时谐 Maxwell 方程组与波动方程的导出

\[ \begin{cases} \nabla\times\vec{E}=-\frac{\partial}{\partial t}\vec{B}\\ \nabla\times\vec{H}=\vec{J}+\frac{\partial}{\partial t}\vec{D}\\ \nabla\cdot \vec{D}=\rho\\ \nabla\cdot \vec{B}=0\\ \end{cases} \]

\[ \vec{E}(\omega)=\mathscr{F}[\vec E(t)]\,,\quad \frac{\partial}{\partial t} = -i\omega \]

\[ \begin{cases} \nabla\times\vec{E}=i\omega\vec{B}\\ \nabla\times\vec{H}=\vec{J}-i\omega\vec{D}\\ \nabla\cdot \vec{D}=\rho\\ \nabla\cdot \vec{B}=0\\ \end{cases} \]

\[ \begin{cases} \hat n\times(\vec{E_2}-\vec{E_1})=0\\ \hat n\times(\vec{H_2}-\vec{H_1})=\vec{J_{sf}}\\ \hat n\cdot(\vec{D_2}-\vec{D_1})=\rho_{sf}\\ \hat n\cdot(\vec{B_2}-\vec{B_1})=0\\ \end{cases} \]