A very good article on the design of a sleeve dipole monopole omni-directional antenna appears in an article in the on-line magazine InCompliance, whose general area is electronic design, testing, and standards.
Sleeve Dipole Antenna Design and Build
https://incompliancemag.com/sleeve-dipole-antenna-design-and-build/
In the article the authors describe how they developed a sleeve dipole monopole antenna for 225-MHz use. The physical configuration is very similar to the MORAD VHF-156: there is a whip radiator upper element and a tubular lower element that encircles the radiator.
The design process began by using theoretical dimensions. These were evaluated using modeling software, which was configured to optimize the VSWR for 50-Ohm transmission line. A physical model was then constructed and tested. Then with some trial and error adjustments, the design was finally optimized for the intended resonant frequency of 225-MHz, as the software modeled antenna was resonant at a difference frequency. The final lengths for the two segments to produce resonance at 225-MHz and provide a 50-Ohm feed point impedance were:
- upper radiator = 312-mm
lower sleeve = 206-mm
Converting the physical dimensions into wavelength λ at 225-MHz = 1332.4-mm
upper radiator = 312/1332.4 = 0.234 λ
lower sleeve = 206/1332.4 = 0.155 λ
Now we compare to the MORAD lengths, which I mention in a separate article:
upper radiator = 0.405 λ
lower sleeve = 0.299 λ
The agreement is not particularly close. Both the radiator and sleeve are longer, roughly by a factor of about 1.7 (radiator length) to 1.8 (overall length) to 1.9 (sleeve length).
Comparing the overall lengths we see
225-MHz Sleeve Dipole = 0.389 λ
MORAD VHF-156 = 0.702 λ
The most interesting inference is the overall MORAD antenna length (0.7 λ) is the region of between 5/8-wavelength (0.625 λ) and 3/4-wavelength (0.75-λ), which in a vertical monopole tends to create gain while maintaining a main lobe with a low angle of radiation. Another inference is the sleeve section is certainly part of an impedance matching section that may also become part of the radiating element.
The strange thing about real world antennas is sometimes their physical design is just developed by trial and error, and then the theory of how it actually works is inferred later by the notion that the antenna, as constructed, somehow works very well so it must follow a particular theory of behavior of radio waves flowing on conductors of certain lengths.
I suspect that the design of the MORAD VHF-156 may have been developed in a similar manner as the 225-MHz sleeve dipole in the cited article. That is, the antenna was physically constructed and then many iterations of lengths and configurations were tested to arrive at the final configuration.
If there is any "secret sauce" in the MORAD VHF-156 antenna, it is probably occurring inside the tube section and at the point where the center conductor of the transmission line is tapped onto the radiating element. Unfortunately, no details are available of the exact configuration of that point inside the sleeve tube.