Update textbook.fr.md

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

@@ -57,42 +57,43 @@ and for all times the tangential component of the electric field $`\overrightarr
 (incident + reflected) and the perpendicular component of the
 induction field $`\overrightarrow{B}_{\perp}`$ (incident + reflected) to be zero:   
 
-$`\overrightarrow{E}_{\parallel}=0\quad\text{and}\quad\overrightarrow{B}_{\perp}=0`$.   
+$`\overrightarrow{E}_{\parallel}=0\quad\text{and}\quad\overrightarrow{B}_{\perp}=0`$.  
 
   
 * For TE modes,   
 this results in the fact that the total electric field
 $`\overrightarrow{E}_{\perp}`$  (it is only tangential by definition) has nodes for 
-$`y=\frac{n\pi}{k\,\cos\theta}=\frac{n\pi}{k_y}=\frac{n\lambda}{2\cos\theta}`$ where 
+$`y=\dfrac{n\pi}{k\,\cos\theta}=\dfrac{n\pi}{k_y}=\dfrac{n\lambda}{2\cos\theta}`$ where 
 $`k_y=k\,\cos\theta`$ is the y component of the wavevector.
 
 * For TM modes,   
-we have instead $`\overrightarrow{E}_{\parallel\,,z}=0`$ as the z component of the
+we have instead $`\overrightarrow{E}_{\parallel\,,\,z}=0`$ as the z component of the
 total electric field represents the tangential component.
 
 As the tangent component of the electric field is zero in the nodal planes,   
-$`y=frac{\pi}{k_y}=frac{2\pi}{k_y}=frac{3\pi}{k_y}=\dots`$ 
+$`y=\dfrac{\pi}{k_y}=\dfrac{2\pi}{k_y}=\dfrac{3\pi}{k_y}=\dots`$ 
 for TE and TM modes, placing a new conducting plate at
 the position of these planes would not disturb the total electric
 field. Let's for instance place a conductive plate at the position of
-the first node $`y=b=\frac{\pi}{k_y}`$ for a TE wave. We can see
+the first node $`y=b=\dfrac{\pi}{k_y}`$ for a TE wave. We can see
 that the incident wave, after striking the first plate will be
 reflected towards the second plate where now will replicate the same
 refection phenomenon with exactly the same incident angle and
 automatically satisfying the boundary conditions for the electric field: we have
 obtained the confinement of the wave, i.e. the wave is guided.
 
-chap4 Rectangular waveguides
+##### Rectangular waveguides
 
 Now, let's add two more conducting plates perpendicularly to the
-previous ones, i.e. two new plates parallel to the *y* axis. The
-boundary conditions to be satisfied at the new surfaces are:
->
-$`\overrightarrow{E}`$*~t~* = 0 ⇒ *E~y~* = 0*, E~z~* = 0;
->
-$`\overrightarrow{B}`$*~n~* = 0 ⇒ *B~x~* = 0
->
-For **TE modes**, according to eq: [4.1](#_bookmark79) we have that
+previous ones, i.e. two new plates parallel to the y axis. The
+boundary conditions to be satisfied at the new surfaces are :   
+
+$`\overrightarrow{E}_{\parallel}=0\quad\Longrightarrow\quad (E_y = 0 , E_z = 0)`$
+
+$`\overrightarrow{B}{\perp}=0\quad\Longrightarrow\quad B_x = 0`$
+
+* 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
 its tangent component must be null, its perpendicular component can
 well be discontinue. For the magnetic field (eq: [4.2),](#_bookmark80)