Boundary Condition
- Fixed boundary condition with hard source
- Fixed boundary condition with soft source
- Absorb boundary condition
Propagation in Medium
- Dielectric Medium
The reflection coefficient is $$ \varGamma = \frac{\sqrt{\epsilon_0} - \sqrt{\epsilon_r}}{\sqrt{\epsilon_0} + \sqrt{\epsilon_r}} $$ and the transmission coefficient is $$ \tau = \frac{2\sqrt{\epsilon_0}}{(\sqrt{\epsilon_0} + \sqrt{\epsilon_r})}. $$ The region where \(x > 100\) has a relative permittivity \(\epsilon_r\) of 4.
- Lossy Dielectric Medium
The region where \(x > 100 \) has a relative permittivity \(\epsilon_r = 4\) and electrical conductivity \(\sigma = 0.04\).
Otherwise, the theoretical decay rate in a lossy dielectric medium is given by $$ \alpha = \frac{\omega}{c_0} \sqrt{\frac{\epsilon_r}{2}} \left[\sqrt{1+\left(\frac{\sigma}{\omega \epsilon_0 \epsilon_r} \right)^2} -1 \right]^{\frac{1}{2}}. $$
