...the roof of the typical vehicle is nowhere near large enough to
provide the ground reflection that is the cause of that difference. The end
result is that the 5/8 wave has a small advantage due to the point of maximum
radiation being higher in the air for the same mounting height.
You're kidding on this, right? The 'average' ground plane requires only a 38 inch diameter circle. Car roofs are always much larger than that--unless you drive a smart car!
No, I'm quite serious, and this leads to a lot of mistaken assumptions about
the behavior of a 5/8 wave antenna.
A 5/8 wave whip is basically a 1/2 wave radiator with the bottom raised
1/8 wavelength above the ground. (There is a slight about of out-of-phase
radiation from the bottom 1/8 wave section, but that just contributes to
higher lobe formation.) A 1/2 wave vertical dipole or a ground plane with
sloping radials (which is basically the same thing) in the same position as
the upper 1/2 wave section will give the same pattern.
While a dipole radiates from all along its length, radiation is maximum from
the center section where the current is highest. For this analysis let's
assume a point source at the point of maximum current: it isn't exact,
but the math is much easier to follow because we don't have to integrate
the current over the element length.
In the 5/8 wave whip the point of maximum current is 3/8 wave up from
the feedpoint, or 1/4 wave down from the top. That's consistent with
the current distribution shown in any of the textbooks. Let's look at
the radiation at an angle of 10 degrees above the horizon: again, we
can repeat the analysis for any angle we want.
The gain of the 5/8 wave antenna is because the direct radiation from
the 1/2 wave element combines in phase with the ground reflection.
In this case, assuming a perfectly reflecting surface, we have one
ray going from the point source up at 10 degrees, and another going
down at 10 degrees, reflecting off the earth, and going back up at
10 degrees parallel to the first ray. This is standard stuff, the same
way we analyze the lobes and nulls of a dipole over ground (except
that reflections are in phase for vertical polarization at low angles
and out of phase for horizontal polarization.)
We can then calculate the point where the ray reflects off the ground
using a bit of geometry as the elevation of the feedpoint divided by the
tangent of the angle. For a feedpoint height of 0.375 wavelengths and
an angle of 10 degrees, tan(10 degrees) = 0.176, so 0.375/0.176 = 2.13
wavelengths. At 146 MHz, that's over 14' from the base of the antenna.
The lower the angle of radiation you want, the further the reflection point
is from the base of the antenna. If the ground plane doesn't extend out
to the reflecting point (and a bit past it, when we are considering the
radiation from the whole 1/2 wave element rather than a point source)
then the reflection doesn't fully reinforce the direct ray, and the 5/8 wave
element won't have the expected gain.
There was a study many years back with regards to antennas on police cars,
which tested various commercial offerings...
You seem to be shovelling it a bit thick here. How about a link to that study...
I think this is it, but can't tell for sure, and my magazines are currently packed...Test Data on 1/4- and 5/8-Wavelength Vertical Antennas for 2m-Mobile
Reed Fisher W2CQH and Bill King W2LTJ, Ham Radio
, May 1976 Page 46