Update textbook.fr.md

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

@@ -22,43 +22,38 @@ and a standing wave pattern along the y axis.
 _TE and TM waves and their corresponding standing wave behaviour along the y axis._
 
 
-* For TE modes (TE = transverse-electric) we have:
-
-_equations_
+* For TE modes (TE = transverse-electric) we have :   
    
 $`\overrightarrow{E}_{\perp}=\underbrace{-2\,E_0\,\sin(k\cdot y\cdot \cos\,\theta)}_{\large{amplitude}}
-\times \sinbig(\underbrace{k\,\sin\,\theta}_{\large{wavevector}}\;z-\omega t\big)\overrightarrow{e_z}`$
+\times \sin\big(\underbrace{k\,\sin\,\theta}_{\large{wavevector}}\;z-\omega t\big)\overrightarrow{e_z}`$    
+   
+and    
+   
+$`\overrightarrow{B}_{\perp}=
+\left(\begin{array}{l}
+0\\
+-\dfrac{2\,E_0}{c}\sin\,\theta \,\sin\,(k\cdot y \cdot \cos\,\theta)\,\sin\,(k\cdot z \cdot \sin\,\theta-\omega t)\\
+-\dfrac{2\,E_0}{c}\cos\,\theta \,\cos\,(k\cdot y \cdot \cos\,\theta)\,\cos\,(k\cdot z \cdot \sin\,\theta-\omega t)
+\end{array}\right)`$
+
+* Similarly for TM modes :  
    
-<!------------
-[]{#_bookmark79 .anchor}$`\overrightarrow{E}`$~⊥~ = −2*E*~0~ sin (*k y* cos *θ*) sin
-*k* sin *θ z* − *ωt***ˆe***~x~* (4.1)
+$`\overrightarrow{E}_{\parallel}=
+\left(\begin{array}{l}
+0\\
+2\,E_0\,\sin\,\theta \,\cos\,(k\cdot y \cdot \cos\,\theta)\,\cos\,(k\cdot z \cdot \sin\,\theta-\omega t)\\
+-2\,E_0\,\cos\,\theta \,\sin\,(k\cdot y \cdot \cos\,\theta)\,\sin\,(k\cdot z \cdot \sin\,\theta-\omega t)
+\end{array}\right)`$    
    
 and   
->
-\'-
->
-[]{#_bookmark80 .anchor}0 2
->
-*amp*..*li*,.*tude* .,
->
-*wa*\'-*ve*..*v*,.*ec*.,*tor*
->
-− 2*E*0 cos *θ* cos (*k y* cos *θ*) cos (*k z* sin *θ* − *ωt*) and
-similarly for TM modes:
    
-0
+$`\overrightarrow{B}_{\parallel}=\underbrace{-2\,E_0\,\sin(k\cdot y\cdot \cos\,\theta)}_{\large{amplitude}}
+\times \sin\big(\underbrace{k\,\sin\,\theta}_{\large{wavevector}}\;z-\omega t\big)\overrightarrow{e_z}`$ 
 
-$`\overrightarrow{E}`$~I/~ = 2*E*~0~ sin *θ* cos(*k y* cos *θ*) cos(*k z* sin *θ* −
-*ωt*)
 
-−2*E*~0~ cos *θ* sin(*k y* cos *θ*) sin(*k z* sin *θ* − *ωt*)
 
-(4.3)
 
-$`\overrightarrow{B}`$~I/~ = −2*E*~0~ sin (*k y* cos *θ*) sin (*k z* sin *θ* − *ωt*)
-**ˆe***~x~* (4.4)
 
-73
 
 ![](media/image199.png)
 ![](media/image201.png){width="1.753584864391951in"