伽利略变换和洛伦兹变换下的有源波动方程

伽利略变换下：

\[ \left( \frac{\partial^2 E}{\partial x’^2} + \frac{\partial^2 E}{\partial y’^2} + \frac{\partial^2 E}{\partial z’^2} \right) - \frac{1}{c^2} \left( v^2 \frac{\partial^2 E}{\partial x’^2} - 2 v \frac{\partial^2 E}{\partial x’ \partial t’} + \frac{\partial^2 E}{\partial t’^2} \right) = 0 \]

洛伦兹变换下：

$$ \gamma^2 \left( \frac{\partial^2 E}{\partial x’^2} - 2\frac{v}{c^2} \frac{\partial^2 E}{\partial x’ \partial t’} + \frac{v^2}{c^4} \frac{\partial^2 E}{\partial t’^2} \right) + \frac{\partial^2 E}{\partial y’^2} + \frac{\partial^2 E}{\partial z’^2} - \frac{1}{c^2} \cdot \gamma^2 \left( \frac{\partial^2 E}{\partial t’^2} - 2v \frac{\partial^2 E}{\partial x’ \partial t’} + v^2 \frac{\partial^2 E}{\partial x’^2} \right) = a $$