The equation in the ARRL Antenna Book is identical
to the equation for rho that is in the Power Wave
literature (see Gonzalez and also see Kurokawa).
Also, numerous literature sources describe how an
open-circuit generator with internal impedance Z0,
connected directly to load ZL, is actually a power
wave setup that leads to a rho formula that is
identical to the formula in the ARRL Antenna Book.
When calculating rho, it is not necessary to fool
around with the wave equations, because frequency
is constant and everything is steady-state.
Bill W0IYH
Also the same conjugate formula in Les Besser's
RF Fundamentals I, and Kurokawa, and the 1992 ARRL
general Handbook, which has NO term for Zo reactance, so
it assumes a purely real Zo.
This page also has the correct general formula:
http://www.zzmatch.com/lcn.html
Slick