Update textbook.fr.md

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

-# Metallic waveguides
+### Metallic waveguides
 
-chap2 Introduction
+<br>
+
+#### Introduction
 
 We propose to study in this chapter the conditions for propagation of
 elm radiation in conductive rectangular waveguides and to identify the
 main char- acteristic of this kind of propagation. We will introduce a
 "practical approach" based on the previous chapter's results.
 
-chap2 Practical approach
+
+#### Practical approach
 
 We have seen in the previous chapter that the oblique incidence of
 plane waves on planar conductive materials gives rise to an
 interference pattern between the incident and reflected waves such
 that the resulting wave has propagating character along the *z* axis
-and a standing wave pattern along the *y* axis (see figure
-[4.1).](#_bookmark81) For TE modes we have:
+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._
+
+For TE modes we have:
 
+<!------------
 []{#_bookmark79 .anchor}$`\overrightarrow{E}`$~⊥~ = −2*E*~0~ sin (*k y* cos *θ*) sin
 *k* sin *θ z* − *ωt***ˆe***~x~* (4.1)
 
@@ -53,14 +61,16 @@ height="1.7708333333333333in"}
 ![](media/image202.png)![](media/image204.png)Figure 4.1:
 []{#_bookmark81 .anchor}TE and TM waves and their corresponding
 standing wave be- haviour along the *y* axis.
->
+---------->
+
+
 These results are dictated by the boundary conditions at the air-metal
 inter- face which impose that for every point on the boundary surface
 and for all times the tangential component of the electric field
 (incident + reflected) and the perpendicular component of the
-induction field (incident + reflected) to be zero: $`\overrightarrow{E}`$*~t~* = 0 and
-$`\overrightarrow{B}`$*~n~* = 0.
->
+induction field (incident + reflected) to be zero: $`\overrightarrow{E}_{\parallel}`$*~t~* = 0 and
+$`\overrightarrow{B}_{\perp}`= 0`$.   
+   
 For TE modes this results in the fact that the total electric field
 $`\overrightarrow{E}`$~⊥~ (it is
 >