The Rest of the Story
 

On Apr 7, 5:31*pm, Cecil Moore wrote:
Keith Dysart wrote:
As long as you stick with simple assertions, followed
by sentences such as, "That is simple physics.",
spoken in a tone which says no further understanding
is necessary, you will be locked in the fruitless
search for the imputed reflected energy flow.

My vision is returning and could turn out to be
the best vision that I've ever had in my life. :-)

You have been asking for the mechanism for storage
and return of the interference energy in the system.
That mechanism is standing waves. Are you aware that
standing waves store energy and return it to the
system every 90 degrees?

More correctly, the energy is stored in the distributed
capacitance and inductance of the transmission line.

In the examples being discussed,
there are standing waves inside the source.

There is no capacitance or inductance in the source to
store energy.

For the 1/8WL
shorted line, there appears to be 125 watts of forward
power and 25 watts of reflected power at points on each
side of the source.

Not if there is no transmission line.

With powers given in average values, the circuit that
you should be using for your instantaneous power
equations is:

* * * * * * * * 50 ohm
----50-ohm----/\/\/\/\----50-ohm----
* * *125w-- * * 100w * * 50w--
* * *--25w * * * * * * * --50w

I will be very surprised if the instantaneous
powers don't balance.

Perhaps. But I don't need more examples where the
powers balance. I already have the one example where
they don't. And it only takes one to disprove an
hypothesis.

Your previous problem is that you were using net
power values on one side of Rs and component power
values on the other side of Rs.

But there are no component powers in the source. It
is a simple circuit element.

...Keith

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On Apr 8, 8:51*am, Cecil Moore wrote:
Roy Lewallen wrote:
Now, I don't know of any way to assign "ownership" to bundles of energy.

One way is to add a unique bit of modulation to each
bundle of wave energy. I am fond of using a TV signal
and observing ghosting on the screen. This, of course,
assumes that the modulation stays with the same component
wave to which it was originally associated.

But as soon as you modulate, you no longer have sinusoidal
steady state.

You can split the signal up into its spectral components
and, using superposition, analyze the circuit for each
spectral component individually, then sum them to obtain
the total system response. But chasing the energy with
one frequency is hard enough. The conundrums that arise
when doing it for several are much worse than the ones
here.

For your enjoyment consider the composite signal
cos(9.95*2*pi*t)+cos(10.05*2*pi*t)
working into 1 ohm.
The imputed average power for each of the components is
0.5 W. The total average power is 1 W as expected.

Consider the 1 second interval from 4.5 to 5.5 seconds.
In this second 0.016393 joules flow for an average
power of 0.016393 W. But the sum of the imputed power
in the two spectral components is 1 W. Where did the
missing energy go?

Just another example of why assigning too much reality
to the imputed powers of the components of superposition
is misleading.

But let's suppose that the energy which flows into the node from the
left side during the "inhalation" part of the cycle is the energy which
flows out to the right during the "exhalation" part of the cycle, and
the energy flowing into the node from the right exits on the left. So
now we've managed to get energy past the node going in both directions
while maintaining zero power and current at the node and conserving
energy as we must.

This agrees with the distributed network model. Since
there is no impedance discontinuity and no impedor at
the node, there can be no reflections at the node.

In other examples, you have suggested inserting a zero length
transmission line to aid analysis. Why not insert a zero length
transmission line with an impedance to produce the desired
reflection? At steady state the reflection cancels but this
would be due to the redistribution of energy according to
your explanations.

The
forward wave flows unimpeded through the node as does
the equal magnitude reflected wave. The net energy flow
is zero. The average energy flow is zero.

Anyone who believes there is zero energy at a standing-
wave current node should touch that point on a transmission
line (which just happens to be the same point as the
maximum voltage anti-node).

No one has said there is zero energy. Only that there is
zero energy flow. For energy flow, one needs simultaneous
voltage and current.

One must be careful not to confuse the net signal with the
component signals.

Agreed. Assigning too much reality to component signals is
seriously misleading. Now actual voltages, currents and
powers, that's a different thing.

One must be careful not to confuse the
average values with the instantaneous values.

This can best be visualized using light waves in free
space. Unimpeded EM waves do not bounce off of each other.

Until one can grasp the simplicity of a transmission line,
moving to the complexity of free space offers nothing but
obfuscation.

...Keith

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On Apr 8, 11:44*am, Cecil Moore wrote:
Keith Dysart wrote:
You seem to be saying that the answers would be
completely different if you chose a different
impedance for the non-existant transmission line.

There you go again, trying to shove your words
into my mouth. (Pattooieee!) Please don't do that.

Did you not say that 50 ohms for the non-existant
line was the correct impedance because other impedances
would yield the 'wrong' answer?

You have apparently done the math and found it
to be valid so, once again, you have to change
the specified conditions in order to try to make
your point.

I don't recall changing anything. I'm still with
Fig 1-1 from your paper, which did not include
non-existant transmission lines.

A Z0 of 50 ohms is the *only* characteristic
impedance that will meet the specified precondition
of zero average interference. Choosing any other
characteristic impedance will move the example
outside of the scope of my Part 1 article.

Did you not say that adding 1 wave length of transmission
line does not alter the conditions? Are you now saying
it does?

I
understand why you want to do such a thing but
obfuscation, diversions, and straw men are not
part of the scientific method. My Part 1 article
has a very narrow scope. Please abide by it.

Yes. That is why I prefer the simplicity of Fig 1-1
without the non-existant transmission lines.

...Keith

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Keith Dysart wrote:
Thus I strongly suggest that Vg, Ig, Pg, represent reality. The
others are a convenient alternative view for the purposes of
solving problems.

Of course they represent *net* reality but we are trying
to determine what is happening at a component wave level.
Defining the component waves out of existence is an un-
acceptable substitute for ascertaining what is happening
in reality.

Typically we see Vg split into Vf and Vr, but why stop at two.
Why not 3, or 4?

Because two is what a directional wattmeter reads. The
two superposed waves, forward and reverse, can be
easily distinguished from one another. Two superposed
coherent forward waves cannot be distinguished from
each other. That's why we stop at two - because it is
foolish to go any farther.

There is power coming from the transmission line. Looking at Pg(t),
some of the time energy flows into the line, later in the cycle
it flows out. The energy transfer would be exactly the same if the
transmission line was replaced by a lumped circuit element. And
we don't need Pf and Pr for an inductor.

OTOH, the distributed network model is a superset of
the lumped circuit model so the inadequate lumped
circuit model might confuse people. Hint: changing
models to make waves disappear from existence doesn't
make the waves disappear.

The lumped circuit model is adequate for lumped circuits.
It is inadequate for a lot of distributed network problems.
If the lumped circuit model worked for everything, we
wouldn't ever need the distributed network model.

I suggest that you take your circuit and apply distributed
network modeling techniques to it including reflection
coefficients and forward and reflected voltages, currents,
and powers at all points in the circuit. Note that the
reflections are *same-cycle* reflections. If the lumped
circuit model analysis differs from the distributed network
model analysis, the lumped circuit analysis is wrong.

It goes up because the impedance presented by the transmission
changes when the reflection returns. This change in impedance
alters the circuit conditions and the power in the various
elements change. Depending on the details of the circuit,
these powers may go up, or they may go down when the reflection
arrives.

That is true, but the impedance is *VIRTUAL*, i.e. not an
impedor, and is therefore only an *EFFECT* of superposition.
We are once again left wondering about the *CAUSE* of the
virtual impedance, i.e. the details of the superposition
process. Ignoring those details will not solve the problem.
--
73, Cecil http://www.w5dxp.com

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On Wed, 9 Apr 2008 03:45:19 -0700 (PDT)
Keith Dysart wrote:

On Apr 7, 12:14*pm, Roger Sparks wrote:
On Sun, 6 Apr 2008 19:21:00 -0700 (PDT)

Keith Dysart wrote:
On Apr 5, 10:06*am, Roger Sparks wrote:
Pg(t) is the result of a standing wave, containing *power from Pf(x) and Pr(x+90). *

This is one way of thinking of it, but it is less misleading to
consider
that Pg(t) describes the actual energy flow, just as Vg(t) describes
the
actual voltage and Ig(t) describes the actual current. Using
superposition
Vf, If, Vr and Ir can be derived and from these Pf and Pr.

Your argument is correct to the extent that the power you describe is passing point Pg(t) at the instant (t). *It is the equivalent statement that an observer watching cars pass on the freeway would make, saying "one blue car moving left and one red car moving right, so two cars are passing". *Not wrong, just "how is the information useful"?

Pg(t) is the actual power at that point in the circuit. It can be
derived by simply multiplying the direct measurement of the actual
voltages and currents at that point in the circuit. One measures
the same voltages and currents regardless of whether it is a
transmission line to the right of point g, or the equivalent
lumped circuit element.

While Vf, Vr, etc. can be used to derive the same information and,
therefore is arguably just a different point of view, Vf and Vr,
If and Ir, etc., must always be used in pairs to arrive at the
actual circuit conditions. It is when one starts to look at them
separately, as if they individually represent some part of reality,
that confusion awaits.

Thus I strongly suggest that Vg, Ig, Pg, represent reality. The
others are a convenient alternative view for the purposes of
solving problems.

Typically we see Vg split into Vf and Vr, but why stop at two.
Why not 3, or 4? Analyzing a two wire telephone line will use
four or more, forward to the east, forward to the west, reflected
to east, reflected to the west, and sometimes many different
reflections. How do we choose how many? Depends on what is
convenient for solving the problem. The power of superposition.
But assigning too much reality to the individual contributors
can be misleading.

Good thoughts.

By breaking Vg into Vf and Vr, we can explain why very long transmission lines, many wavelengths long, have repeating patterns of inductive and capacitive reactance as if they were lumped components. If Vf and Vr work for long lines, they should work for short lines.

So far as breaking Vg into many sequential/different Vf and Vr, we usually need to do that. Cecil chose our simple example to prevent re-reflection (reflection of the reflection) but even then it is apparent that the voltage source will have a reactive component.

If we can't account for the power, it is because we are doing the accounting incorrectly.

And the error in the accounting may be the expectation that the
particular set of powers chosen should balance. Attempting to
account for Pr fails when Pr is the imputed power from a partial
voltage and current because such computations do not yield powers
which exist.

If we remove the transmission line from the circuit, we have an open circuit with no current. *Without current, there can be no power. How can power arrive at Rs if there is no power coming through the transmission line? *

There is power coming from the transmission line. Looking at Pg(t),
some of the time energy flows into the line, later in the cycle
it flows out. The energy transfer would be exactly the same if the
transmission line was replaced by a lumped circuit element. And
we don't need Pf and Pr for an inductor.

But this flow is quite different than the flow suggested by Pf and
Pr. These suggest a continuous flow in each direction. It is only
when they are summed that it becomes clear that flow is first in
one direction and then other.

I understand the delemma here. It is like trying to both fill and empty the bottle at the same time. We can't do that with physical objects and we like to think of energy as if it were a physical object. So how can energy seemingly flow into a line at the same time it flows out?

Of course one way would be if Vf actually did reflect from Vr. A reflection beginning at the point Vg, then propagating down the line as an artifact of the original wave. So far as I have been able to figure, the result is the same as when we think of both Vf and Vr as actual waves, which are much easier to follow and calculate.

Would it help to consider that before the "reflection from the short" arrives, power arrives via the transmission line path but the impedance is 100 ohms for our example, composed of Rs = 50 ohms and transmission line = 50 ohms? *After the "reflection from the short" arrives, the impedance drops to 70.7 ohms so the power to the circuit goes up (assuming a constant voltage source). *How can this happen if power is not carried via the "reflection from the short"?

It goes up because the impedance presented by the transmission
changes when the reflection returns. This change in impedance
alters the circuit conditions and the power in the various
elements change. Depending on the details of the circuit,
these powers may go up, or they may go down when the reflection
arrives.

Your comment almost makes the altered impedance sound like a resistance, probably not quite the picture you want to convey. I think of power to Rs coming via two paths, one longer than the other. In my mind, the changed impedance is the result of two power streams merging back together.

The impedance found when the reflection returns is dependent upon the line impedance, length of line, and conditons at the point of reflection. The length of line is measured in terms of time and wave velocity. While this is strong evidence supporting Vf and Vr, it does not rule out reflection between wave components.

I don't know how many people have seen an railroad engine starting a train from stop, when there is a small gap between each of the cars. You can hear each of the cars bumping the adjacent car in a chain reaction going from engine to the end of the train. Clearly, the reaction has a velocity of travel. Our EM waves could do the same thing but we would never measure anything except the the resultant wave.

--
73, Roger, W7WKB

The Rest of the Story
 

Keith Dysart wrote:
Cecil Moore wrote:
There is no capacitance or inductance in the source to
store energy.

"In" is an oxymoron for the lumped circuit model.
The lumped reactance exists *at* the same point as
the source because everything is conceptually lumped
into a single point.

In the real world, circuits are never single points
and there exists a frequency at which distributed
network effects cannot be ignored. In reality,
distributed network effects occur for all real
circuits but they can often be ignored as negligible.

The two inches of wire connecting the source to the
source resistor has a characteristic impedance and
is a certain fraction of a wavelength long. If it is
not perfectly matched, reflections will occur, i.e.
there will exist forward power and reflected power on
that two inches of wire.

For the 1/8WL
shorted line, there appears to be 125 watts of forward
power and 25 watts of reflected power at points on each
side of the source.

Not if there is no transmission line.

Aha, there's your error. What would a Bird directional
wattmeter read for forward power and reflected power?
Consider that short pieces of 50 ohm coax are used
to connect the real-world components together.

Or chose any characteristic impedance and do the math.
You will discover something about the real world, i.e.
that you have been seduced by the lumped circuit model.

Perhaps. But I don't need more examples where the
powers balance. I already have the one example where
they don't.

And that one example is outside the scope of the
preconditions of my Part 1 article. Let me help you
out on that one. There are an infinite number of
examples where the reflected power is NOT dissipated
in the source resistor but none of those examples,
including yours, satisfies the preconditions specified
in my Part 1 article. Therefore, they are irrelevant
to this discussion.

But there are no component powers in the source. It
is a simple circuit element.

No wonder your calculations are in error.
Perform your calculations based on the readings of
an ideal 50 ohm directional wattmeter and get back to us.
Hint: Mismatches cause reflections, even in real-world
circuits. The reflections happen to be *same-cycle*
reflections. The simplified lumped circuit model, that
exists in your head and not in reality, ignores those
reflections and thus causes confusion among the
uninitiated who do not understand its real-world
limitations.
--
73, Cecil http://www.w5dxp.com

The Rest of the Story
 

Keith Dysart wrote:
On Apr 8, 8:51 am, Cecil Moore wrote:
Roy Lewallen wrote:
Now, I don't know of any way to assign "ownership" to bundles of energy.
One way is to add a unique bit of modulation to each
bundle of wave energy. I am fond of using a TV signal
and observing ghosting on the screen. This, of course,
assumes that the modulation stays with the same component
wave to which it was originally associated.

But as soon as you modulate, you no longer have sinusoidal
steady state.

You know and I know that is a copout diversion to avoid
your having to face the technical facts.

Consider the 1 second interval from 4.5 to 5.5 seconds.
In this second 0.016393 joules flow for an average
power of 0.016393 W. But the sum of the imputed power
in the two spectral components is 1 W. Where did the
missing energy go?

Hint: Missing energy is impossible except in your mind.
Just because you are ignorant of where the energy goes
doesn't mean it is missing. It just means that you fail
to understand interference. Have you not read Hecht's
Chapter 9 on "Interference"?

Obviously, interference is present and there is *NO*
missing energy. I have previously listed the possibilities
at least four times so will not bother listing them again.

Just another example of why assigning too much reality
to the imputed powers of the components of superposition
is misleading.

Just another example of ignorance in action. Waves
possess energy that cannot be destroyed. Just because
you cannot track it doesn't mean it cannot be tracked.

In other examples, you have suggested inserting a zero length
transmission line to aid analysis. Why not insert a zero length
transmission line with an impedance to produce the desired
reflection?

What would be the characteristic impedance of a length of
transmission that caused a reflection coefficient of 1.0?

No one has said there is zero energy. Only that there is
zero energy flow. For energy flow, one needs simultaneous
voltage and current.

Vfor/Ifor = Z0, Vfor*Ifor = Pfor = EforxHfor
If an EM wave exists, it is moving at the speed of
light and transferring energy. For Z0 purely resistive,
Vfor cannot exist without Vfor/Z0 = Ifor. Vfor is
always in phase with Ifor.

Assigning too much reality to component signals is
seriously misleading.

Assigning reality to the components of superposition
is seriously misleading???? Can we therefore throw
out the entire principle of superposition?

Until one can grasp the simplicity of a transmission line,
moving to the complexity of free space offers nothing but
obfuscation.

It is obvious that you have many things you desire to
hide inside that black transmission line to which we
are not even allowed to attach a directional wattmeter.

Since you are incapable of explaining what happens in
free space for all to see, why should we believe that
you have figured out what is happening inside a
transmission line where everything is hidden from view?

Until one can grasp the transparency of free space,
moving to the opaque transmission line where all kinds
of important things are hidden from view offers nothing
but obfuscation.
--
73, Cecil http://www.w5dxp.com

The Rest of the Story
 

Keith Dysart wrote:
Did you not say that 50 ohms for the non-existant
line was the correct impedance because other impedances
would yield the 'wrong' answer?

ABSOLUTELY NOT! I said that other impedances do
not meet the specified preconditions for my
Part 1 article so the answers are irrelevant,
not wrong.

If a 50 ohm Z0 and 50 ohm load is specified, do
other Z0s yield wrong answers? Of course not.
Those answers are merely irrelevant to the
specified preconditions.

I don't recall changing anything. I'm still with
Fig 1-1 from your paper, which did not include
non-existant transmission lines.

You have not proved any errors exist in my Part 1
article. You keep trying to change the specified
preconditions from average power to instantaneous
power but that is simply unethical.

Did you not say that adding 1 wave length of transmission
line does not alter the conditions? Are you now saying
it does?

The set(A) conditions are not altered. The set(B)
conditions are altered. Exactly what conditions
are you referring to?

Yes. That is why I prefer the simplicity of Fig 1-1
without the non-existant transmission lines.

Of course, you absolutely avoid using any tool that
would prove you wrong. So what's new?
--
73, Cecil http://www.w5dxp.com

The Rest of the Story
 

Keith Dysart wrote:
On Apr 8, 8:51 am, Cecil Moore wrote:
. . .
The
forward wave flows unimpeded through the node as does
the equal magnitude reflected wave. The net energy flow
is zero. The average energy flow is zero.

Anyone who believes there is zero energy at a standing-
wave current node should touch that point on a transmission
line (which just happens to be the same point as the
maximum voltage anti-node).

No one has said there is zero energy. Only that there is
zero energy flow. For energy flow, one needs simultaneous
voltage and current.
. . .

In the interesting case of a current node on an infinite-SWR line, it
appears we do have energy flow without any current, and therefore
without power. Energy flows into the node from both directions in equal
amounts at the same time, and out to both directions in equal amounts at
the same time. What we don't have is *net* energy flow at the node.
Likewise, there's charge flowing into the node from both directions, and
out in both directions, which results in the zero net current. I don't
believe that's the same as saying there's no energy or charge flow at
all, even though the power and current are zero. And it's not necessary
to separately consider forward and reverse waves of current, energy, or
power in order to observe this -- it can be seen from looking only at
the total charge or energy.

Roy Lewallen, W7EL

The Rest of the Story
 

On Apr 9, 12:51*pm, Roger Sparks wrote:
On Wed, 9 Apr 2008 03:45:19 -0700 (PDT)

Keith Dysart wrote:
On Apr 7, 12:14*pm, Roger Sparks wrote:
On Sun, 6 Apr 2008 19:21:00 -0700 (PDT)

Keith Dysart wrote:
On Apr 5, 10:06*am, Roger Sparks wrote:
Pg(t) is the result of a standing wave, containing *power from Pf(x) and Pr(x+90). *

This is one way of thinking of it, but it is less misleading to
consider
that Pg(t) describes the actual energy flow, just as Vg(t) describes
the
actual voltage and Ig(t) describes the actual current. Using
superposition
Vf, If, Vr and Ir can be derived and from these Pf and Pr.

Your argument is correct to the extent that the power you describe is passing point Pg(t) at the instant (t). *It is the equivalent statement that an observer watching cars pass on the freeway would make, saying "one blue car moving left and one red car moving right, so two cars are passing". *Not wrong, just "how is the information useful"?

Pg(t) is the actual power at that point in the circuit. It can be
derived by simply multiplying the direct measurement of the actual
voltages and currents at that point in the circuit. One measures
the same voltages and currents regardless of whether it is a
transmission line to the right of point g, or the equivalent
lumped circuit element.

While Vf, Vr, etc. can be used to derive the same information and,
therefore is arguably just a different point of view, Vf and Vr,
If and Ir, etc., must always be used in pairs to arrive at the
actual circuit conditions. It is when one starts to look at them
separately, as if they individually represent some part of reality,
that confusion awaits.

Thus I strongly suggest that Vg, Ig, Pg, represent reality. The
others are a convenient alternative view for the purposes of
solving problems.

Typically we see Vg split into Vf and Vr, but why stop at two.
Why not 3, or 4? Analyzing a two wire telephone line will use
four or more, forward to the east, forward to the west, reflected
to east, reflected to the west, and sometimes many different
reflections. How do we choose how many? Depends on what is
convenient for solving the problem. The power of superposition.
But assigning too much reality to the individual contributors
can be misleading.

Good thoughts. *

By breaking Vg into Vf and Vr, we can explain

I am not sure that 'explain' is the correct word. It certainly
provides
a convenient technique for computing the voltate and current along the
line, but there are other ways to compute the result; differential
equations being one other way, though less convenient. But I am not
convinced that a convenient technique for solving the problem is
necessarily an 'explanation of why'.

why very long transmission lines, many wavelengths long, have repeating patterns of inductive and capacitive reactance as if they were lumped components. *If Vf and Vr work for long lines, they should work for short lines.

This is true. But when we descend to zero length lines, as some have
done, the rationale becomes quite a bit weaker.

So far as breaking Vg into many sequential/different Vf and Vr, we usually need to do that. *Cecil chose our simple example to prevent re-reflection (reflection of the reflection) but even then it is apparent that the voltage source will have a reactive component.

I still think of a voltage source as just being a voltage source, not
something
with resitance, reactance or impedance.

If we can't account for the power, it is because we are doing the accounting incorrectly.

And the error in the accounting may be the expectation that the
particular set of powers chosen should balance. Attempting to
account for Pr fails when Pr is the imputed power from a partial
voltage and current because such computations do not yield powers
which exist.

If we remove the transmission line from the circuit, we have an open circuit with no current. *Without current, there can be no power. How can power arrive at Rs if there is no power coming through the transmission line? *

There is power coming from the transmission line. Looking at Pg(t),
some of the time energy flows into the line, later in the cycle
it flows out. The energy transfer would be exactly the same if the
transmission line was replaced by a lumped circuit element. And
we don't need Pf and Pr for an inductor.

But this flow is quite different than the flow suggested by Pf and
Pr. These suggest a continuous flow in each direction. It is only
when they are summed that it becomes clear that flow is first in
one direction and then other.

I understand the delemma here. *It is like trying to both fill and empty the bottle at the same time. *We can't do that with physical objects and we like to think of energy as if it were a physical object. *So how can energy seemingly flow into a line at the same time it flows out?

Of course one way would be if Vf actually did reflect from Vr. *A reflection beginning at the point Vg, then propagating down the line as an artifact of the original wave. *So far as I have been able to figure, the result is the same as when we think of both Vf and Vr as actual waves, which are much easier to follow and calculate.

I agree that the final results are the same. The intermediate results
can
mislead in different ways.

Would it help to consider that before the "reflection from the short" arrives, power arrives via the transmission line path but the impedance is 100 ohms for our example, composed of Rs = 50 ohms and transmission line = 50 ohms? *After the "reflection from the short" arrives, the impedance drops to 70.7 ohms so the power to the circuit goes up (assuming a constant voltage source). *How can this happen if power is not carried via the "reflection from the short"?

It goes up because the impedance presented by the transmission
changes when the reflection returns. This change in impedance
alters the circuit conditions and the power in the various
elements change. Depending on the details of the circuit,
these powers may go up, or they may go down when the reflection
arrives.

Your comment almost makes the altered impedance sound like a resistance,

Impedance does have some similarity to resistance, but only for
single
frequency sinusoidal excitation, though I was not trying to say that.

probably not quite the picture you want to convey. *I think of power to Rs coming via two paths, one longer than the other. *In my mind, the changed impedance is the result of two power streams merging back together.

The impedance found when the reflection returns is dependent upon the line impedance, length of line, and conditons at the point of reflection. *The length of line is measured in terms of time and wave velocity. *While this is strong evidence supporting Vf and Vr,

The technique of breaking actual voltage into Vf and Vr certainly
works. But
I would not say this is evidence for the existance of Vf and Vr,
merely agreement
with superposition.

One of my text books drags the reader through the solution using
differential
equations, and then introduces Vf and Vr as a simpler way to solve the
problem.
The student is truly happy from learning that diffyQs will not be
required.

it does not rule out reflection between wave components.

I don't know how many people have seen an railroad engine starting a train from stop, when there is a small gap between each of the cars. *You can hear each of the cars bumping the adjacent car in a chain reaction going from engine to the end of the train. *Clearly, the reaction has a velocity of travel. *

Some what like hole flow in semiconductors. Electrons going forward
make the
holes flow backwards.

Our EM waves could do the same thing but we would never measure anything except the the resultant wave.

...Keith