达朗贝尔方程

在经典电动力学中，将描述电磁波的势所满足的一个微分方程组称作达朗贝尔方程（英文：d'Alembert equation）。达朗贝尔方程以数学家让·勒朗·达朗贝尔的名字命名，他于 1747 年将其作为振动弦问题的解决方案推导出来。

达朗贝尔方程是一个非齐次（英语：Homogeneity and heterogeneity）的波动方程。^[1]

形式

达朗贝尔方程的形式如下：

${\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}=-\mu _{0}{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

其中${\displaystyle {\boldsymbol {A}}}$为磁矢势，${\displaystyle \varphi }$为电势，${\displaystyle c}$为真空光速。^[1]

推导

经典电动力学中的麦克斯韦方程组如下所示

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {E}}=-{\frac {\partial {\boldsymbol {B}}}{\partial t}}}$

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {H}}={\frac {\partial {\boldsymbol {D}}}{\partial t}}+{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {D}}=\rho }$

${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {B}}=0}$

且有${\displaystyle {\boldsymbol {D}}=\varepsilon _{0}{\boldsymbol {E}},{\boldsymbol {B}}=\mu _{0}{\boldsymbol {H}}}$。

由${\displaystyle {\boldsymbol {B}}}$的无源性可以引入磁矢势${\displaystyle {\boldsymbol {A}}}$，有${\displaystyle {\boldsymbol {B}}={\boldsymbol {\nabla }}\times {\boldsymbol {A}}}$，代入麦克斯韦方程组的第一式得${\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}\right)=0}$。这说明矢量${\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}}$是无旋场，可以用标量势${\displaystyle \varphi }$的负梯度描述：

${\displaystyle {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {A}}}{\partial t}}=-{\boldsymbol {\nabla }}\varphi }$

也即${\displaystyle {\boldsymbol {E}}=-{\boldsymbol {\nabla }}\varphi -{\frac {\partial {\boldsymbol {A}}}{\partial t}}}$。

因此

${\displaystyle {\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\times {\boldsymbol {A}}\right)=\mu _{0}{\boldsymbol {J}}-\mu _{0}\varepsilon _{0}{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\varphi \right)-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}}$

${\displaystyle -{\boldsymbol {\nabla }}^{2}\varphi -{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}\right)={\frac {\rho }{\varepsilon _{0}}}}$

而${\displaystyle \mu _{0}\varepsilon _{0}={\frac {1}{c^{2}}}}$，代入并整理得

${\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}-{\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=-\mu _{0}{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}^{2}\varphi +{\frac {\partial }{\partial t}}\left({\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}\right)=-{\frac {\rho }{\varepsilon _{0}}}}$

采用洛伦茨规范，即${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {A}}+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0}$，可得

${\displaystyle {\boldsymbol {\nabla }}^{2}{\boldsymbol {A}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {A}}}{\partial t^{2}}}=-\mu _{0}{\boldsymbol {J}}}$

${\displaystyle {\boldsymbol {\nabla }}^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

此即达朗贝尔方程，其自由项为电流密度和电荷密度。