Update textbook.fr.md

12.temporary_ins/96.electromagnetism-in-media/20.reflexion-refraction-at-interfaces/20.metallic-waveguides/10.main/textbook.fr.md

@@ -165,100 +165,34 @@ with $`k=\sqrt{k_x^2+k_y^2+k_z^2}`$.
 * The wave is propagating along the $`z`$-axis and it is the __nature of__ $`k_z`$
 which defines the type of propagation.
 
-In summary, for $`TE`$ waves:
->
-⇒ Existence of a cut-off frequency. The dispersion relation is:
+In summary, for $`TE`$ waves:   
+$`\Longrightarrow`$ Existence of a cut-off frequency. The dispersion relation is:
 
-*k~z~* =
+   * $`k_z=\sqrt{\dfrac{\omega^2}{c^2}-\dfrac{\pi^2 n^2}{b^2}}`$ for a $`TE_{0,n}`$ mode.
 
-*ω*2
->
-*c*2 −
->
-*π*2*n*2 for a TE mode
->
-*b*
+   * $`k_z=\sqrt{\dfrac{\omega^2}{c^2}-\dfrac{\pi^2 m^2}{a^2}}`$ for a $`TE_{m,0}`$ mode.
 
-*k~z~* =
+   * $`k_z=\sqrt{\dfrac{\omega^2}{c^2}-\dfrac{\pi^2 n^2}{b^2}-\dfrac{\pi^2 m^2}{a^2}}`$ for a $`TE_{m,n}`$ mode.
 
 We have for:
->
-*k~z~* =
->
-*ω*2
->
-*c*2 −
 
-*ω*2
+* *a $`TE_{0,n}`$ mode :
+   * if $`\omega\gt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{R}e`$   
+     $`\quad\Longrightarrow`$ propagation without absorption for mode $`TE_{0,n}`$.
+   * if $`\omega\lt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{I}m`$   
+     $`\quad\Longrightarrow`$ evanescent wave for mode $`TE_{0,n}`$.
+
+* *a $`TE_{0,n}`$ mode :
+   * if $`\omega\gt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{R}e`$   
+     $`\quad\Longrightarrow`$ propagation without absorption for mode $`TE_{0,n}`$.
+   * if $`\omega\lt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{I}m`$   
+     $`\quad\Longrightarrow`$ evanescent wave for mode $`TE_{0,n}`$.
+
+
+
 
-*c*2 −
 
-*π*2*n*2
 
-*b*2 −
->
-*π*2*m*2 for a TE mode
->
-*a*
->
-*π*2*m*2 for a TE mode
->
-*a*
->
-**a TE**~0*,n*~ **mode**
->
-if *ω \ω~c~*
->
-if *ω \< ω~c~*
->
-= *[nπc]{.underline}*
->
-*b*
->
-= *[nπc]{.underline}*
->
-*b*
->
-→ *k~z~*
->
-→ *k~z~*
->
-∈ Re → Propagation without absorption for mode TE~*n,*0~
->
-∈ Im → Evanescent wave for mode TE~0*,n*~
->
-**a TE**~0*,n*~ **mode**
->
-if *ω \ω~c~*
->
-if *ω \< ω~c~*
->
-= *[mπc]{.underline}*
->
-*a*
->
-= *[mπc]{.underline}*
->
-*a*
->
-→ *k~z~*
->
-→ *k~z~*
->
-∈ Re → Propagation without absorption for mode TE~*m,*0~
->
-∈ Im → Evanescent wave for mode TE~*m,*0~
->
-**a TE***~m,n~* **mode**
->
-if = *mπc* 2 + *nπc* 2
->
-Re Propagation without absorption for mode TE
->
-if = *mπc* 2 + *nπc* 2
->
-Re Evanescent wave for mode TE
->
 where *ω~c~* is the cut-off angular frequency. So we can propagate and
 ELM wave as a TE wave in rectangular waveguides only for frequencies
 larger than the cut-off one: the waveguide acts as a high-pass filter.