David wrote:
"I am a little confused with calculation of appropriate length of metal
tubing used in a sleeve antenna design (the 1/4 wave section)."

It should approximate the free-space 1/4-wave valie. The coax exterior,
or the cylinder exterior,,= is the radiator, so interior velocity
factor is not involved.

See the 3rd edition of Kraus` "Antennas", page 721 There are many other
texts that may have better construction information, but everyone should
have the Kraus book, or go get it now. "Transmission Lines, Antennas,
and Wave Guides" by King, Mimno, and Wing is very good for sleeves as I
recall, but I don`t have it at hand at the moment.

The commercial and homemade versions I have used had (2) 1/4-wave
secrions. The antenna was actually a center-fed vertical 1/2-wave dipole
in which the coax was fed through the lower tubular section. The top
section was just a rod or wire.

These worked as well as ground plane =antennas, so we used them at VHF
on small boats since there were no radials to get in the way or poke
anyone in the eye.

The cylinder is slightly shorter than the rod at a 1/4-wave resonant
length. You can experiment to find the resonant lengths, then put them
back to back to make your 1/2-wave dipole.

Best regards, Richard Harrison, KB5WZI