Thus far your math is correct so keep going and perhaps we will arrive at the point of your question.

Ok actually I finished with the math. This is how I see things:

- Whatever the reflection at the load, we will always have part of the amplitude of the incident wave which will "combine" with the reflected wave to form a standing wave with nodes and nulls fixed in the transmission line of course. However it doesn't mean that the drawing (representation or animation) of

**the total** voltage along the line is a standing wave. Actually it is a superposition of standing wave and a propagating wave based on the equations that I wrote and that we agreed on.

**We can note that the amplitude of the reflected and the standing wave depends on the incident wave amplitude and the reflection coefficient.**
We can ask our-self few questions:

1) Since the result is not a standing wave but a combination of standing wave and propagating wave, what is the speed of the nodes/nulls in this case ?

2) At which speed is transmitted the power (yes I think that the speed of the nodes/nulls are different than the speed of the energy) ?

3) What is the quantity of power which is transmitted ?

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By trying to answer to

**question 2 and 3**, we notice that :

a) In the case of propagating wave, the speed of nodes/nulls is the same as the speed of transmission of energy.

b) The difference between the speed of transmission of power and the speed of nodes/nulls can be seen in the standing wave case (full reflection: reflected coefficient = 1). In the case of standing wave, the speed of nulls/nodes is zero (we all agree on that) but actually the power flows from the generator to the load

**AND** the same

**quantity of power** flows from the load to the generator. To me it`s not the same thing at all as if we would have considered a "standing power". I don`t believe in such a concept at least in this case.

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By trying to answer to

**question 1** (which drove me crazy for 5 months now), we notice that :

a) If we have a case which is very close to standing wave, where the C value of my derivation is very low compared to Vi (or Vr), the nodes and nulls will definitely move/oscillate slightly around its original value (the one obtained by its standing part (Simpon formula of my previous derivation)). The speed of this move/oscillation depends on the amplitude of the transmitted wave (in other words it depends on C in my equations). In the perfect case (full reflection) no transmitted wave exist and the nodes/nulls are not moving anymore.

b) The speed of the nodes/nulls increase little by little as reflection as load becomes weaker until the moment for which this is no reflection, then the nulls/nodes move at the speed of light (if the dielectric in the transmission line is the air). We are now in the propagating case only. It would be interesting to quantify mathematically the evolution of the speed of the nodes/nulls from 0 (standing wave case = full reflection) to the speed of light (propagating wave = no reflection). I didn`t quantify that yet but I plan to do that when I have time.

Before I thought that there is a discontinuity of speed between the standing wave case and the case of "small" transmitted wave.

**Anyway there is no discontinuity of any speed (neither the nodes/nulls speed nor the speed of propagation of power).**
I think I`m done with the explanation of what I think. Before writing that it wasn`t so clear but it became clearer and clearer by writing it... Let me know what you think about it.

I forgot to say that I put myself in the simplest case: lossless line, resistive load (real coefficient of reflection) and non radiative transmission line since Quasi-TEM propagation with fringing fields are the reality for micro strip line but let's stay in the simple case just to understand...

PS: I attached the propagating wave case with it`s model of cascaded parallel self and capacity and I tried to display the voltage and current at t=0 for example to see what happens for the energy in the simple case of propagation. The resonant circuits will resonate and a transmission of energy will go from self to capacity then from capacity to self. The speed of transmission (of nodes or energy) is given by 1/LC of course. The same graph can be done with 4 curves (partially reflected case).