K7JEB wrote:
"But....whatever, I think Herr Doktor Einstein would approve of the
derivation from first principles found on---."

No doubt, as that illustrates it is a problem involving geometry. But in
all cases the distance to the horizon is inexact due to constant
variations in refraction of the atmosphere. Most often the earth appears
to have a radius of about 4/3 the actual which means the earth appears
flatter than it is so that radio waves range farther than many
predictions. When propagation for line-of-sight signals gets tough in
the early am under still air conditions, the earth can apper to have 2/3
its actual radius or even less. Bad news out on the fringes!

Terman says:
"Theoretical analysis indicates that the earth curvature reduces the
received signal below the value calculated by Eq. (219) by the factor
given by Fig. 362. This factor takes into account that refraction in the
atmosphere and also the diffraction of the energy around the curved
surface. Under practical conditions the reduction factor of Fig. 362 is
negligible as long as a straight line path exists, but at greater
distances it decreases rapidly and the signals soon become unusable
because of fading, as mentioned below."

Terman also has a height versus distance chart similar to that in the
ARRL Antenna Book. Fact is that the experimentally determined formula is
related to the geometric calculations and is plenty close enough for
practice. I`ve used it commercially many times and for more than half a
century and never been embarrassed by inaccuracy causing excess expense
nor excess outage time. It is a good indicator of the radio distance to
the horizon under "usual" propogation conditions. It is easy to remember
and simple to apply.

Best regards, Richard Harrison, KB5WZI

Terman also says the radio horizon as seen from a ground-mounted vertical
antenna is at distance of -

50 / Cuberoot(FreqMhz) miles.

This does not mean the ground wave suddenly weakens at this distance but
that Earth curvature and atmospheric refraction begin to have a significant
effect on ground-wave propagation.

At MF and LF, useful ground-wave propagation can occur at distances
considerably greater than the radio horizon. At HF the skywave can provide
stronger signals.
----
Reg, G4FGQ