Update cheatsheet.fr.md

12.temporary_ins/69.waves/30.n3/20.overview/cheatsheet.fr.md

@@ -1672,7 +1672,7 @@ $`\quad\quad \;\; = \cdots`$
                \right\}\Longrightarrow\\
         \quad\quad exp(ia)\,+\,exp(-ia)\,=\,2\,cos\,a}}`$  
   <br>
-  **$`\quad\;\; = 2\;cos(\theta_{12})\; e^{\,i\,\theta_{moy}}`$**   
+  **$`\quad\;\; = 2\;cos\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}`$**   
   <br>
   de même   
   <br>
@@ -1687,7 +1687,7 @@ $`\quad\quad \;\; = \cdots`$
                \right\}\Longrightarrow\\
         \quad\quad exp(ia)\,-\,exp(-ia)\,=\,-2i\,sin\,a}}`$  
   <br> 
-  **$`\quad\;\; = -2i\;sin(\theta_{12})\; e^{\,i\,\theta_{moy}}`$**   
+  **$`\quad\;\; = -2i\;sin\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}`$**   
   <br>
 
 * Nous obtenons l'**expression complexe** de l'onde résultante de la *superposition de deux OPPH* quelconques :   
@@ -1697,25 +1697,25 @@ $`\quad\quad \;\; = \cdots`$
   $`\begin{array}\quad = &A_1\,exp\big[\,i\,\big(\underbrace{\omega_1 t + \overrightarrow{k}_1\cdot\overrightarrow{r}+\varphi_1}_{\color{blue}{\theta_1(\vec{r},t)}}\big)\big]\\
   &+ A_2\,exp\big[\,i\,\big(\underbrace{\omega_2 t + \overrightarrow{k}_2\cdot\overrightarrow{r}+\varphi_2}_{\color{blue}{\theta_2(\vec{r},t)}}\big)\big]\end{array}`$   
   <br>
-  **$`\begin{array}\quad = &+\,2\,A_{moy}\,cos(\theta_{12})\; e^{\,i\,\theta_{moy}}\\
-   &-\,2i\,\Delta A_{1-2}\,sin(\theta_{12})\; e^{\,i\,\theta_{moy}}\end{array}`$**   
+  **$`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}\\
+   &-\,2i\,\Delta A_{1-2}\,sin\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}\end{array}`$**   
   <br>
 
  * L'**onde réelle** est la *partie réelle de l'onde complexe* :    
   <br>
   $`U(\overrightarrow{r},t)\,=\,\mathscr{R}e\big[\underline{U}(\overrightarrow{r},t)\big]`$   
   <br>
-  $`\begin{align}\; = &\mathscr{R}e\big[\,2\,A_{moy}\,cos(\theta_{12})\; e^{\,i\,\theta_{moy}}\\
+  $`\begin{align}\; = \;&\mathscr{R}e\big[\,2\,A_{moy}\,cos(\theta_{12})\; e^{\,i\,\theta_{moy}}\\
    &-\,2i\,\Delta A_{1-2}\,sin(\theta_{12})\; e^{\,i\,\theta_{moy}}\big]\end{align}`$   
   <br>
   $`\color{blue}{\scriptsize{\quad\text{Utilisons }\;exp(ia)\,=\, cos\,a\,+\,i\, sin\,a}}`$  
   <br>   
-  $`\begin{align}\;= &\mathscr{R}e \big[\,2\,A_{moy}\,cos(\theta_{12})\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\\
+  $`\begin{align}\;= \;&\mathscr{R}e \big[\,2\,A_{moy}\,cos(\theta_{12})\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\\
    &-\,2i\,\Delta A_{1-2}\,sin(\theta_{12})\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\big]\end{align}`$  
   <br>
-  $`\begin{align}\; = &2\times\mathscr{R}e \big[\\
-  \,(\,A_{moy}\,cos(\theta_{12})\;cos\,\theta_{moy}\,+\,\Delta A_{1-2}\,sin(\theta_{12}\;sin\,\theta_{moy}\,) \\
-  \,+\,i\,(\,A_{moy}\,cos(\theta_{12})\;sin\,\theta_{moy}\,-\,\Delta A_{1-2}\,sin(\theta_{12}\;sin\,\theta_{moy}\,)\big] \\
+  $`\begin{align}\; = &2\times\mathscr{R}e \big[ \\
+  \,(\,A_{moy}\,cos\,\theta_{12}\;cos\,\theta_{moy}\,+\,\Delta A_{1-2}\,sin\,\theta_{12}\;sin\,\theta_{moy}\,) \\
+  \,+\,i\,(\,A_{moy}\,cos\,\theta_{12}\;sin\,\theta_{moy}\,-\,\Delta A_{1-2}\,sin\,\theta_{12}\;sin\,\theta_{moy}\,)\big] \\
   \end{align}`$