Jim Kelley wrote:
"The error I find is in your notation that virtual shorts and opens
cause reflections, and that real reflection coefficients may be
calculated from virtual impedances."

Impedance is a voltage to current ratio. A low impedance allows much
current from a low impressed voltage. A high impedance only allows a low
current from a high impressed voltage.

A transmission line with a mismatched load, has a reverse or reflected
wave traveling back from the load in addition to the incident wave
traveling toward and impressed on the load. Both waves, incident and
reflected, have the same voltage to current ratio. It equals the Zo of
the line and is enforced by the construction of the assumed uniform
line.

Superposition of the forward or incident wave and the reverse or
reflected wave produces periodic variations in the combined volts and
amps along the line. The rms values of the constituent volts and amps
are likely very steady. It`s their combination which varies.

High voltage points are high impedance points. Low voltage points are
low impedance points.

Severity of voltage and impedance variations along the line depends on
how different the load impedance is from Zo.

A short or an open on the line can produce segments analogous with
series and parallel resonant LC circuits (which behave as shorts and
opens with respect to impedance for example).

Input impedance of an open circuited line of length less than a quarter
wavelength is capacitive. Input impedance of an open line of length
greater than a quarter wavelength but less than a half wavelength is
inductive.

An open-circuited quarter wavelength of line is practically a short
circuit at its input.

A quarter wavelength back from a line short, its impedance is is an open
circuit, and as above, a quarter wavelength back from an open circuit,
the line impedance is a short circuit.

At a line short circuit, incident and reflected current phasors are
in-phase while the incident and reflected voltage phasors are
out-of-phase.

A quarter wave back from the short, the incident and reflected current
phasors are out-of-phase while the the incident and reflected voltage
phasors are in-phase.

A quarter wave back from a hard short on a good line, the reflected
voltage is equal to and of the same phase as the incident voltage.
Therefore there is no potential difference between the incident and
reflected voltages at this point and the current is zero. This is
analogous to connecting identical battery cells in parallel. No current
flows between them. The same can be said of connecting correctly phased
identical transformer windings in parallel.

High voltage and almost zero net current means the impedance is nearly
infinite. This is similar to a good parallel resonant circuit. It is
almost an open circuit. Quarter wave shorted stubs have been used as
"metal insulators" for line support and other purposes.

I have no problem with "virtual shorts and opens". We so called them and
used them in school when I was there over a half century ago.

Examine RADAR TR and anti-TR circuits for examples of virtual shorts and
opens which automatically route the energy to the right places and keep
it out of the wrong places.

Best regards. Richard Harrison, KB5WZI