What is Zin?

So listen, there's this formula:

$$Z_{in} = Z_o \frac{Z_L + Z_o tan(\beta l)}{Z_o + Z_L tan(\beta l)}$$

I have no idea what this means. What's more,

$$\beta = \frac{2\pi}{\lambda} = \frac{2\pi f}{v}$$

Apparently this is input impedance. Ok, whatever.

So now, what does $\beta$ mean and why is it there? Looks like the $f$ is the frequency of the voltage moving in the circuit. And $v$ is the ... speed of the wave in that medium? Maybe.

There was also this other formula which looks kind of like this: $v = \frac{c}{\epsilon_r}$ though I have no idea where I saw it. It might have been a dream.

More details

Looks like when you generalise the above $Z_{in}$ to non-lossless mediums, you get this, where $tan$ is replaced by $tanh$ and $\beta$ by $\gamma$ (the full propagation constant). I don't get how $tan$ becomes $tanh$ but I don't have the kind of time needed to investigate.

$$Z_{in} = Z_o \frac{Z_L + Z_o tanh (\gamma l)}{Z_o + Z_L tanh (\gamma l)}$$

I have made a mistake

The formula was supposed to have a $j$ in it. Like so:

$$Z_{in} = Z_o \frac{Z_L + j Z_o tan(\beta l)}{Z_o + j Z_l tan(\beta l)}$$

Same, ofc, for the non-lossless medium formula.

The dream formula

Remember $v = \frac{c}{\epsilon_r}$? That wasn't a dream (but was incorrect). Here's that, corrected, with some related things.

The general formula:

$$v = \frac{1}{\sqrt{\epsilon\mu}}$$

For the speed of light in free space ($c$), we have

$$c = \frac{1}{\sqrt{\epsilon_o\mu_o}}$$

For some other random medium we have

$$v = \frac{1}{\sqrt{\epsilon_r\epsilon_o \mu_r\mu_o}}$$

Breaking this down we get

$$v = \frac{1}{\sqrt{\epsilon_o\mu_o}\sqrt{\epsilon_r\mu_r}} = \frac{c}{\sqrt{\epsilon\mu}}$$