# Propagation Constant Expression in a Transmission Line

let's consider a generic transmission line:

Its wave vector Kz is equal to:

where

Then, I read also that it is true that:

where does this relation come from?

The wavenumber $$\k\$$ is the result of solving the differential equations of the (general) electromagnetic wave for $$\E\$$ and $$\H\$$ in e.g. free space. The result for $$\k\$$ consists of the material characteristics $$\\mu,\epsilon,\sigma\$$.
When solving the telegrapher equations for the coaxial cable, you also get $$\k\$$, but this time $$\k\$$ consists of an R,L,G & C expression.
So the wavenumber $$\k\$$ comes either from the definition of a general wave or from the definition of a generic transmission line like e.g. a coaxial cable.
(The phase constant $$\\beta\$$ is just the imaginary component of $$\k\$$.)