I can see how that works for two sections (coil and whip). What do you do for three sections, e.g. your stub made of 50/450/50 sections?
The principle is the same with one interesting difference. While the phase shift is positive at the 450-50 ohm impedance junction at the top of the 450 ohm section, the phase shift at the 50-450 ohm impedance discontinuity at the bottom of the 450 ohm section is negative and subtracts from the 15 degrees of 50 ohm line. In the previous example, the negative phase shift subtracts about 13 degrees from the 15 degree 50 ohm section tied to the feedpoint.
Have you ever noticed how little effect adding length to the base section under the coil has on the resonance frequency vs how much effect adding length to the stinger changes the resonant frequency? The negative phase shift at the bottom of the coil and the positive phase shift at the top of the coil are responsible for that phenomenon.
It is well known that for a fixed length mobile antenna, moving the loading coil from the base to the center of the antenna requires considerable increase in the inductance of the loading coil. That is because we are making the stinger shorter.
If we keep the same long stinger and simply add a base section under the loading coil, the resonant frequency doesn't change much because the negative phase shift at the bottom of the loading coil cancels out a large number of the added degrees. Adding 15 degrees of bottom section to the physical length of a resonant base-loaded antenna only adds about two degrees to the electrical length of the antenna.
Hoping that people would see the similarity between shortened mobile antennas and dual Z0 shortened stubs, I wrote the following article about five years ago but nobody seems to have had an epiphany because of it.http://w5dxp.com/shrtstub.htm
I know there's nothing new under the sun concerning antennas but as far as I know, the above information is original, i.e. I've never seen it explained anywhere else.