Let's take a look at the energy in pulses and sine waves. At the end of
the day, the energy all has to be accounted for, whether it superposes
or not. It's really not all that difficult to do the analysis, as long
as we're careful not to fall into the traps which seem to have tripped
up quite a few others.

First a rectangular pulse. The energy E it takes to launch a pulse of
voltage Vp and duration T (seconds) on a transmission line of
characteristic impedance (assumed purely real) Z0 is Pp * T = Vp * Ip *
T where

Pp is the constant power applied as the pulse as created
T is the length of time the power was applied
Vp and Ip are the voltage and current of the traveling pulse

The pulse is a traveling wave, so for a forward traveling pulse, Ip = Vp
/ Z0; consequently, E = Vp^2 * T / Z0 = Ip^2 * T * Z0. Note that it's
essential to assume a purely resistive Z0 for this simple time-domain
analysis, since a reactive Z0 would cause a distortion of the pulse shape.

Once launched onto the line, we don't have any guarantee that all the
energy will stay within the spatial boundaries of the pulse -- all we
know for sure is how much total energy we've put into the line. But we
can conceptually freeze the pulse at any instant and see where the
energy is. Let's do that.

The obvious way to determine the energy in the pulse is to integrate the
power, which we can easily calculate. This is, after all, what we did to
find the energy we put into the line in the first place. But we're
interested in the energy distribution as a function of physical position
at an instant of time, so we can't find it by integrating the power.
(This is a mistake that seems to be commonly made.) Why not? Well, first
of all, energy is the *time* integral of the power. If we integrate the
power over a time interval of zero, the result is zero. We could look at
a single position on the line and integrate the power during the time it
takes for the wave to move by, to get the amount of energy which went by
during the time interval. But that's an indirect way of seeing where the
energy is on the line at a given time, and can easily lead to invalid
results. There are at least two potential problems with integrating the
power over a period of time to get the energy which passes a point. The
first is that we assign a sign to power, negative when energy is
traveling one way and positive when traveling the other. Consequently,
the result of the integral can be positive or negative. Although the
concept of negative potential energy is a valid one, I don't believe it
really applies to this situation, so one would have to be very careful
in interpreting and dealing with the sign resulting from the
integration. The second potential problem is that an integral never
produces a unique answer, but only an answer that's correct to within a
constant which has to be separately determined. Careless evaluation of
the constant or ignoring it altogether can produce invalid results.

So what I'm going to do is to evaluate the stored energy *per unit
length* of the line at each position along the line. The meaning of this
is that if we were to choose some sufficiently short segment length, the
amount of energy stored on each segment will be proportional to the
energy per unit length evaluated at that segment. In other words, I'll
evaluate the energy density as a function of position, or the energy
distribution along the line. This tells us where the energy is at the
instant of evalulation. I'm going to use the convention that the stored
or potential energy of a discharged line (V and I = 0) is zero.

The energy per unit length stored in the electric field, or line
capacitance, is C'V^2/2, where C' is the capacitance per unit length and
V is the voltage on the segment of line being evaluated. V is assumed to
not vary significantly over the segment length. We can let the segment
length approach zero as a limit, and say that the energy per unit length
is this value at any particular point along the line, where V is the
voltage at that point.

Likewise, the energy per unit length stored in the magnetic field, or
line inductance, is L'I^2/2 where L' is the inductance per unit length.
The total energy stored per unit length at any point is

E' = (C'V^2 + L'I^2)/2

On our line with purely real Z0, Z0 = sqrt(L'/C'), so L' = Z0^2 * C' and

E' = C'(V^2 + (Z0*I)^2)/2

where
E' is the total stored energy per unit length (or energy density) at
some point
V is the voltage at that point
I is the current at that point
Z0 is the (purely real) line characteristic impedance
C' is the capacitance per unit length

Length units for E' and C' can be anything as long as they're the same
for both.

Now let's look at a traveling pulse. We'll freeze it at some instant
while it's traveling down the line.

At any point to the left or right of the pulse, V and I are zero, so the
energy density is zero except where the pulse is. Where the pulse is, V
is the pulse voltage Vp and I the pulse current Ip, so

E' = C'(Vp^2 + (Z0*Ip)^2)/2

For a traveling wave, I = V/Z0, so

E' = C'Vp^2

This energy density is constant over the whole length of the pulse,
since Vp and Ip are constant over that distance. The total energy in the
pulse is then

E = E' * len = C'Vp^2 * len where len is the length of the pulse in
the same length units as C' and E'.

Because the energy density beyond the pulse in both directions is zero,
this is also the total energy in the line, which must equal the amount
we put in originally. So

E = C'Vp^2 * len = Vp^2 * T / Z0

from which we can calculate C' = T / (Z0 * len). Some manipulation of
this gives

T / len = sqrt(L'C') which relates line delay to L' and C', a result
which can be derived by other means.

All the energy in the line is accounted for -- it's traveling along with
the pulse, confined to the width of the pulse as we'd expect.

Ok, now let's fire another pulse at it from the other end of the line,
and see what happens when they completely overlap. Call the pulse 1 and
2 voltages Vp1 and Vp2, and currents Ip1 and Ip2. Assume that both have
the same duration T and therefore the same length len.

Voltages and currents (or E and H fields) add in the overlap region, so
the total V and I are the sum of the individual pulses' V and I. The
energy density in the overlap region is then:

E' = C'((Vp1 + Vp2)^2 + (Z0*(Ip1 + Ip2))^2)/2 * len

= C'(Vp1^2 + Vp2^2 + 2*Vp1*Vp2 + Z0(Ip1^2 + Ip2^2 + 2*Ip1*Ip2))/2

But what's the simple sum of the energy densities of the two pulses?

E1' + E2' = C'(Vp1^2 + Vp2^2 + Z0*(Ip1^2 + Ip2^2))/2

Oops! The energy density of the sum of the two pulses isn't the same as
the sum of the energy densities of the two pulses! And Since the overlap
region length is the same as the single pulse length, the same holds
true for the total energy. The problem is the two additional terms in
the total energy density 2*Vp1*Vp2 and 2*Z0*Ip1*Ip2.

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._

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.

Well, we ducked that bullet. But what if the two pulses are traveling in
the same direction? What then? The two troublesome terms don't cancel,
so some energy ends up getting created or destroyed. But before worrying
too much about that, try to imagine how you'd accomplish it. The
propagation speed is the same for all pulses, so there' no way one can
catch up with another if both are fired from the input. I believe you
can contrive a situation where two pulses can be generated, one from
each end which, if long enough, will partially overlap when going the
same direction, after reflection. But the overlap and energy
calculations will be different than for this example, and I'm sure the
energy of the summed pulses will equal the total energy in the line. I'd
appreciate seeing an analysis from anyone who thinks he can show
differently. Remember that this analysis assumed that no other pulse was
present at the input while the pulse was being generated. If one is, the
amount of energy going into the pulse will be different. It also assumed
a constant line Z0 and velocity factor (constant L' and C'), so a
different analysis would have to be used if that condition is violated.

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.

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!

As always, I appreciate any corrections to either the methodology or the
calculations.

Roy Lewallen, W7EL

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

Cecil Moore wrote:
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.

I certainly misspoke there. The bright interference rings
are four times the intensity of one of the two equal
waves. The dark interference rings are, of course,
zero intensity.

If the intensity of one wave is P, the intensity of the
bright rings will be 4P and the intensity of the dark
rings will be zero. The average intensity will, of course,
be 2P, the sum of the two wave intensities.
--
73, Cecil http://www.w5dxp.com

Roy Lewallen wrote:
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!

An example from optics will make the situation clear.

http://www.w5dxp.com/thinfilm.GIF

At t3, when the 0.009801 watt internal reflection arrives
to interfere with the 0.01 watt external reflection, what
is the resulting reflected power toward the source?

Anyone who can answer that simple question from the field
of optics will understand what happens to the energy in
a transmission line.

Hint: the reflected power is *not* 0.01w - 0.009801w.
--
73, Cecil http://www.w5dxp.com

Cecil Moore wrote:


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



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



This is a distinction with no technical value. Waves in the same
location are subject to the usual rules of linear superposition of the
fields. Whether you want to call this "interference" is simply a
philosophical choice. There is a whole gamut of results resulting from
the superposition, ranging from zero field to a maximum of all the field
magnitudes combined. The terms "destructive" and "constructive" are
sometimes used to denote the extreme cases, but those terms are not so
well defined for the more intermediate cases.

There is utterly no scientific distinction that applies to "signals
traveling in opposite directions." The mathematical results may look
special in the opposite direction case, but the same basic equations
apply in all cases.

73,
Gene
W4SZ

On Jan 23, 8:35*am, Cecil Moore wrote:
Yes, signals traveling in opposite directions don't interfere.
Call this assertion A.

Consider two antennas several wavelengths apart and driven with
the same frequency. Exploring the field strength far from the
antennas we find regions with zero field strength (nulls) and
regions with increased field strength. This variation in field
strength is usually ascribed to interference and the pattern
of variation is often called an interference pattern.

Similar results can be observed with light (google "two slit
experiment").

Locate one of these nulls far from the antennas and follow it
back towards the antennas. Eventually you will be on a line
between the two antennas.

From assertion A above, is it your contention that far from
the antennas it is "interference" that causes the variation
in field strength, but that on the line drawn between the two
antennas some other mechanism is responsible?

If so, what is the other mechanism? And does it only work
exactly on the line, or does it start working when you get
close to the line? How close?

Now I suggest that interference works just as well on the
line drawn between the antennas as it does every where
else and the conditions along that line are not a special
case.

That said, when we look at the two slit experiment, it is
generally agreed that the photons are redistributed such
that there are no photons in dark regions and more photons
in the bright regions.

On the line drawn between the two antennas, there are dark
regions and bright regions (the standing wave). By analogy,
there are no photons in the dark regions and more in the
bright regions. But the photons from the two sources were
travelling towards each other. What is the mechanism that
redistributes the photons such that there are none in the
dark regions? Do the photons stop and not enter the dark
region? Or do they turn into 'dark photons' as they
transit the dark regions? What are 'dark photons'?

...Keith

Gene Fuller wrote:
Cecil Moore wrote:
Yes, signals traveling in opposite directions don't interfere.

This is a distinction with no technical value. Waves in the same
location are subject to the usual rules of linear superposition of the
fields. Whether you want to call this "interference" is simply a
philosophical choice.

Not so. Here's what Eugene Hecht says: "... optical
interference corresponds to the interaction of two
or more [plane] light waves yielding a resultant
irradiance that deviates from the sum of the component
irradiances."

Superposition can occur with or without interference. If
P1 and P2 are the power densities for two plane waves:

If Ptot = P1 + P2, there is no interference because the
resultant power density does not deviate from the sum of
the component power densities.

If Ptot P1 + P2, there exists interference because
the resultant irradiance does deviate from the sum of
the component power densities.

There is utterly no scientific distinction that applies to "signals
traveling in opposite directions."

Interference only occurs when coherent, collinear waves
are traveling in the same direction. When they are
traveling in opposite directions, standing waves are
the result. Let's limit our discussion to plane waves.

The mathematical results may look
special in the opposite direction case, but the same basic equations
apply in all cases.

Yes, but boundary conditions apply. The phasors of the plane
waves traveling toward each other are rotating in opposite
directions so interference is impossible. Here is a slide
show about interference which only occurs when the waves are
traveling in the same direction.

http://astro.gmu.edu/classes/a10594/...8/l08s025.html
--
73, Cecil http://www.w5dxp.com

Keith Dysart wrote:
From assertion A above, is it your contention that far from
the antennas it is "interference" that causes the variation
in field strength, but that on the line drawn between the two
antennas some other mechanism is responsible?

Of course not - please don't be ridiculous. If the two
antenna elements were isotropic point sources, on a
line drawn between them, there could be no interference
and there would be only standing waves in free space
along that line assuming no reflections from nearby
objects, etc.

Everywhere else there are components of waves traveling
in the same direction so interference is possible anywhere
except on that line between the point sources. When the
sources are not a point, seems to me, interference could
occur at any and all points in space.

My "assertion A above" was about transmission lines,
an essentially one-dimensional context. Two waves in
a transmission line are either traveling in opposite
directions or in the same direction.

Incidentally, I came across another interesting quote
from one of my college textbooks, "Electrical Communication",
by Albert. "Such a plot of voltage is usually referred to
as a *voltage standing wave* or as a *stationary wave*. Neither
of these terms is particularly descriptive of the phenomenon.
A *plot* of the effective values of voltage ... is *not a wave*
in the usual sense. However, the term "standing wave" is in
wide-spread use." [Emphasis is the author's]
--
73, Cecil http://www.w5dxp.com

Cecil Moore wrote:
Gene Fuller wrote:
Cecil Moore wrote:
Yes, signals traveling in opposite directions don't interfere.

This is a distinction with no technical value. Waves in the same
location are subject to the usual rules of linear superposition of the
fields. Whether you want to call this "interference" is simply a
philosophical choice.

Not so. Here's what Eugene Hecht says: "... optical
interference corresponds to the interaction of two
or more [plane] light waves yielding a resultant
irradiance that deviates from the sum of the component
irradiances."

Superposition can occur with or without interference. If
P1 and P2 are the power densities for two plane waves:

Why do you attribute such magic to the word "interference"? Do you think
that Hecht's "interaction" is any different than superposition?

What if the waves are not quite anti-parallel, say at an angle of 179
degrees? Is interference now possible?

Suppose the waves are only 1 degree from parallel. Does that negate the
interference?

Repeating: This is a distinction with no technical value.

73,
Gene
W4SZ

On Jan 23, 1:12*pm, Cecil Moore wrote:
Keith Dysart wrote:
From assertion A above, is it your contention that far from
the antennas it is "interference" that causes the variation
in field strength, but that on the line drawn between the two
antennas some other mechanism is responsible?

Of course not - please don't be ridiculous. If the two
antenna elements were isotropic point sources, on a
line drawn between them, there could be no interference
and there would be only standing waves in free space
along that line assuming no reflections from nearby
objects, etc.

Everywhere else there are components of waves traveling
in the same direction so interference is possible anywhere
except on that line between the point sources. When the
sources are not a point, seems to me, interference could
occur at any and all points in space.

OK. So it is your contention that "far from the antennas
it is "interference" that causes the variation in field
strength, but that on the line drawn between the two
antennas some other mechanism is responsible".

But why do you say "Of course not" and then proceed to
paraphrase my statement?

When the mechanism abruptly changes from interference
when off the line to "standing wave" when EXACTLY (how
exact?) on the line, is there any discontinuity in
the observed field strengths?

...Keith