This rather oversimplified analysis overlooks an important factor. The
field radiated upward from an antenna seen at long distances (that is,
the sky wave as contrasted to the short-range surface wave) consists of
a vector sum of two components: one radiated directly, and one which is
inintially radiated downward then reflected from the ground. The ground
reflection alters both the magnitude and phase of the reflected
component depending on ground characteristics and the polarization of
the wave. At low angles, horizontally polarized waves are reflected very
well even when the ground is quite poor; vertically polarized waves
react differently. The resulting fields can fairly easily be calculated
manually if desired using simple geometry, equations for reflection
coefficient which can be found in Kraus and other references, and vector
addition. One thing you'll quickly discover is that the field from a
vertical does NOT monotonically increase as the elevation angle
decreases, but decreases below a moderate angle determined by the ground
characteristics. EZNEC (including the free demo) and other modeling
programs clearly show this important effect.

Roy Lewallen, W7EL

Reg Edwards wrote:
There is much discussion about the relative merits of the simple
vertical versus horizontal dipole antennas.

Their radiation patterns are well known. They are very broad in both
the vertical and horizontal planes. Both have have a null.

We need consider only the broadside, maximum, radiation from a dipole.

Most of the arguments can be settled by considering the elevation
angle of the path taken by the radio wave between the transmitting and
receiving stations. Followed by a little elementary geometry or
trigonometry. For present purposes a flat Earth can be assumed.

At an elevation angle of around 45 degrees the strength of radiation
received from vertical and horizontal antennas are about equal. (This
has nothing to do with Eznec take-off angles.)

The heights of the Ionospheric reflecting layers are -

E-layer = 70 miles, daylight only.
F1-layer = 140 miles, occasionally, in daylight only.
F2-layer = 190 miles, night-time.
F2-layer = 250 miles, in daylight.

From flat-Earth geometry, at an elevation angle of 45 degrees, the
distance between transmitting and receiving stations is twice the
height of the reflecting layer. Therefore, at this distance the
received signal strength can be expected to be about the same from
both types of antenna.

As the elevation angle decreases, the distance increases and radiation
from the vertical antenna increases. The radiation from the dipole
decreases. There is an extra propagation loss due to an increase in
radio path length but this equally affects radiation from both antenna
types.

As the elevation angle increases towards the vertical, distance
decreases, radiation from the dipole increases and radiation from the
vertical antenna decreases in strength. The radio path loss decreases
but the difference in pattern between the two antenna types is
maintained at the receiver.

With a spherical Earth, in daylight, using the F2-layer, at elevation
angles around 5 degrees, one-hop distances of 3,500 miles can occur.
With two hops, at angles of around 12 degrees, distances of 5,000
miles can occur.

For each additional hop there is loss in the layer and loss in the
reflection in the ground. Some parts of the radio path may be in
daylight and others in darkness. More than one layer may be involved.
Muli-path distortion occurs. Peculiar things happen and much depends
on frequency.

The low-angle performance of a half-wave dipole, even when radiating
broadside towards the receiver, is very poor in comparison with a
simple vertical.

On the other hand, a simple vertical does reasonably well when working
just across county because of the short propagation path, almost
straight up and down again, or even via the groundwave for very short
distances.
----
Reg, G4FGQ.