Update textbook.fr.md

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

+#### Chapter 4
+
 ### Metallic waveguides
 
 <br>
 
-#### 1 - Introduction
+#### 4.1 - Introduction
 
 We propose to study in this chapter the conditions for propagation of
 elm radiation in conductive rectangular waveguides and to identify the
@@ -10,7 +12,7 @@ main char- acteristic of this kind of propagation. We will introduce a
 "practical approach" based on the previous chapter's results.
 
 
-#### 2 - Practical approach
+#### 4.2 - Practical approach
 
 We have seen in the previous chapter that the oblique incidence of
 plane waves on planar conductive materials gives rise to an
@@ -19,7 +21,7 @@ that the resulting wave has propagating character along the z axis
 and a standing wave pattern along the y axis. 
 
 ![](TE-and-TM-waves-1_L1200.jpg)   
-_TE and TM waves and their corresponding standing wave behaviour along the y axis._
+_Figure 4.1 : TE and TM waves and their corresponding standing wave behaviour along the y axis._
 
 
 * For TE modes (TE = transverse-electric) we have :   
@@ -112,11 +114,11 @@ simplicity only the case of TE modes.
 
 
 ![](rectangular_waveguide_with_a_TE_mode.jpg)   
-_Rectangular waveguide with a TE mode._
+_Figure 4.2 : Rectangular waveguide with a TE mode._
 
 <br>
 
-#### 3 $`\mathbf{TE}`$ modes
+#### 3 TE modes
 
 
 ##### Mode numbering
@@ -215,7 +217,7 @@ $`v_{\varphi}=\dfrac{\omega}{k_z}=\dfrac{c}{\sqrt{1-\frac{\omega_c^2}{\omega^2}}
 It is a dispersive medium.
 
 ![](dispersion-relation-for-a-mode_temp_L1200.jpg)   
-_Fig. 4.4: The geometric interpretation of the dispersion._
+_Fig. 4.4 : The geometric interpretation of the dispersion._
 
 Similarly we obtain for the group velocity:
 
@@ -224,7 +226,7 @@ $`v_{\varphi}=\dfrac{\partial \omega}{\partial k_z}=c\,\sqrt{1-\dfrac{\omega_c^2
 __Geometrical interpretation__
 
 ![](geometrical-interpretation-of-the-dispersion_temp_L1200.jpg)   
-_Fig. 4.5: The geometric interpretation of the dispersion._
+_Fig. 4.5 : The geometric interpretation of the dispersion._
 
 We can understand the propagating behaviour of the $`TE`$ wave using a
 simple interpretation:
@@ -235,7 +237,7 @@ simple interpretation:
     progression) and $`v_g=0`$. In regions close to $`\omega_c`$ the
     waveguides is highly dispersive.
 
-* For $`\omega\longrightarrow\infty\,,k_z\longrightarrow\frac{\omega}{c}•,\theta\longrightarrow\pi/2`$
+* For $`\omega\longrightarrow\infty\,,k_z\longrightarrow\frac{\omega}{c}\,\theta\longrightarrow\pi/2`$
   and $`v_g\longrightarrow c`$.   
   The $`TE`$ waves tends to have very large incidence angles and the guide
     behaves essentially as vacuum, i.e. dispersionless.
@@ -247,20 +249,20 @@ simple interpretation:
   <br>
   where $`\lambda_z=\lambda / \sin\theta`$ and $`\lambda_y=\lambda / \cos\theta`$, or again
   <br>
-  $`\dfrac{1}{\lambda_z}=\sqrt{\big(\dfrac{1}{\lambda}\big)^2-\big(\dfrac{n}{\2b}\big)^2}`$   
+  $`\dfrac{1}{\lambda_z}=\sqrt{\big(\dfrac{1}{\lambda}\big)^2-\big(\dfrac{n}{2b}\big)^2}`$   
   <br>
 from which it is easy to confirm (see definition of $`\omega_c`$) that the
 cut-off wavelength for a $`TE_{0,n}`$ mode is $`\lambda_c=\frac{2b}{n}`$
 
 ![](variation-phase-group-velocity-wersus-omega-guided-mode_temp_L1200.jpg)    
-_Fig. 4.6: The phase and group velocity variations vs angular frequency for a guided mode._
+_Fig. 4.6 : The phase and group velocity variations vs angular frequency for a guided mode._
 
 !!!!! *Exercice 4.1 : Refractive index*   
 !!!!! Calculate and plot the refractive index vs the angular frequency for a $`TE`$ wave.
 
 
 
-#### 4. Power flow
+#### 4.4 - Power flow
 
 The power density (power per unit surface, units $`[W:m^2]`$)
 traversing the waveguide can be evaluated from the time-averaged