film coating on glass.

Sine waves are another problem -- there, we can easily have overlapping
waves traveling in the same direction, so we'll run into trouble if
we're not careful. I haven't worked the problem yet, but when I do, the
energy will all be accounted for. Either the energy ends up spread out
beyond the overlap region, or the energy lost during reflections will
account for the apparent energy difference between the sum of the
energies and the energy of the sum. You can count on it!

There is no problem. Optical physicists figured it out long
before any of us were born.

www.mellesgriot.com/products/optics/oc_2_1.htm

"If the two [out-of-phase] reflections are of equal amplitude,
then this amplitude (and hence intensity) minimum will be
zero."

This applies to reflections toward the source at a Z0-match
in a transmission line.

"... the principle of
conservation of energy indicates all 'lost' reflected intensity
[in the reflected waves] will appear as enhanced intensity in
the transmitted [forward wave] beam."

i.e. All the energy seemingly "lost" during the cancellation
of reflected waves toward the source at a Z0-match in a
transmission line, is recovered in the forward wave toward
the load.

That is exactly what happens when we match our systems. We
cause destructive interference toward the source in order
to eliminate reflections toward the source. The "lost"
energy joins the forward wave toward the load making the
forward power greater than the source power.
--
73, Cecil http://www.w5dxp.com