Planar Sinusoidal Wave

E=E^cos⁡(ωt−k⋅r+ϕ)eEinstantanious value\bm E = \hat{E}\cos(\omega t - \bm k\cdot\bm r + \phi)\bm e_E \quad \textup{instantanious value}
E=E0e−jk⋅reEcomplex value\bm E = E_0e^{-j\bm k\cdot\bm r}\bm e_E \quad \textup{complex value}
E0=E^ejϕtop value scaleE_0 = \hat{E}e^{j\phi} \quad \textup{top value scale}
E0=E^2ejϕeffective value scaleE_0 = \frac{\hat{E}}{\sqrt 2}e^{j\phi} \quad \textup{effective value scale}

Propagation Rate

v=1μ0μrϵ0ϵrv=ωkk=∣k∣v = \frac 1{\sqrt{\mu_0\mu_r\epsilon_0\epsilon_r}} \quad v = \frac {\omega}{k} \quad k = |\bm k|

Wave Impedance Non-Conductive Space

η=μrμ0ϵrϵ0\eta = \sqrt{\frac{\mu_r\mu_0}{\epsilon_r\epsilon_0}}

Rule of Right-Hand Systems

ek=eE×eHE=ηHek=eE×eBE=vB\bm e_k = \bm e_E\times\bm e_H \quad E = \eta H \qquad \bm e_k = \bm e_E\times\bm e_B \quad E = vB

Planar Wave in Space with Condctivity

E=E0eγzex\bm E = E_0e^{\gamma z} \bm e_x

Complex Propagation Constant

γ=jωμrμ0(σ+jωϵrϵ0)γ=αjβ\gamma = \sqrt{j\omega\mu_r\mu_0(\sigma + j\omega\epsilon_r\epsilon_0)} \quad \qquad \gamma = \alpha j\beta

Waveinpedance, Space With Given Conductivity

η=jωμrμ0σ+jωϵrϵ0\eta = \sqrt{\frac{j\omega\mu_r\mu_0}{\sigma + j\omega\epsilon_r\epsilon_0}}

Penetration Depth

δ=2ωμrμ0σ\delta = \sqrt{\frac 2{\omega\mu_r\mu_0\sigma}}