Update textbook.fr.md

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

@@ -38,7 +38,7 @@ $`\left(\begin{array}{l}
 
 <br>
 
-* Similarly for TM modes :  
+* Similarly for TM modes (TM = transverse-magnetic) :  
    
 $`\overrightarrow{E}_{\parallel}=`$
 $`\left(\begin{array}{l}
@@ -51,28 +51,11 @@ $`\overrightarrow{B}_{\parallel}=-2\,E_0\,\sin\,(k\; y\, \cos\theta)
 \cdot \sin\big(k\;z\,\sin\,\theta -\omega t\big)\overrightarrow{e_z}`$ 
 
 
-
-
-
-
-![](media/image199.png)
-![](media/image201.png){width="1.753584864391951in"
-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.
----------->
-
-* Similarly for TM modes (TM = transverse-magnetic)
-
-_equations_
-
 These results are dictated by the boundary conditions at the air-metal
 interface which impose that for every point on the boundary surface
-and for all times the tangential component of the electric field
+and for all times the tangential component of the electric field $`\overrightarrow{E}_{\parallel}`$
 (incident + reflected) and the perpendicular component of the
-induction field (incident + reflected) to be zero:   
+induction field $`\overrightarrow{B}_{\perp}`$ (incident + reflected) to be zero:   
 
 $`\overrightarrow{E}_{\parallel}=0\quad\text{and}\quad\overrightarrow{B}_{\perp}=0`$.   
 
@@ -80,31 +63,23 @@ $`\overrightarrow{E}_{\parallel}=0\quad\text{and}\quad\overrightarrow{B}_{\perp}
 * 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 
-$`y=\frac{}{}=\frac{}{}=\frac{}{}`$ where $`k_y=k\,\cos\theta`$ is the y component 
-of the wavevector.
+$`y=\frac{n\pi}{k\,\cos\theta}=\frac{n\pi}{k_y}=\frac{n\lambda}{2\cos\theta}`$ 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
 total electric field represents the tangential component.
 
 As the tangent component of the electric field is zero in the nodal planes,   
-$`y=dfrac{\pi}{k_y}=dfrac{\pi}{k_y}=dfrac{\pi}{k_y}
->
-*y* = *[π]{.underline} ,* [2*π*]{.underline} *,* [3*π*]{.underline}
->
-. . . for TE and TM modes, placing a new conducting plate at
->
-*ky ky ky*
->
+$`y=frac{\pi}{k_y}=frac{2\pi}{k_y}=frac{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* = *[π]{.underline}* for a TE wave. We can see
+the first node $`y=b=\frac{\pi}{k_y}`$ 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
+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