David wrote:
Image theory is for a perfect groundplane e.g. large area metal sheet. The
wave emitted by the vertical radiating element is reflected by the ground
plane.

Image theory as I see it follows. Wave emitted by vertical element is the
incident wave that hits ground plane, inducing currents in the ground plane.
Currents flowing in skin depth of ground plane emit a wave of opposite
polarity to cancel out the wave at the boundary of the ground plane, thus
making the electric field in the ground plane zero. The wave of opposite
polarity is the reflected wave. The reflected wave appears to be coming from
an image antenna. Image theory is a mathematical model for solving antenna
simulations where there is a monopole over a ground plane.

How do the radials reflect the wave? If they are not a good enough ground
plane because of the gap, how do they reflect? I cannot see the transition
from ground plane to radials, when looking at image theory.

Picture a half wave disk of metal as the ground plane, producing the
inverted image of the vertical. Then imagine thin radial slots spread
around the vertical. Since these slots do not cross any current path
that is needed to produce the image, they have little effect on the
image. Widen those slots, and decrease the number of them, and
eventually you get to a ground radial system with only a few radials.
There has to be a transition point, where the radials are only a
poor approximation of the original disk. The question is, how well
must you approximate the disk to get a reasonable approximation of the
far field radiation pattern it would have helped produce?