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**）。达朗贝尔方程是一个非齐次（英语：）的波动方程。

形式
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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\boldsymbol{A}为磁矢势，\varphi 为电势，c为真空光速。

推导
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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\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\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。

参考资料
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1. 1 2  郭硕鸿. 《电动力学（第三版）》. 北京: 高等教育出版社. 2008. ISBN 978-7-04-023924-9.