Изменение во времени дивергенции вектора Пойнтинга

Last modified by Alexey Popov on 2019/03/12 13:20

$-{\frac {\partial^2 u}{\partial t^2}}=\nabla \cdot \frac{\partial {\mathbf {S}}}{\partial t}+\frac{\partial {\mathbf {J}}\cdot {\mathbf {E}}}{\partial t}$

$\frac{\partial {\mathbf {S}}}{\partial t}=\frac{1}{\mu_0}\frac{\partial \mathbf E \times \mathbf B}{\partial t} =\frac{1}{\mu_0} \frac{\partial \mathbf E }{\partial t} \times \mathbf B + \frac{1}{\mu_0}\mathbf E \times \frac{\partial \mathbf B}{\partial t}=\frac{c^2}{\mu_0}(\nabla \times \mathbf B ) \times \mathbf B - \frac{c^2}{\mu_0}\mathbf J \times \mathbf B - \frac{1}{\mu_0}\mathbf E \times (\nabla \times \mathbf E) =$

$= -\frac{c^2}{\mu_0} \mathbf B\times (\nabla \times \mathbf B ) - \frac{1}{\mu_0}\mathbf E \times (\nabla \times \mathbf E)-\frac{c^2}{\mu_0}\mathbf J \times \mathbf B = -\frac{1}{2}\frac{c^2}{\mu_0}\nabla (\mathbf B\cdot\mathbf B) + \frac{c^2}{\mu_0}(\mathbf B\cdot\nabla)\mathbf B-\frac{1}{2}c^2\varepsilon_0\nabla (\mathbf E\cdot\mathbf E) + c^2\varepsilon_0(\mathbf E\cdot\nabla)\mathbf E-\frac{c^2}{\mu_0}\mathbf J \times \mathbf B$

$-\frac{1}{2}\frac{\varepsilon_0}{c^2}\frac{\partial^2 (\mathbf E\cdot\mathbf E)}{\partial t^2}-\frac{1}{2}\frac{1}{\mu_0}\frac{\partial^2 (\mathbf B\cdot\mathbf B)}{\partial t^2}+\frac{1}{2}\varepsilon_0\Delta(\mathbf E\cdot\mathbf E)+\frac{1}{2}\frac{1}{\mu_0}\Delta(\mathbf B\cdot\mathbf B)=\varepsilon_0\nabla\cdot((\mathbf E\cdot\nabla)\mathbf E)+\frac{1}{\mu_0}\nabla\cdot((\mathbf B\cdot\nabla)\mathbf B)-\frac{1}{\mu_0}\nabla \cdot (\mathbf J \times \mathbf B)+\frac{1}{c^2}\frac{\partial {\mathbf {J}}\cdot {\mathbf {E}}}{\partial t}$

Acoustic wave equation

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Created by Alexey Popov on 2019/03/11 18:50