Jim Kelley wrote:
But when there is a change in the amplitude of the
standing wave in x, and I max is known, then determining the shift in
phase from the equation is a no brainer, Cecil.

Exactly! You and I are agreed that the *amplitude* holds
the key to the phase shift *through a straight wire* so
you are preaching to the choir. The phase of the total
current on a standing-wave antenna is unrelated to the
delay through the loading coil.

Unfortunately, the current "bulge" through a loading
coil causes an error in the simple "no-brainer"
calculation that you are suggesting. Some other method
of determining the phase-shift/delay through a coil is
needed.

That other method is to load the loading coil with
its characteristic impedance and measure the phase
shift in the resulting traveling wave.

For instance, in the coil426.EZ file from my web
page, the current at the bottom of the coil is
1.0168 amps at 0.00 degrees. The current at the
top of the coil is 0.8179 amps at -0.06 degrees.
The maximum current in the middle of the coil
is 1.1092 amps at -0.04 degrees.

The total current equation through a loading coil
is not a simple cosine function like it is through
a thin-wire. Determining the actual delay through a
loading coil is apparently NOT a no brainer.
--
73, Cecil, IEEE, OOTC, http://www.w5dxp.com