How much gain does a 5/8th wave mobile antenna really have?
As with many things in life the answer is "it depends".
A 5/8th wave is claimed to have 3dB gain over a 1/4 wl antenna. Let's see if that is true.
Above....Perfect infinite ground 1/4 WL. 5.15 dBi gain
Above pattern 5/8th wave over perfect infinite flat ground. Peak gain 8dBi. Gain over 1/4 wave is 2.85db assuming ZERO matching coil loss.
The same infinite-groundplane antenna, with a matching coil Q of 100, the 5/8th wave has 2.6dB gain over the reference 1/4 wave.
This is over an infinite size perfect ground.
Mounted on a roof to increase height above ground:
A 1/4 wave centered on a 1 meter radius plate five feet high (that's a big, flat, and high vehicle roof) has 4.4dBi gain at zero degrees. Notice the pattern change. This is over infinite perfect earth. Not a real antenna.
A 5/8th wave centered on a 1 meter radius plate five feet high (that's a big, flat, and high vehicle roof). Five feet over perfect lossless earth. This antenna has
6.81 dBi gain at zero degrees. Notice a similar pattern change. A vertical antenna' pattern is largely created by the reflection from earth around the antenna.
The 5/8th wave gain is 2.4dB dB better than the 1/4 wave in this case.
A 5/8th wave centered on a large flat roof over good flat soil has -4.56dBi gain at 2 degrees.
Maximum gain is now only 3.07dBi.
At 2 degrees takeoff the 1/4 wl over good soil with perfectly flat terrain has -6.75 dBi gain. Maximum gain is now only 3.08 dBi.
The gain of a 5/8th wave over a 1/4 wl antenna at 2 degrees is now 2.19 dB, but the maximum gain is only .01dB different than the maximum gain of a 1/4 wave antenna!
With a normal mid-size roof contour 5/8th wave gain is now -4.64dBi at 2 degrees elevation. Maximum gain is now up slightly to 3.2dBi.
With a normal sloped roof shape and mid-car size ground, the 1/4 wave has -6.59dBi at 2 degrees TO. Maximum gain is 3.05dBi.
5/8th wave gain is now 1.95 dB over a 1/4 wave antenna at 2 degrees, but the peak gain is only .15dB better!
The smaller the roof and/or the less centered the antenna, the less gain advantage for a 5/8th wave.
Lower height roofs also have less gain difference, and shape of the roof greatly affects pattern and gain.
The 2.85dB theoretical gain is only for a perfect lossless loading coil antenna over perfect infinite flat groundplane.
In the real world the actual gain difference between a 1/4 wave antenna and a 5/8th wave can be anywhere from the same up to a maximum of 2 dB. The 5/8th wave is never really worse, explaining why people have no problems using them. This is true for any repeater antenna height.