Update cheatsheet.fr.md

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

@@ -1653,43 +1653,46 @@ $`\quad\quad \;\; = \cdots`$
   <br>
   $`\quad\;\; =\;e^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,+\,e^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
   <br>
-  $`\quad\;\; \color{blue}{\scriptsize{exp\,(a+b)\,=\, exp\,a\,+\, exp\,b}}`$  
+  $`\quad\quad \color{blue}{\scriptsize{exp\,(a+b)\,=\, exp\,a\,+\, exp\,b}}`$  
   <br>
   $`\quad\;\; = e^{\,i\,\theta_{moy}}\,e^{\,i\,\theta_{12}}\,+\,e^{\,i\,\theta_{moy}}\,e^{-\,i\,\theta_{12}}`$   
   <br>
   $`\quad\;\; = e^{\,i\,\theta_{moy}}\,\big(e^{\,i\,\theta_{12}}\,+\,e^{-\,i\,\theta_{12}}\big)`$   
   <br>
-  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp\,ia\,=\, cos\,a\,+\,i\, sin\,a\\
-                     &exp\,- ia\,=\, cos\,a\,-\,i\, sin\,a\end{align}
+  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp(ia)\,=\, cos\,a\,+\,i\, sin\,a\\
+                     &exp(- ia)\,=\, cos\,a\,-\,i\, sin\,a\end{align}
                \right\}\Longrightarrow\\
-        \quad\quad exp\,ia\,+\,exp\,-ia\,=\,2\,cos\,a}}`$  
+        \quad\quad exp(ia)\,+\,exp(-ia)\,=\,2\,cos\,a}}`$  
   <br>
-  **$`\quad\;\; = 2\,cos(\theta_{12})\, e^{\,i\,\theta_{moy}}`$**   
+  **$`\quad\;\; = 2\;cos(\theta_{12})\; e^{\,i\,\theta_{moy}}`$**   
   <br>
   de même   
   <br>
   **$`exp\,i\theta_1\,-\,exp\,i\theta_2`$**   
   <br>
-  $`\quad\;\; =\;e^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,-\,e^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
+  $`\quad\quad =\;e^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,-\,e^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
   <br>
   $`\quad\;\; = e^{\,i\,\theta_{moy}}\,\big(e^{\,i\,\theta_{12}}\,-\,e^{-\,i\,\theta_{12}}\big)`$   
   <br>
-  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp\,ia\,=\, cos\,a\,+\,i\, sin\,a\\
-                     &exp\,- ia\,=\, cos\,a\,-\,i\, sin\,a\end{align}
+  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp(ia)\,=\, cos\,a\,+\,i\, sin\,a\\
+                     &exp(-ia)\,=\, cos\,a\,-\,i\, sin\,a\end{align}
                \right\}\Longrightarrow\\
-        \quad\quad exp\,ia\,-\,exp\,-ia\,=\,-2i\,cos\,a}}`$  
+        \quad\quad exp(ia)\,-\,exp(-ia)\,=\,-2i\,sin\,a}}`$  
   <br> 
-  **$`\quad\;\; = -2i\,sin(\theta_{12})\, e^{\,i\,\theta_{moy}}`$**   
+  **$`\quad\;\; = -2i\;sin(\theta_{12})\; e^{\,i\,\theta_{moy}}`$**   
   <br>
 
 * Nous obtenons l'**expression finale** de l'onde résultante de la *superposition de deux OPPH* quelconques :   
   <br>
-  **$`U(\overrightarrow{r},t)`$**  
+  **$`\underline{U}(\overrightarrow{r},t)`$**  
   <br>
-  *$`\begin{array}\quad = &A_1\,cos\big(\underbrace{\omega_1 t + \overrightarrow{k}_1\cdot\overrightarrow{r}+\varphi_1}_{\color{blue}{\theta_1(\vec{r},t)}}\big)\\
-  &+ \;A_2\,cos\big(\underbrace{\omega_2 t + \overrightarrow{k}_2\cdot\overrightarrow{r}+\varphi_2}_{\color{blue}{\theta_2(\vec{r},t)}}\big)
-  \end{array}`$*   
+$`\begin{array}\quad = &A_1\,exp\big[\,i\,\big(\underbrace{\omega_1 t + \overrightarrow{k}_1\cdot\overrightarrow{r}+\varphi_1}_{\color{blue}{\theta_1(\vec{r},t)}}\big)\big]\\
+  &+ A_2\,exp\big[\,i\,\big(\underbrace{\omega_2 t + \overrightarrow{k}_2\cdot\overrightarrow{r}+\varphi_2}_{\color{blue}{\theta_2(\vec{r},t)}}\big)\big]\end{array}`$   
   <br>
+
+
+
+
   $`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{moy}\;cos\,\Delta\theta_{12}\\
    &-\,2\,\Delta A_{1-2}\,\,sin\,\theta_{moy}\;sin\,\Delta\theta_{12}\end{array}`$   
   <br>