On Jun 29, 9:46*am, Cecil Moore wrote:
On Jun 28, 5:27*pm, Keith Dysart wrote:

And yet, you have not located any errors in the math or the models.

Superposition of power *IS* an error!

You have really latched on to that haven't you. I remember the good
ol'
days when you were superposing powers and the very same people with
whom you are arguing today were trying to convince you that
superposing
power was an invalid operation.

Now you have bought in to the rule and apply it even where it does
not apply.

Just because a sum is being computed does not mean that superposition
is at work.

When you add up the energies to validate that energy is not created
or destroyed, you are not superposing energy.

When I add up the energy flows for the same purpose, that does not
mean that superposition is at work.

From a basic calculus, if the energy balances, so must the energy
flow:

f(t), g(t), etc. are functions that describes the energy in
individual entities of a system with respect to time. Then

f(t) + g(t) + h(t) = k

is the expression that says the sum of all the energies must be
a constant, for energy is neither created nor destroyed.

From basic calculus

d(f(t) + g(t) + h(t))/dt = d(k)/dt
d(f(t) + g(t) + h(t))/dt = 0
d(f(t))/dt + d(g(t))/dt + d(h(t))/dt = 0

d(f(t))/dt is the derivitive with respect to time of the energy
(i.e. energy flow, or power) for the entity.

So it is completely valid to sum these energy flows to
track the energy within the system; as my example does.

You add and subtract powers
willy-nilly as if that mathematical step were valid which it is not.
Two coherent 50w waves do not add up to a 100w wave, even using
average powers, except for the special case of zero interference where
the waves are 90 degrees out of phase with each other.

True. You must always use the actual energy flows in the entity and
not the powers that are not real.

In order to use power as an energy tracking tool, we must be very
careful to ensure that there is a one-to-one correspondence between
energy and power, i.e. every joule passing a point in one second must
result in one watt of power (no VARs allowed).

Also true. And as is done in my example.

If that one-to-one
correspondence doesn't exist, no valid conclusion can be drawn from
tracking the power and any valid conclusion must be based on tracking
the energy which is no small task. The key is that there is no such
thing as imaginary energy. All energy is real. Some "power" is not
real.

Agreed. Especially the power in the waves that are being superposed.
That is why you need a fudge factor when you sum your partial powers
from your waves. I carefully do not sum partial powers from
superposed waves.

A one-to-one correspondence does not exist in a standing wave.
Therefore, tracking power as if it were equivalent to energy in
standing waves is invalid. You have made that error for years.

Please provide an example of such an error, preferably from one
of my expositions.

One-to-one correspondence also does not exist over a fraction of a
wave.

Of course it does. Energy must always balance; not just at the end
of the cycle.

Therefore, instantaneous power is irrevelent in tracking the
energy. That's your latest error which is the same conceptual error as
before. In general, average power in the traveling waves over at least
one complete cycle (or over many cycles) has a one-to-one
correspondence to the average energy in the traveling waves. But that
one-to-one correspondence is more often than not violated within a
fraction of each cycle.

Now you are allowing us to create and destroy energy as long as it
balances at the end of the cycle? Are you sure you want to do this?
It does not align with the well understood principle of conservation
of energy.

If I follow that rule, I just make the cycle long enough, say, my
lifetime, and I am golden.

But I do not hold out any hope for this.

Here's a quote from "Optics", by Hecht, concerning power density
(irradiance).

"If however, the 'T' is now divided out, a highly practical quantity
results, one that corresponds to the average energy per unit area per
unit time, namely 'I'." - where 'I' is the irradiance (*AVERAGE* power
density).

There you go again, assuming that the 'practical' challenges of optics
also apply to transmission lines. On a transmission line, we can
measure the voltage and the current. We do not suffer from the same
constraints as optics.

Use the tools that are available for transmission lines. Don't tie
two hands behind your back just because those tools do not work in
the optical domain.

....Keith