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# 平面波

## 数学表述

3D平面波的动画。 每种颜色表示波的不同的相位

${\displaystyle \nabla ^{2}f-{\frac {1}{v^{2))}{\frac {\partial ^{2}f}{\partial t^{2))}=0}$

${\displaystyle \nabla ^{2}{\tilde {\psi ))-{\frac {1}{v^{2))}{\frac {\partial ^{2}{\tilde {\psi ))}{\partial t^{2))}=0}$

${\displaystyle {\tilde {\psi ))(\mathbf {x} ,t)={\tilde {A))e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)))$

${\displaystyle \operatorname {Re} \((\tilde {\psi ))(\mathbf {x} ,t)\}=|{\tilde {A))|\cos(\mathbf {k} \cdot \mathbf {x} -\omega t+\arg {\tilde {A)))}$

${\displaystyle \mathbf {k} \cdot \mathbf {x} -\omega t_{0}+\arg {\tilde {A))=c_{1))$

${\displaystyle \mathbf {k} \cdot \mathbf {x} =c_{2))$

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {1}{v^{2))}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2))}=0}$
${\displaystyle \nabla ^{2}\mathbf {B} -{\frac {1}{v^{2))}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2))}=0}$

${\displaystyle {\tilde {\boldsymbol {\psi ))}(\mathbf {x} ,\ t)={\tilde {\mathbf {A} ))e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)))$

${\displaystyle v_{p}=\omega /k}$

${\displaystyle v_{g}={\frac {\partial \omega }{\partial \mathbf {k} ))}$

## 参考文献

1. ^ Hecht, Eugene, Optics 4th, United States of America: Addison Wesley, 2002, ISBN 0-8053-8566-5 （英语）

• J. D. Jackson, Classical Electrodynamics (Wiley: New York, 1998 )。