Electrical Length Analysis

How ?mathematically? does adding a capacitor (series or shunt) change the electrical length of a coaxial resonator?

https://en.wikipedia.org/wiki/Electr...gth_by_loading

A reactive load shifts the resonance frequency, it can't change the electrical length of the resonator itself.

Intuitive explanation, determine equivalent lumped L and C representing the coaxial resonator, calculate the new resonance with reactive load.

Exact solution, calculate the complex input impedance of the transmission line stub, determine the length to achieve resonance with reactive load.

Zin = j Z0 tan(? l)

OK, thank you.