Roger Sparks wrote:
"It is not too hard to use the concept of traveling waves to derive the
familiar reflection coefficient to SWR relationship."

Yes. Terman has done it for us in his 1955 opus. He just uses the letter
S to represent SWR.

On page 86:
"The voltage and current of the incident wave at the load must satisfy
Eq. (4-8) =
Eprime / Iprime = Zo."

And at the top of page 86:
"The reflected wave.----is identical with the incident wave except that
it is traveling toward the generator. Eq. (4-11) =
Edouble prime / Idouble prime = -Zo."

"The load voltage is the sum of the voltages of the incident and
reflected waves at the load,....
The load current is the sum of the currents of the incident and
reflected waves at the load,....
The vector ratio EL / IL must equal the load impedance ZL."

"The vector ratio E2 / E1 of the voltage of the reflected wave to the
voltage of the incident wave at the load is termed the reflection
coefficient of the load.

Reflection coefficient = rho = E2/E1 =
(ZL/Zo)-1 / (ZL/Zo)+1."

And on page 97:
"Standing-wave ratio = S = Emax/Emin, Eq.(4-20)
And: S= [E1] + [E2] / [E1] - [E2], Eq. (4-21)"

The standing-wave ratio S is one means of expressing the magnitude of
the reflection coefficient; the exact relation between the two is S =
1+[rho] / 1- [rho]

or [rho] = S-1 / S +1
This relationship is illustrated graphically in Fig. 4-9.

Best regards, Richard Harrrison, KB5WZI