Keith Dysart wrote:
So yes, the phrase "conservation of power" is
appropriately descriptive and follows from
conservation of energy.

You have a contradiction built into your concepts.
You have argued that the instantaneous power dissipated
in the source resistor is not equal to the instantaneous
forward power component plus the instantaneous reflected
power because power must be conserved at each instant of
time. That is simply not true.

I'm telling that energy must be conserved at each instant
of time but power does not have to be conserved at each
instant of time. Energy can obviously be stored in a
battery or network reactance for dissipation later in
time.

Do you require that the power used to charge a battery
be instantaneously dissipated in the battery? Of course
not! That's true for a dummy load but NOT for a battery.
There is no reason to require dissipation of power at
each instant of time to balance. Since energy can be
stored, there is no such thing as conservation of
instantaneous power, only of instantaneous energy.

Where is "cos(theta)" in this?
And what "theta" is to be used?

How many times do I have to explain this? For instantaneous
values of voltage, if the sign of the two interfering voltages
are the same, theta is zero degrees and the cosine of theta is
+1.0. If the sign of the two interfering voltages are opposite,
theta is 180 degrees and the cosine of theta is -1.0.

Is it the same theta that was used to conclude that
this was a "no interference" example?
See "A Simple Voltage Source - No Interference" at
http://www.w5dxp.com/intfr.htm

That theta is the phase angle of the phasor. If you
would switch over to phasor notation for your values
of instantaneous voltages, it might make things clearer
for you. You have been dealing only with real part of
the phasors which limits theta to either zero or 180
degrees.

So the plus/minus in the equation aboves means that
for part of the cycle we should add and part of the
cycle we should subtract.
This equation is ambiguous and is therefore incomplete
since it does not tell us when we should add and when
we should subtract.

I have told you about five times now. If the sign of the
two interfering voltages are the same, the interference
term has a positive sign (constructive). If the sign of
the two interfering voltages are opposite, the interference
term has a negative sign (destructive). Instead of me
having to keep posting this over and over how about you
keep reading it over and over until you understand it?

Could you provide a mathematically precise description
of when we should do each?

See above. If we deal with phasors, theta is the phase
angle between the two interfering voltages. If we deal
only with real values, theta is limited to either zero
(cosine = +1.0) or 180 degrees (cosine = -1.0).

Where is this destructive interference energy stored
while waiting to be dissipated constructively later?

In our 45 degree shorted stub example, it is stored in
the equivalent inductive reactance of the stub. For each
instant in which destructive interference energy is stored,
there is an instant in time 90 degrees later when that same
energy is dissipated as constructive interference power
in the source resistor. That's why 100% of the average
reflected power is dissipated in the source resistor for
these special cases of zero average interference.
--
73, Cecil http://www.w5dxp.com