Update textbook.fr.md

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

@@ -24,17 +24,15 @@ _TE and TM waves and their corresponding standing wave behaviour along the y axi
 
 * For TE modes (TE = transverse-electric) we have :   
    
-$`\overrightarrow{E}_{\perp}=`$$`\underbrace{-2\,E_0\,\sin(k\cdot y\cdot \cos\,\theta)}_{\large{amplitude}}
-\times \sin\big(\underbrace{k\,\sin\,\theta}_{\large{wavevector}}\,z-\omega t\big)\overrightarrow{e_z}`$    
+$`\overrightarrow{E}_{\perp}=`$$`\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}`$    
 
    
 $`\overrightarrow{B}_{\perp}=`$
 $`\left(\begin{array}{l}
 0\\
-\\
-\underbrace{-\dfrac{2\,E_0}{c}\sin\theta \,\sin\,(k\cdot y \cdot \cos\theta)}_{\large{amplitude}}\,\sin\,(\underbrace{k\,\sin\theta}_{\large{wavevector}}\,z -\omega t)\\
-\\
-\underbrace{-\dfrac{2\,E_0}{c}\cos\,\theta \,\cos\,(k\cdot y \cdot \cos\,\theta)}_{\large{amplitude}}\,\cos\,(\underbrace{k\,\sin\theta}_{\large{wavevector}}\,z -\omega t)
+-\dfrac{2\,E_0}{c}\sin\theta \,\sin\,(k\cdot y \cdot \cos\theta)\,\sin\,(k\,\sin\theta\,z -\omega t)\\
+-\dfrac{2\,E_0}{c}\cos\,\theta \,\cos\,(k\cdot y \cdot \cos\,\theta)\,\cos\,(k\,\sin\theta\,z -\omega t)
 \end{array}\right)`$
 
 <br>
@@ -44,13 +42,13 @@ $`\left(\begin{array}{l}
 $`\overrightarrow{E}_{\parallel}=`$
 $`\left(\begin{array}{l}
 0\\
-2\,E_0\,\sin\,\theta \,\cos\,(k\cdot y \cdot \cos\,\theta)\,\cos\,(k\,\sin\,\theta\;z -\omega t)\\
--2\,E_0\,\cos\,\theta \,\sin\,(k\cdot y \cdot \cos\,\theta)\,\sin\,(k\,\sin\,\theta\;z -\omega t)
+2\,E_0\,\sin\theta \,\cos\,(k\cdot y \cdot \cos\theta)\,\cos\,(k\,\sin\theta\;z -\omega t)\\
+-2\,E_0\,\cos\theta \,\sin\,(k\cdot y \cdot \cos\theta)\,\sin\,(k\,\sin\theta\;z -\omega t)
 \end{array}\right)`$    
    
    
 $`\overrightarrow{B}_{\parallel}=\underbrace{-2\,E_0\,\sin(k\cdot y\cdot \cos\,\theta)}_{\large{amplitude}}
-\times \sin\big(\underbrace{k\,\sin\,\theta}_{\large{wavevector}}\,z-\omega t\big)\overrightarrow{e_z}`$ 
+\times \sin\big(\underbrace{k\,\sin\,\theta}_{\large{wavevector}} z-\omega t\big)\overrightarrow{e_z}`$