One of the earlier postings suggested that the quarterwave vertical antenna
with radials was elementary and easy to understand. I have never found this
antenna easy to understand.

RF experts on this newsgroup cannot agree on whether i) the radials reflect
the wave or ii) the field from the radials cancels out. The standard
academic books show that the principle behind the vertical ground plane
antenna is that the vertical radiating element emits the wave, and is
reflected by the ground plane.

You can view a conductor as having current pushed through it by a RF source,
or the current can be induced in the conductor by the wave. This is a
boundary condition in Maxwell's equations, referred to in theory of
transmission lines and guided waves.

You can view the radials as reflecting the wave and having current induced
in them, or they can have current pushed through them by the RF source. This
is probably the same thing, due to the arrangement of all antenna parts
forming the antenna impedance. In image theory, the impedance comes from
both the self impedance and the mutual impedance.

It appears that a single counterpoise wire is connected to the RF ground
side to provide a conductor for that side and be a form of dipole. If a
proper RF ground is not provided, the result may be RF in the shack e.g. the
RF tries to return via mains wiring. Does connecting several wires make the
RF ground side less live i.e. occupying a larger area to be more of a
reflector and thus dissipative? If a RF ground is live, it can be dangerous
to touch it. Do you increase the area of RF ground to make it less dangerous
to touch e.g. radials under a carpet when relatives and pets are about?

The theory behind the quarterwave vertical is the monopole above a ground
plane, where the ground plane reflects the wave emitted by the vertical.
The monopole is explained using image theory. In practice, the ground plane
is
replaced by radials. Do the radials reflect the wave then?

The reflecting element on a Yagi manages to reflect most of the wave. The
reflecting element on a Yagi is a parasitic element that has an impedance
to cause the wave emitted by the driven element to flow in a particular
direction. A Yagi normally has only one reflector. Although the reflector
is in the near field of the Yagi, can a comparison be made with the radials
of
a quarterwave vertical antenna? The reflector on a Yagi is usually a thin
tube with lots of air (gap) around it. Even though it occupies a small
area, it still manages to reflect most of the wave. Yagi has a Front to
Back
ratio in dB.

Radials can be tuned. Some antennas have loading coils in the radials.


Antenna theory is often about wires and metallic items reflecting waves,
and the phase of the reflected wave. The phase of the reflected wave can be
constructive or destructive, affecting the impedance of the antenna. If an
antenna is mounted too close to the ground, the reflected wave cancels out
the emitted wave.

Because a ground plane reflects the wave, the impedance of an antenna can
vary with height.

Parastic elements on a Yagi have a mutual impedance to each other. Would
you regard the radials on a quarterwave vertical as having a mutual
impedance?

The radials increase the conductivity below the radiating element,
decreasing ground losses. The radials are regarded as a finite or imperfect
ground plane.

References:
"Antenna Theory and Design" by Warren Stutzman and Gary Thiele. pages 66 to
68. Practical monopole with radial wires to simulate a ground plane.
"Antenna Engineering Handbook" by Richard C. Johnson. Radials suppress
currents from flowing on outside of coax. p 28. If the ground is imperfect,
the perfect reflected image is mutiplied by a complex ground reflection
coefficient. The ground has a mutual impedance.
"Antenna Theory" by Professor Constantine Balanis. Second Edition p 165. A
ground plane formed by a perfect conductor completely reflects the wave. If
the ground is finite i.e. not as conductive, it still reflects the wave but
not as well. The conductivity determines the quality of the reflection.