Only plane waves in the far field exhibit the
characteristic impedance of free space, which is exactly:
$$Z_0 = \frac{\left|\vec{E}\right|}{\left|\vec{H}\right|} = \sqrt{\frac{\mu_0}{\epsilon_0}} = \mu_0\cdot c_0 \approx 376.73,\Omega$$
| where:
| $c_0 = 299,792,458,\frac{\text{m}}{\text{s}}$: the speed of light in free space
| $\mu_0 = 4\pi\cdot10^{-7}\frac{\text{H}}{\text{m}}$: the free space permeability
| $\epsilon_0 = \frac{1}{\mu_0 c_0^2}$: the absolute permittivity of free space
| $Z_0$: the characteristic impedance of free space
Euler's formula:
$$\e{\j\omega t} = \cos{\omega t} + \j \sin{\omega t}$$
The input impedance $Z_\text{in,$,$short}$ of a transmission line stub terminated in a short circuit is given by:
$$Z_\text{in,$,$short} = Z_\text{c} \tanh{(\gamma\ell)} \approx \j\tan{(\beta\ell)},Z_\text{c}$$
| where:
| $\gamma = \alpha + \j\beta$ is the propagation constant $\gamma$,
| $\alpha$ is the attenuation constant, and
| $\beta$ is the phase constant.