Keith Dysart wrote:
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After considerable thought, I think the math you presented above is for one of two cases of reflective waves, the reflection from a higher impedance load. When the load is less than the Zo of the line, the currents add but voltages subtract. Right?



I don't think so. Vt = Vf + Vr, It = If - Ir, Vf = If * Z0 and Vr = Ir * Z0 are the fundamental equations defining forward and reverse waves. Perhaps you arrive at two choices because sometimes Vr and Ir are negative, which after simplification appears to give an alternate form?



The end result is the same for both cases.



This is good. If you chase the signs, though, I think you will find that there is only one case. We probably should not toss Power into the mix until agreement is reached on this. Power is fraught with issues which seriously confuse some. ....Keith

I can see that I need to further explain. 

My analysis always begins with the source because the first formation of the wave comes from the source, then travels through the transmission line system.  The source defines the wave only until the wave reaches any discontinuity(s) or the line end.   Thereafter, discontinuities and end conditions define the system,.

Why might I say that?   Initiation of the wave at the source results in a sine wave with the impedance of the transmission line, and the power and frequency of the  source.   This is a steady state condition until the first discontinuity or reflection point is reached by the traveling wave.   Each successive reflection point (discontinuity) reflects power which travels back to the source and changes the feed point impedance conditions.   The most distant possible reflection point is the end of the transmission line (ignoring reflections which might occur on the antenna) and might be an open circuit, a reactive resistance, or a short circuit.   Any power reflected from the end will change the measured impedance found at any point on the transmission line all the way back to the source, and will define the steady state conditions of the system.

If we accept that the steady state conditions are defined by the load, then we should examine the conditions on the source side of the load, assuming it is the end of the transmission line.  The forward wave spawns the reflective wave in one of two ways, one way of  load resistance higher than line impedance, and a second way of  load resistance lower than line impedance.   In both cases the power of both forward and reflective wave add, but the voltages and currents both add and subtract.   (Cecil explained it very well in his follow up postings.   Thanks Cecil.)  I presented the power equations to illustrate the two conditions.

It is convenient that both cases result in the same math for the directional watt meter.  

73, Roger, W7WKB