Update textbook.fr.md

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

@@ -116,7 +116,7 @@ _Rectangular waveguide with a TE mode._
 
 <br>
 
-#### TE modes
+#### $`\mathbf{TE}`$ modes
 
 
 ##### Mode numbering
@@ -125,56 +125,47 @@ In the previous example, we have chose the orientation of the frame to
 describe the rectangular waveguide such that the successive reflection
 of the electric field intervene between the conducting plates placed
 at $`y=0`$ and $`y=b`$ where $`b=\dfrac{n\lambda}{2\,\cos\theta}`$.
-We will describe this wave as a TE~0*,n*~ which signifies that the reflection
-occurs between the plates placed at *y* = 0 and *y* = *b* and that
->
-![](media/image212.png) ![](media/image213.png)
->
-![](media/image214.png) ![](media/image218.png)
+We will describe this wave as a $`TE_{0,n}`$ which signifies that the reflection
+occurs between the plates placed at $`y=0`$ and $`y=b`$ and that there are $`n`$
+antinodes of the electric field standing wave between
+these plates.
 
-Figure 4.3: The mode numbering convention.
+![](mode-numbering-convention_temp_2_L1200.jpg)    
+_Figure 4.3: The mode numbering convention._
 
-there are *n* antinodes of the electric field standing wave between
-these plates. Obviously the situation will be formally identical if
+Obviously the situation will be formally identical if
 now we decide to force the successive refections along the other
-conducting plates placed now at *x* = 0
-
-and *x* = *a* where *a* = *[mλ]{.underline}* if the wave is TE for
-**that reflection**: The electric
-
-field must be now oriented along the *y* direction in order to be a TE
-wave **for these new plates**.We will describe this wave as a
-TE~*m,*0~ which signifies that the reflection occurs between the
-plates placed at *x* = 0 and *x* = *a* and that there are *m*
+conducting plates placed now at $`x=0`$ and $`x=a`$ where $`a=\frac{m\lambda}{2\cos\theta}`$
+if the wave is $`TE`$ for __that reflection__: The electric
+field must be now oriented along the $`y`$  direction in order to be a $`TE`$
+wave __for these new plates__.We will describe this wave as a
+$`TE_{m,0}`$ which signifies that the reflection occurs between the
+plates placed at $`x=0`$ and $`x=a`$ and that there are $`m`$
 antinodes of the electric field standing wave between these plates. If
-two TE waves are simultaneously excited in the system (each one being
-TE for its respective incidence), than we will describe this wave as a
-TE*~m,n~*.
+two $`TE`$ waves are simultaneously excited in the system (each one being
+$`TE`$ for its respective incidence), than we will describe this wave as a
+$`TE_{m,n}`$.
 
-chap4 Properties of the confined modes
 
-Let's try to find the dispersion relation of a rectangular waveguides
-with a few simple considerations on TE~0*,n*~ waves in a waveguide
-with dimensions *y* = *b* and *x* = *a*:
-
--   The wave is propagating in air (or vacuum for simplicity). The wave
-    must satisfy the usual wave equation and dispersion relation *k* =
-    *[ω]{.underline}* with
+##### Properties of the confined modes
 
-*k* = j*k*2 + *k*2 + *k*2.
+Let's try to find the dispersion relation of a rectangular waveguides
+with a few simple considerations on $`TE_{0,n}`$ waves in a waveguide
+with dimensions $`y=b`$ and $`x=a`$ :
 
--   We now consider that the boundary conditions impose restrictions on
-    the possible values of *k~y~*. For what seen before, the
-    propagation condition reads *k~y~* = *k* cos *θ* =
-    *[nπ]{.underline}* , and being a TE~0*,n*~ wave we have *k~x~* =
-    0.
+*  The wave is propagating in air (or vacuum for simplicity). The wave
+    must satisfy the usual wave equation and dispersion relation $`k=\frac{\omega}{c}`$
+with $`k=\sqrt{k_x^2+k_y^2+k_z^2}`$.
 
--   the wave is propagating along the *z* axis and it is the **nature
-    of** *k~z~*
+*  We now consider that the boundary conditions impose restrictions on
+    the possible values of $`k_y`$. For what seen before, the
+    propagation condition reads $`k_y=k\,\cos\theta=frac{n\pi}{b}`$ , 
+    and being a $`TE_{0,n}`$ wave we have $`k_x=0`$.
 
+* The wave is propagating along the $`z`$-axis and it is the __nature of__ $`k_z`$
 which defines the type of propagation.
 
-In summary, for TE waves:
+In summary, for $`TE`$ waves:
 >
 ⇒ Existence of a cut-off frequency. The dispersion relation is: