Update textbook.fr.md

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

@@ -264,26 +264,27 @@ _Fig. 4.6 : The phase and group velocity variations vs angular frequency for a g
 
 #### 4.4 - Power flow
 
-The power density (power per unit surface, units $`[W:m^2]`$)
+The power density (power per unit surface, units $`[W/m^2]`$
 traversing the waveguide can be evaluated from the time-averaged
 Poynting vector :
 
-$`\langle\overrightarrow{P}=\Big\langle\dfrac{\overrightarrow{E}\times\overrightarrow{B}}{\mu}\Big\rangle`$
+$`\langle\overrightarrow{P}\rangle=\Big\langle\dfrac{\overrightarrow{E}\times\overrightarrow{B}}{\mu}\Big\rangle`$
 
+Let's consider a $`TE_{0,n}`$ mode as described by equations
+[4.1] and [4.2.]. We can omit the pre-factor
+"-2" from the amplitudes and use the fact that 
+$`k\,\cos\theta=k_y=n\pi/b`$ and $`k\,\sin\theta=k_z`$. We get
+
+$`\overrightarrow{E}_{\perp}=`$$`E_0\,\sin\Big(\dfrac{n\pi}{b}\,y\Big)\,\sin(k_z\,z-\omega\,t})\overrightarrow{e_x}`$
+
+and
+
+
+
+\cdot \sin\big(\underbrace{k\,\sin\theta}_{\large{wavevector}} z-\omega t\big)\overrightarrow{e_z}`$$`\quad(eq.1)`$    
 
 
 
->
-(**P**) = ( [$`\overrightarrow{E}`$ × $`\overrightarrow{B}`$]{.underline} )*.* (4.10)
->
-Let's consider a TE~0*,n*~ mode as described by equations
-[4.1and](#_bookmark79) [4.2.](#_bookmark80) We can omit the pre-factor
-"-2" from the amplitudes and use the fact that *k* cos *θ* = *k~y~* =
-*nπ/b* and *k* sin *θ* = *k~z~*. We get
->
-$`\overrightarrow{E}`$ = *E*~0~ sin *[nπ]{.underline}y* sin(*k z* − *ωt*)*\_e*
->
-(4.11)
 
 and