Update cheatsheet.fr.md

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

@@ -1602,7 +1602,7 @@ $`\quad\quad \;\; = \cdots`$
   $`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{moy}\;cos\,\Delta\theta_{12}\\
    &-\,2\,\Delta A_{1-2}\,\,sin\,\theta_{moy}\;sin\,\Delta\theta_{12}\end{array}`$   
   <br>
-  **$`\mathbf{\boldsymbol{\begin{array}\quad = &+\,2\,A_{moy}\,cos\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
+  **$`\begin{array}\mathbf{\boldsymbol{\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)
@@ -1652,7 +1652,7 @@ $`\quad\quad \;\; = \cdots`$
 <br>
 **$`\quad\quad\;\;=\,\theta_{moy}(\overrightarrow{r},t)\,-\,\theta_{12}(\overrightarrow{r},t)`$**  
 
-@@@@@@@@@@@@@@@
+
 
 * *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 $`\underline{U}(\vec{r},t)`$ :   
@@ -1678,16 +1678,16 @@ $`\quad\quad \;\; = \cdots`$
   <br>
   **$`exp\,i\theta_1\,-\,exp\,i\theta_2`$**   
   <br>
-  $`\quad\quad =\;e^{\,i\,(\theta_{moy}+\Delta\theta_{12})}\,-\,e^{\,i\,(\theta_{moy}-\Delta\theta_{12})}`$   
+  $`\quad\;\; =\;e^{\,i\,(\theta_{moy}+\theta_{12})}\,-\,e^{\,i\,(\theta_{moy}-\theta_{12})}`$   
   <br>
   $`\quad\;\; = e^{\,i\,\theta_{moy}}\,\big(e^{\,i\,\theta_{12}}\,-\,e^{-\,i\,\theta_{12}}\big)`$   
   <br>
   $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &exp(ia)\,=\, cos\,a\,+\,i\, sin\,a\\
                      &exp(-ia)\,=\, cos\,a\,-\,i\, sin\,a\end{align}
                \right\}\Longrightarrow\\
-        \quad\quad exp(ia)\,-\,exp(-ia)\,=\,-2i\,sin\,a}}`$  
+        \quad\quad exp(ia)\,-\,exp(-ia)\,=\,+ 2i\,sin\,a}}`$  
   <br> 
-  **$`\quad\;\; = -2i\;sin\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}`$**   
+  **$`\quad\;\; = 2i\;sin\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}`$**   
   <br>
 
 * Nous obtenons l'**expression complexe** de l'onde résultante de la *superposition de deux OPPH* quelconques :   
@@ -1698,40 +1698,37 @@ $`\quad\quad \;\; = \cdots`$
   &+ 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>
   **$`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}\\
-   &-\,2i\,\Delta A_{1-2}\,sin\,\theta_{12}\cdot e^{\,i\,\theta_{moy}}\end{array}`$**   
+   &+\,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>
   **$`\mathbf{U(\overrightarrow{r},t)\,=\,\mathscr{R}e\big[\underline{U}(\overrightarrow{r},t)\big]}`$**   
   <br>
-  $`\begin{align}\; = \;&\mathscr{R}e\big[\,2\,A_{moy}\,cos(\theta_{12})\; e^{\,i\,\theta_{moy}}\\
-   &-\,2i\,\Delta A_{1-2}\,sin(\theta_{12})\; e^{\,i\,\theta_{moy}}\big]\end{align}`$   
+  $`\begin{align}\; = \;&\mathscr{R}e\big[\,2\,A_{moy}\,cos\,\theta_{12}\; e^{\,i\,\theta_{moy}}\\
+   &+\,2i\,\Delta A_{1-2}\,sin\,\theta_{12}\; e^{\,i\,\theta_{moy}}\big]\end{align}`$   
   <br>
   $`\color{blue}{\scriptsize{\quad\text{Utilisons }\;exp(ia)\,=\, cos\,a\,+\,i\, sin\,a}}`$  
   <br>   
-  $`\begin{align}\;= \;&\mathscr{R}e \big[\,2\,A_{moy}\,cos(\theta_{12})\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\\
-   &-\,2i\,\Delta A_{1-2}\,sin(\theta_{12})\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\big]\end{align}`$  
+  $`\begin{align}\;= \;&\mathscr{R}e \big[\,2\,A_{moy}\,cos\,\theta_{12}\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\\
+   &+\,2i\,\Delta A_{1-2}\,sin\,\theta_{12}\;(cos\,\theta_{moy}\,+\,i\,sin\,\theta_{moy})\big]\end{align}`$  
   <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] \\
+  $`\; \begin{align}= & 2\times\\
+  &\mathscr{R}e \big[(\,A_{moy}\,cos\,\theta_{12}\;cos\,\theta_{moy}\,-\,\Delta A_{1-2}\,sin\,\theta_{12}\;sin\,\theta_{moy}\,) \\
+  &\,+\,i\,(\,A_{moy}\,cos\,\theta_{12}\;sin\,\theta_{moy}\,+\,\Delta A_{1-2}\,sin\,\theta_{12}\;cos\,\theta_{moy}\,)\big] \\
   \end{align}`$    
 
-
-
-
-  **$`\begin{array}\quad = &+\,2\,A_{moy}\,cos\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
+  **$`\begin{array}\mathbf{\boldsymbol{\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>
 
-@@@@@@@@@@@@@@@
-
+---------------------
 
+<br>
 
 #### Retour sur quelques cas particuliers