Update textbook.fr.md

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

@@ -166,6 +166,7 @@ with $`k=\sqrt{k_x^2+k_y^2+k_z^2}`$.
 which defines the type of propagation.
 
 In summary, for $`TE`$ waves:   
+
 $`\Longrightarrow`$ Existence of a cut-off frequency. The dispersion relation is:
 
    * $`k_z=\sqrt{\dfrac{\omega^2}{c^2}-\dfrac{\pi^2 n^2}{b^2}}`$ for a $`TE_{0,n}`$ mode.
@@ -176,45 +177,44 @@ $`\Longrightarrow`$ Existence of a cut-off frequency. The dispersion relation is
 
 We have for:
 
-* *a $`TE_{0,n}`$ mode* :
+* a $`TE_{0,n}`$ mode :
    * if $`\omega\gt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{R}e`$   
      $`\Longrightarrow`$ propagation without absorption for mode $`TE_{0,n}`$.
    * if $`\omega\lt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{I}m`$   
      $`\Longrightarrow`$ evanescent wave for mode $`TE_{0,n}`$.
 
-* *a $`TE_{0,n}`$ mode :
+* a $`TE_{m,0}`$ mode :
    * if $`\omega\gt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{R}e`$   
-     $`\Longrightarrow`$ propagation without absorption for mode $`TE_{0,n}`$.
+     $`\Longrightarrow`$ propagation without absorption for mode $`TE_{m,0}`$.
    * if $`\omega\lt\omega_c=\dfrac{n\pi c}{b}\;\Longrightarrow\;k_z\in\mathscr{I}m`$   
-     $`\Longrightarrow`$ evanescent wave for mode $`TE_{0,n}`$.
+     $`\Longrightarrow`$ evanescent wave for mode $`TE_{m,0}`$.
 
+* a $`TE_{m,n}`$ mode :
+   * if $`\omega\gt\omega_c=\sqrt{\dfrac{n\pi c}{b}^2+\dfrac{n\pi c}{b}^2}\;\Longrightarrow\;k_z\in\mathscr{R}e`$   
+     $`\Longrightarrow`$ propagation without absorption for mode $`TE_{m,n}`$.
+   * if $`\omega\lt\omega_c=\sqrt{\dfrac{n\pi c}{b}^2+\dfrac{n\pi c}{b}^2}\;\Longrightarrow\;k_z\in\mathscr{I}m`$   
+     $`\Longrightarrow`$ evanescent wave for mode $`TE_{m,n}`$.
 
+where $`\omega_c`$ is the cut-off angular frequency. So we can propagate an
+electromagnetic 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.
 
+$`\Longrightarrow`$ Propagation in the waveguide is dispersive:
+>
+For TEM waves in free space we have: $`k=\omega / c`$ and $`v_{\varphi}=c`$,
+which is independent of $`\omega`$: non-dispersive medium. For $`TE`$ waves in
+rectangular waveguides, we have in the case of a $`TE_{0,n}`$ mode: (the
+propagation is along the $`z`$-axis):
 
+$`k_z=\sqrt{\dfrac{\omega^2}{c^2}-\dfrac{\pi^2 n^2}{b^2}}`$
 
+and   
+
+$`v_{\varphi}=\dfrac{\omega}{k_z}=\dfrac{c}{\sqrt{1-\frac{\omega_c^2}{\omega^2}}}`$
 
-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.
->
-Propagation in the waveguide is dispersive:
->
-For TEM waves in free space we have: *k* = *ω/c* and *v~ϕ~* = *c*,
-which is independent of *ω*: non-dispersive medium. For TE waves in
-rectangular waveguides, we have in the case of a TE~0*,n*~ mode: (the
-propagation is along the *z* axis):
 
-and
 
-[]{#_bookmark83 .anchor}*k~z~* =
->
-*ω*
->
-*ω*2
->
-*c*2 −
 
-*c*
 
 *π*2*n*2 (4.5)
 >