# 平面波

## 數學表述

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

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

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

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

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

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

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

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

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

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

## 參考文獻

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 )。