# 包络

## 拍频波

{\displaystyle {\begin{aligned}F(x,\ t)&=\sin \left[2\pi \left({\frac {x}{\lambda -\Delta \lambda }}-(f+\Delta f)t\right)\right]+\sin \left[2\pi \left({\frac {x}{\lambda +\Delta \lambda }}-(f-\Delta f)t\right)\right]\\[6pt]&\approx 2\cos \left[2\pi \left({\frac {x}{\lambda _{\rm {mod}}}}-\Delta f\ t\right)\right]\ \sin \left[2\pi \left({\frac {x}{\lambda }}-f\ t\right)\right]\end{aligned}}}

${\displaystyle {\frac {1}{\lambda \pm \Delta \lambda }}={\frac {1}{\lambda }}\ {\frac {1}{1\pm \Delta \lambda /\lambda }}\approx {\frac {1}{\lambda }}\mp {\frac {\Delta \lambda }{\lambda ^{2}}}.}$

${\displaystyle \lambda _{\rm {mod}}={\frac {\lambda ^{2}}{\Delta \lambda }}\ .}$

### 相速度与群速度

${\displaystyle \xi _{C}=\left({\frac {x}{\lambda }}-f\ t\right)\ ,}$
${\displaystyle \xi _{E}=\left({\frac {x}{\lambda _{\rm {mod}}}}-\Delta f\ t\right)\ ,}$

${\displaystyle \left({\frac {x}{\lambda }}-f\ t\right)=\left({\frac {x+\Delta x}{\lambda }}-f(t+\Delta t)\right)\ ,}$

${\displaystyle v_{\rm {p}}={\frac {\Delta x}{\Delta t}}=\lambda f\ .}$

${\displaystyle v_{\rm {g}}={\frac {\Delta x}{\Delta t}}=\lambda _{\rm {mod}}\Delta f=\lambda ^{2}{\frac {\Delta f}{\Delta \lambda }}\ .}$

${\displaystyle k={\frac {2\pi }{\lambda }}\ .}$

${\displaystyle \Delta k=\left|{\frac {dk}{d\lambda }}\right|\Delta \lambda =2\pi {\frac {\Delta \lambda }{\lambda ^{2}}}\ ,}$

${\displaystyle v_{\rm {g}}={\frac {2\pi \Delta f}{\Delta k}}={\frac {\Delta \omega }{\Delta k}}\ ,}$

${\displaystyle v_{\rm {g}}={\frac {d\omega (k)}{dk}}\ .}$

${\displaystyle \omega =c_{0}k}$

## 函数近似

${\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} )\ ,}$

${\displaystyle \psi (\mathbf {r} )=\sum _{\mathbf {k} }F(\mathbf {k} )e^{i\mathbf {k\cdot r} }u_{\mathbf {k} }(\mathbf {r} )\ ,}$

${\displaystyle \psi (\mathbf {r} )\approx \left(\sum _{\mathbf {k} }F(\mathbf {k} )e^{i\mathbf {k\cdot r} }\right)u_{\mathbf {k} =\mathbf {k} _{0}}(\mathbf {r} )=F(\mathbf {r} )u_{\mathbf {k} =\mathbf {k} _{0}}(\mathbf {r} )\ .}$

## 衍射图样

${\displaystyle I_{1}=I_{0}\sin ^{2}\left({\frac {\pi d\sin \alpha }{\lambda }}\right)/\left({\frac {\pi d\sin \alpha }{\lambda }}\right)^{2}\ ,}$

${\displaystyle I_{q}=I_{1}\sin ^{2}\left({\frac {q\pi g\sin \alpha }{\lambda }}\right)/\sin ^{2}\left({\frac {\pi g\sin \alpha }{\lambda }}\right)\ ,}$

## 参考文献

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6. ^ Peter Y. Yu; Manuel Cardona. Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes. Fundamentals of Semiconductors: Physics and Materials Properties 4th. Springer. 2010: 111. ISBN 978-3642007095.
7. ^ V. Cerveny; Vlastislav Červený. §2.2.9 Relation between the phase and group velocity vectors. Seismic Ray Theory. Cambridge University Press. 2005: 35. ISBN 0521018226.
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9. Christian Schüller. §2.4.1 Envelope function approximation (EFA). Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer. 2006: 22. ISBN 3540365257.
10. ^ For example, see Marco Fanciulli. §1.1 Envelope function approximation. Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures. Springer. 2009: 224 ff. ISBN 978-3540793649.
11. Kordt Griepenkerl. Intensity distribution for diffraction by a slit and Intensity pattern for diffraction by a grating. John W Harris; Walter Benenson; Horst Stöcker; Holger Lutz (编). Handbook of physics. Springer. 2002: 306 ff. ISBN 0387952691.
12. ^ Envelope Extraction - MATLAB & Simulink. MathWorks. 2021-09-02 [2021-11-16]. （原始内容存档于2023-10-19）.