Update cheatsheet.fr.md

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

@@ -1589,7 +1589,7 @@ $`\quad\quad \;\; = \cdots`$
 
 * Nous obtenons l'**expression finale** de l'onde résultante de la *superposition de deux OPPH* quelconques :   
   <br>
-  **$`U(\overrightarrow{r},t)`$**  
+  **$`\mathbf{U(\overrightarrow{r},t)}`$**  
   <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)
@@ -1598,11 +1598,11 @@ $`\quad\quad \;\; = \cdots`$
   $`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{moy}\;cos\,\Delta\theta_{12}\\
    &-\,2\,\Delta A_{1-2}\,\,sin\,\theta_{moy}\;sin\,\Delta\theta_{12}\end{array}`$   
   <br>
-  **$`\begin{array}\quad = &+\,2\,A_{moy}\,cos\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
+  **$`\mathbf{\boldsymbol{\begin{array}\quad = &+\,2\,A_{moy}\,cos\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
     &\quad\times\,cos\big(\Delta \omega_{12} t + \Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}\big)\\
    &-\,2\,\Delta A_{1-2}\,sin\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
     &\quad\times\,sin\big(\Delta \omega_{12} t + \Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}\big)
-    \end{array}`$**
+    \end{array}}}`$**
 
 <br>
 
@@ -1646,28 +1646,41 @@ $`\quad\quad \;\; = \cdots`$
 
 @@@@@@@@@@@@@@@
 
-* *Travaillons* les **termes $`(exp\,i\theta_1\,+\,exp\,i\theta_2)`$ et $`(exp\,i\theta_1\,-\,exp\,i\theta_2)`$** qui interviennent
+* *Travaillons* les **termes $`exp\,i\theta_1\,+\,exp\,i\theta_2\;`$ et $`\;exp\,i\theta_1\,-\,exp\,i\theta_2`$** qui interviennent
   dans l'expression de $`\underline{U}(\vec{r},t)`$ :   
   <br>
   **$`exp\,i\theta_1\,+\,exp\,i\theta_2`$**   
-  $`\quad\;\; =\;exp^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,+\,exp^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
   <br>
-  $`\color{blue}{\scriptsize{exp\,(a+b)\,=\, exp\,a\;+\; exp\,b}}`$  
+  $`\quad\;\; =\;e^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,+\,e^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
   <br>
-  **$`\quad\;\; = 2\,cos\,\theta_{moy}\;cos\,\Delta\theta_{12}`$**   
+  $`\quad\;\; \color{blue}{\scriptsize{exp\,(a+b)\,=\, exp\,a\,+\, exp\,b}}`$  
+  <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>
+  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp\,ia\,=\, cos\,a\,+\,i\, sin\,a\\
+                     &exp\,- ia\,=\, cos\,a\,-\,i\, sin\,a\end{align}
+               \right\}\Longrightarrow\\
+        \quad\quad exp\,ia\,+\,exp\,-ia\,=\,2\,cos\,a}}`$  
+  <br>
+  **$`\quad\;\; = 2\,cos(\theta_{12})\, e^{\,i\,\theta_{moy}}`$**   
   <br>
   de même   
   <br>
-  **$`cos\theta_1-cos\theta_2`$**   
-  $`\quad\;\; = \;cos(\theta_{moy}+\Delta\theta_{12})-cos(\theta_{moy}-\Delta\theta_{12})`$   
+  **$`exp\,i\theta_1\,-\,exp\,i\theta_2`$**   
   <br>
-  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\
-                     &cos(a-b)=cos(a)cos(b)+-sin(a)sin(b)\end{align}
+  $`\quad\;\; =\;e^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,-\,e^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
+  <br>
+  $`\quad\;\; = e^{\,i\,\theta_{moy}}\,\big(e^{\,i\,\theta_{12}}\,-\,e^{-\,i\,\theta_{12}}\big)`$   
+  <br>
+  $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp\,ia\,=\, cos\,a\,+\,i\, sin\,a\\
+                     &exp\,- ia\,=\, cos\,a\,-\,i\, sin\,a\end{align}
                \right\}\Longrightarrow\\
-        \quad\quad cos(a+b)\,-\,cos(a-b)\,=\,-\,2\,sin(a)\,sin(b)}}`$  
+        \quad\quad exp\,ia\,-\,exp\,-ia\,=\,-2i\,cos\,a}}`$  
+  <br> 
+  **$`\quad\;\; = -2i\,sin(\theta_{12})\, e^{\,i\,\theta_{moy}}`$**   
   <br>
-  **$`\quad\;\; = -\,2\,sin\,\theta_{moy}\;sin\,\Delta\theta_{12}`$**
-
 
 * Nous obtenons l'**expression finale** de l'onde résultante de la *superposition de deux OPPH* quelconques :   
   <br>