Update textbook.fr.md

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

@@ -2,7 +2,7 @@
 
 <br>
 
-#### Introduction
+#### 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 +10,7 @@ main char- acteristic of this kind of propagation. We will introduce a
 "practical approach" based on the previous chapter's results.
 
 
-#### Practical approach
+#### 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
@@ -26,7 +26,7 @@ _TE and TM waves and their corresponding standing wave behaviour along the y axi
    
 
 $`\overrightarrow{E}_{\perp}=`$$`\underbrace{-2\,E_0\,\sin(k\; y\, \cos\theta)}_{\large{amplitude}}
-\cdot \sin\big(\underbrace{k\,\sin\theta}_{\large{wavevector}} z-\omega t\big)\overrightarrow{e_z}`$    
+\cdot \sin\big(\underbrace{k\,\sin\theta}_{\large{wavevector}} z-\omega t\big)\overrightarrow{e_z}`$$`\quad(eq.1)`$    
 
    
 $`\overrightarrow{B}_{\perp}=`$
@@ -34,7 +34,7 @@ $`\left(\begin{array}{l}
 0\\
 -\dfrac{2\,E_0}{c}\sin\theta \cdot \sin\,(k\; y \, \cos\theta)\cdot \sin\,(k\; z\,\sin\theta -\omega t)\\
 -\dfrac{2\,E_0}{c}\cos\theta \cdot\cos\,(k\; y \, \cos\theta)\cdot\cos\,(k\; z\,\sin\theta -\omega t)
-\end{array}\right)`$
+\end{array}\right)`$$`\quad(eq.2)`$   
 
 <br>
 
@@ -45,10 +45,10 @@ $`\left(\begin{array}{l}
 0\\
 +2\,E_0\,\sin\theta \cdot\cos\,(k\;y \, \cos\theta)\cdot\cos\,(k\;z\,\sin\theta -\omega t)\\
 -2\,E_0\,\cos\theta \cdot\sin\,(k\; y \, \cos\theta)\cdot\sin\,(k\;z\,\sin\theta -\omega t)
-\end{array}\right)`$    
+\end{array}\right)`$$`\quad(eq.3)`$    
    
 $`\overrightarrow{B}_{\parallel}=-2\,E_0\,\sin\,(k\; y\, \cos\theta)
-\cdot \sin\big(k\;z\,\sin\,\theta -\omega t\big)\overrightarrow{e_z}`$ 
+\cdot \sin\big(k\;z\,\sin\,\theta -\omega t\big)\overrightarrow{e_z}`$$`\quad(eq.4)`$ 
 
 
 These results are dictated by the boundary conditions at the air-metal
@@ -93,19 +93,24 @@ $`\overrightarrow{E}_{\parallel}=0\quad\Longrightarrow\quad (E_y = 0 , E_z = 0)`
 $`\overrightarrow{B}{\perp}=0\quad\Longrightarrow\quad B_x = 0`$
 
 * For TE modes,   
-according to eq: [4.1](#_bookmark79) we have that
-*E\_ x* and the only condition imposed on the electric field is that
+according to $`(eq.1)`$ we have that $`\overrightarrow{E}\;\parallel\;x`$
+and the only condition imposed on the electric field is that
 its tangent component must be null, its perpendicular component can
-well be discontinue. For the magnetic field (eq: [4.2),](#_bookmark80)
-*B~x~* = 0 is already satisfied. In summary, nothing changes for the
+well be discontinue.   
+For the magnetic field $`(eq.2)`$
+$`b_x = 0`$ is already satisfied.   
+In summary, nothing changes for the
 electric or magnetic fields, but we have totally confined the
-radiation in the *x* and *y* directions.
->
-For **TM modes** (at least for TM~*m,*0~ or TM~0*,n*~, see later for
+radiation in the x and y directions.
+
+* For TM modes,
+(at least for TM_m_,_0 or TM_0_,_n, see later for
 the definition of these modes), the boundary conditions cannot be
 satisfied (try to show it). In what follows we will consider for
 simplicity only the case of TE modes.
 
+@@@@@@@@@@@@@@@@@@@
+
 ![](media/image206.png)
 
 Figure 4.2: Rectangular waveguide with a TE mode.