It's sad to see that the response to my analysis and equations is insult
and derision rather than any coherent counter argument, but I'm
unfortunately not surprised. I don't see in it any evidence that my
posted calculation was in error -- the only objections I see are that it
doesn't support a flawed theory, so it therefore must be wrong. The
calculations I made are based on solid theory which has been
successfully used for more than a century, and you won't be able to make
any measurement which will refute them. They're also entirely self
consistent with all other transmission line phenomena which can be
calculated or measured.

So I wouldn't bother to respond at all except that it does provide the
opportunity to elaborate a bit on what I posted. If further responses
are as devoid of substance as this one, I'll probably end up plonking
Dave as I did Cecil some time ago, for the same reason.

Dave wrote:
so roy has correctly calculated the standing wave 'power' to be zero at two
points on the line. i am sure that yuri will take great exception to this
result showing there is no power in the standing wave.

I did what I claimed to be able to do -- correctly calculate the power
from v(t) * i(t). Yuri takes exception to many things I say, but frankly
that bothers me not in the least.

but he missed the definition of pr and pf...

Indeed I did. Can you define them for me please?

p(t) = v(t) * i(t)
v(t) = vf(t) + vr(t)
i(t) = if(t) + ir(t)

Therefore,

p(t) = [vf(t) + vr(t)] * [if(t) + ir(t)]
= vf(t) * if(t) + vr(t) * ir(t) + vf(t) * ir(t) + vr(t) * if(t)

Of these four terms, which, if any, are pf(t) and pr(t)? What are the
two remaining terms called?

dave in problem statement:
now, calculate vf(t), if(t),pf(t), vr(t), ir(t), pr(t) at that point,
where
the 'f' terms are the forward wave, the 'r' terms are the reflected wave.

so he conveniently skipped that step and instead writes this cop out:

No, I did calculate all except pf(t) and pr(t), which you didn't define.
As soon as you do (see the question above), I'll be glad to calculate
them also. Or you could do it for us -- it won't involve more than
simple arithmetic.

roy:
I haven't seen a definition of pr and pf, but they're not relevant to
the discussion. If you get a different result for power than zero by
using whatever you take them to mean, then the concept is invalid.

pr and pf are, as i stated, the power in the forward and reflected waves.

What would that be, then, vr(t) * ir(t) and vf(t) * if(t)? Where does
the power in those remaining two terms come from or go?

There is of course power in these two waves and it is indeed 'sloshing' back
and forth in the line.

Please note that I didn't say that power was "sloshing" back and forth.
I said that energy was. Power is not the same as energy -- they bear the
same relationship as speed and distance.

These are the waves that can be measured by any of
the simple devices such as neon bulbs or bird watt meters that clearly show
equal and opposite powers in the waves. so you can indeed have power in the
traveling waves, but no power in the standing waves... which will always be
the case.

Unfortunately, people assume that the units indicated on a meter are the
quantity actually being measured, which often they're not. But this has
been explained many times before here.

i will give him this point as being correct for a lossless open (or shorted)
line:

There
is no average power leaving the source and no average power being
dissipated in the load(*). So there had better be no average power
anywhere in the line.

but then he loses it again:

There will be non-zero instantaneous power
everywhere along the line except at the input, far end, and midway, but
its average value will be zero,

the traveling waves will have power EVERYWHERE on the line, the special
cases are are just the ones where the standing wave is most easily shown as
having no power. obviously if there is power in a wave at one point on a
line it is not going to stop and bypass the quarter wave points, the forward
and reflected waves continue end to end and their power goes with them... it
is at those 'special' points where the voltage or currents in the forward
and reflected waves always cancel each other so if you measure with a simple
tool you will see the voltage or current nulls at those points. that does
not mean there is no power passing those points, only that the voltage or
current in the traveling waves has conveniently canceled each other out at
those points.

Here's just one of the problems with assigning powers to the traveling
waves, attempting to keep track of them separately, and applying
superposition to nonlinear quantities. The conclusion that there is
power at the ends of the line, for example, is demonstrably not true.
There is no current at the far end of the line at any time, and
therefore no power, as I showed. My comment at the end further shows why
the power at the end of the line must be zero.

and then he has to end up with an obvious contradiction:

indicating the movement of energy back
and forth but no net energy flow.

how does energy not flow if it is moving back and forth???

Hopefully most of the readers were more astute than this and noticed the
word "net". If equal amounts of energy flow in each direction during a
cycle (as indicated by a power waveform with no offset), there is no net
energy flow. It means that energy is being stored at some location in
one direction, then returned during the other half of the power cycle.
This is true in any purely reactive circuit, for example a tank circuit,
where exactly the same calculation I made can be done with the same result.

Consider our open ended line for a moment, and imagine it laid out from
left to right with the open end to the right, so I can name directions.
There's no place to the right of the line to store energy, so no energy
can be moved past that point to the right. If we consider the positive
direction of energy flow as being to the right, this means that the
power at the end of the line can never go positive, even for an instant
-- if it did, it would mean energy is moving past the end of the line
during that time. But since there's no storage mechanism beyond the end,
this can't happen. And since we can't have net energy moving past the
end, either, the power therefore can't go negative at any time either.
So the power at the end of the line must be zero at all times. This is
of course the result I got from v(t) * i(t).

Roy Lewallen, W7EL