plane wave
calculate the parameters of a plane wave in a homogeneous media
`\vec E=E_0 exp(-jkz)`
frequency
relative permitivity
electric amplitude
length
angler frequency
\( \omega=2\pi f\)
period
\( \omega T=2\pi\)
wave number
\(k^2=\omega^2 \epsilon_0 \mu_0\)
reflect index
\(n_0=\frac{ck}{\omega}=1\)
wave length
\(k \lambda=2\pi\,\,\,\, \lambda_0=\frac{2\pi}{k}\)
phase velocity
\(k r-\omega t=const\,\,\,\,\,k v_p-\omega=0\,\,\,\, v_p=\frac{\omega}{k}=\frac{\omega}{\omega\sqrt{\epsilon_0\mu_0}}=c\)
wave impendance
`Z_0=E/H=k/(omega epsilon_0)=sqrt(mu_0/epsilon_0)`
averaged energy flux density
\(W=\frac{1}{2}\epsilon_0 E_0^2\)
wave number
\(k^2=\omega^2 \epsilon_r\epsilon_0\mu_0 \,\,\, k=k_0 \sqrt{\epsilon_r}=k_0 n\)
reflect index
\(\mathbf n=\frac{c \mathbf k}{\omega}=\sqrt{\epsilon_r}\)
wave length
\(k\lambda=2\pi\,\,\,\, \lambda=\frac{\lambda_0}{n}\)
electric length `l_E`
`l/lambda`
phase velocity
\(v_p=\frac{\omega}{k}=\frac{v_{p0}}{n}\)
wave impendance
\(Z=\frac{E}{H}=\frac{k}{\omega\epsilon_0\epsilon_r}=\frac{Z_0}{n}\,\,\,\,|E|=Z H=Z\frac{|B|}{\mu_0}=\frac{c|B|}{n}\)
averaged energy flux density
\(w=\frac{1}{2}\epsilon_0\epsilon_r E_0^2=w_0n^2\)
slowly varying critical
\(\frac{dn}{dz}\frac{1}{k_0 n^2}\ll1\)
electrical field
\(E_y=\frac{A}{\sqrt n} e^{\pm k_0 \int^z n dz}\)
magnetic field
\(cB_x=\mp A \sqrt n e^{\pm k_0 \int^z n dz}\)