Update cheatsheet.fr.md

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

@@ -1499,21 +1499,21 @@ deux célérités différentes pour $`{\omega_1}`$ et $`{\omega_1}`$. Ainsi le c
 
 * Pour chaque grandeur physique, la façon de *gérer ces valeurs indépendantes* est de les **réexprimer** en fonction de ce qu'elles ont en commun,
 une **valeur moyenne**, et de leurs **écarts respectifs** par rapport à la valeur moyenne. Ainsi :   
-   * **$`A_1 =`$**$`\; \dfrac{A_1 + A_2}{2} +  \dfrac{A_1 - A_2}{2} = `$ *$`\; A_{moy} + \Delta A_{1-2}`$*    
+   * **$`A_1 =`$**$`\; \dfrac{A_1 + A_2}{2} +  \dfrac{A_1 - A_2}{2} = `$ *$`\; A_{moy} + \Delta A_{12}`$*    
      <br>
-     **$`A_2 =`$**$`\; \dfrac{A_1 + A_2}{2} -  \dfrac{A_1 - A_2}{2} = `$ *$`\; A_{moy} - \Delta A_{1-2}`$*   
+     **$`A_2 =`$**$`\; \dfrac{A_1 + A_2}{2} -  \dfrac{A_1 - A_2}{2} = `$ *$`\; A_{moy} - \Delta A_{12}`$*   
 
-   * **$`\omega_1 =`$**$`\; \dfrac{\omega_1 + \omega_2}{2} +  \dfrac{\omega_1 - \omega_2}{2} = `$ *$`\; \omega_{moy} + \Delta \omega_{1-2}`$*   
+   * **$`\omega_1 =`$**$`\; \dfrac{\omega_1 + \omega_2}{2} +  \dfrac{\omega_1 - \omega_2}{2} = `$ *$`\; \omega_{moy} + \Delta \omega_{12}`$*   
      <br>
-     **$`\omega_2 =`$**$`\; \dfrac{\omega_1 + \omega_2}{2} -  \dfrac{\omega_1 - \omega_2}{2} = `$ *$`\; \omega_{moy} - \Delta \omega_{1-2}`$*  
+     **$`\omega_2 =`$**$`\; \dfrac{\omega_1 + \omega_2}{2} -  \dfrac{\omega_1 - \omega_2}{2} = `$ *$`\; \omega_{moy} - \Delta \omega_{12}`$*  
 
-   * **$`\overrightarrow{k}_1 =`$**$`\; \dfrac{\overrightarrow{k}_1 + \overrightarrow{k}_2}{2} +  \dfrac{\overrightarrow{k}_1 - \overrightarrow{k}_2}{2} = `$ *$`\; \overrightarrow{k}_{moy} + \Delta \overrightarrow{k}_{1-2}`$*    
+   * **$`\overrightarrow{k}_1 =`$**$`\; \dfrac{\overrightarrow{k}_1 + \overrightarrow{k}_2}{2} +  \dfrac{\overrightarrow{k}_1 - \overrightarrow{k}_2}{2} = `$ *$`\; \overrightarrow{k}_{moy} + \Delta \overrightarrow{k}_{12}`$*    
      <br>
-     **$`\overrightarrow{k}_2 =`$**$`\; \dfrac{\overrightarrow{k}_1 + \overrightarrow{k}_2}{2} -  \dfrac{\overrightarrow{k}_1 - \overrightarrow{k}_2}{2} = `$ *$`\; \overrightarrow{k}_{moy} - \Delta \overrightarrow{k}_{1-2}`$*   
+     **$`\overrightarrow{k}_2 =`$**$`\; \dfrac{\overrightarrow{k}_1 + \overrightarrow{k}_2}{2} -  \dfrac{\overrightarrow{k}_1 - \overrightarrow{k}_2}{2} = `$ *$`\; \overrightarrow{k}_{moy} - \Delta \overrightarrow{k}_{12}`$*   
 
-   * **$`\varphi_1 =`$**$`\; \dfrac{\varphi_1 + \varphi_2}{2} +  \dfrac{\varphi_1 - \varphi_2}{2} = `$ *$`\; \varphi_{moy} + \Delta \varphi_{1-2}`$*    
+   * **$`\varphi_1 =`$**$`\; \dfrac{\varphi_1 + \varphi_2}{2} +  \dfrac{\varphi_1 - \varphi_2}{2} = `$ *$`\; \varphi_{moy} + \Delta \varphi_{12}`$*    
      <br>
-     **$`\varphi_2 =`$**$`\; \dfrac{\varphi_1 + \varphi_2}{2} -  \dfrac{\varphi_1 - \varphi_2}{2} = `$ *$`\; \varphi_{moy} - \Delta \varphi_{1-2}`$*    
+     **$`\varphi_2 =`$**$`\; \dfrac{\varphi_1 + \varphi_2}{2} -  \dfrac{\varphi_1 - \varphi_2}{2} = `$ *$`\; \varphi_{moy} - \Delta \varphi_{12}`$*    
 
 * *Travaillons* d'abord avec les **termes d'amplitude** :   
 <br>
@@ -1524,23 +1524,23 @@ $`\begin{array}\quad = &A_1\,cos\big(\underbrace{\omega_1 t + \overrightarrow{k}
 
 $`\quad\;\, = A_1\,cos\,\theta_1 + A_2\,cos\,\theta_2`$   
 
-$`\quad\;\, = (A_{moy}+ \Delta A_{1-2})\,cos\,\theta_1 + (A_{moy} - \Delta A_{1-2})\,cos\,\theta_2`$   
+$`\quad\;\, = (A_{moy}+ \Delta A_{12})\,cos\,\theta_1 + (A_{moy} - \Delta A_{12})\,cos\,\theta_2`$   
 
-**$`\quad\;\, = A_{moy}\,(cos\,\theta_1 + cos\,\theta_2) + \Delta A_{1-2}\,(cos\theta_1 - cos\theta_2)`$**
+**$`\quad\;\, = A_{moy}\,(cos\,\theta_1 + cos\,\theta_2) + \Delta A_{12}\,(cos\theta_1 - cos\theta_2)`$**
 
 * *Travaillons* les **termes de phase** :   
 <br>
 **$`\theta_1(\overrightarrow{r},t)`$**$`\; = \omega_1 t + \overrightarrow{k}_1\cdot\overrightarrow{r}+\varphi_1`$
 
-$`\begin{align}\quad\;\;= \big[ (\omega_{moy}&+ \Delta \omega_{1-2})\, t + \big(\overrightarrow{k}_{moy}+ \Delta \overrightarrow{k}_{1-2}\big)\cdot\overrightarrow{r}\\
-&+( \varphi_{moy} + \Delta \varphi_{1-2})\big]\end{align}`$
+$`\begin{align}\quad\;\;= \big[ (\omega_{moy}&+ \Delta \omega_{12})\, t + \big(\overrightarrow{k}_{moy}+ \Delta \overrightarrow{k}_{12}\big)\cdot\overrightarrow{r}\\
+&+( \varphi_{moy} + \Delta \varphi_{12})\big]\end{align}`$
 
 $`\begin{align}
 \quad\;\,&= \big[ \big(\underbrace{\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}}_{\color{blue}{\theta_{moy}(\overrightarrow{r},t)}}\big) \\
-&\quad + \big(\underbrace{\Delta \omega_{1-2} t + \Delta \overrightarrow{k}_{1-2}\cdot\overrightarrow{r}+\Delta\varphi_{1-2}}_{\color{blue}{\theta_{1-2}(\overrightarrow{r},t)}}\big)\big]
+&\quad + \big(\underbrace{\Delta \omega_{12} t + \Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}}_{\color{blue}{\theta_{12}(\overrightarrow{r},t)}}\big)\big]
 \end{align}`$
 
-**$`\quad\;\;=\,\theta_{moy}(\overrightarrow{r},t)\,+\,\theta_{1-2}(\overrightarrow{r},t)`$**   
+**$`\quad\;\;=\,\theta_{moy}(\overrightarrow{r},t)\,+\,\theta_{12}(\overrightarrow{r},t)`$**   
 <br>
 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;et de la même façon nous obtenons :   
 <br>
@@ -1548,31 +1548,31 @@ $`\begin{align}
 <br>
 $`\quad\quad \;\; = \cdots`$   
 <br>
-**$`\quad\quad\;\;=\,\theta_{moy}(\overrightarrow{r},t)\,-\,\theta_{1-2}(\overrightarrow{r},t)`$**  
+**$`\quad\quad\;\;=\,\theta_{moy}(\overrightarrow{r},t)\,-\,\theta_{12}(\overrightarrow{r},t)`$**  
 
 
 * *Travaillons* les **termes $`(cos\theta_1+cos\theta_2)`$ et $`(cos\theta_1-cos\theta_2)`$** qui interviennent
   dans l'expression de $`U(\vec{r},t)`$ :   
   <br>
-  **$`cos\theta_1+cos\theta_2 =`$**$`\;cos\,(\theta_{moy}+\Delta\theta_{1-2})+cos\,(\theta_{moy}-\Delta\theta_{1-2}`$   
+  **$`cos\theta_1+cos\theta_2 =`$**$`\;cos(\theta_{moy}+\Delta\theta_{12})+cos(\theta_{moy}-\Delta\theta_{12})`$   
   <br>
   $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\
                      &cos(a-b)=cos(a)cos(b)+-sin(a)sin(b)\end{align}
                \right\}\Longrightarrow\\
         \quad\quad cos(a+b)\,+\,cos(a-b)\,=\,2\,cos(a)\,cos(b)}}`$  
   <br>
-  **$`\quad\;\; = 2\,cos\,\theta_{moy}\;cos\,\Delta\theta_{1-2}`$**   
+  **$`\quad\;\; = 2\,cos\,\theta_{moy}\;cos\,\Delta\theta_{12}`$**   
   <br>
   de même   
   <br>
-  **$`cos\theta_1-cos\theta_2 =`$**$`\;cos\,(\theta_{moy}+\Delta\theta_{1-2})-cos\,(\theta_{moy}-\Delta\theta_{1-2}`$   
+  **$`cos\theta_1-cos\theta_2 =`$**$`\;cos(\theta_{moy}+\Delta\theta_{12})-cos(\theta_{moy}-\Delta\theta_{12})`$   
   <br>
   $`\color{blue}{\scriptsize{\left.\begin{align} \quad\quad &cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\
                      &cos(a-b)=cos(a)cos(b)+-sin(a)sin(b)\end{align}
                \right\}\Longrightarrow\\
         \quad\quad cos(a+b)\,-\,cos(a-b)\,=\,-\,2\,sin(a)\,sin(b)}}`$  
   <br>
-  **$`\quad\;\; = -\,2\,sin\,\theta_{moy}\;sin\,\Delta\theta_{1-2}`$**
+  **$`\quad\;\; = -\,2\,sin\,\theta_{moy}\;sin\,\Delta\theta_{12}`$**
 
 
 * Nous obtenons l'**expression finale** de l'onde résultante de la *superposition de deux OPPH* quelconques :   
@@ -1583,13 +1583,13 @@ $`\quad\quad \;\; = \cdots`$
   &+ \;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}`$*   
   <br>
-  $`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{moy}\;cos\,\Delta\theta_{1-2}\\
-   &-\,2\,\Delta A_{1-2}\,\,sin\,\theta_{moy}\;sin\,\Delta\theta_{1-2}\end{array}`$   
+  $`\begin{array}\quad = &+\,2\,A_{moy}\,cos\,\theta_{moy}\;cos\,\Delta\theta_{12}\\
+   &-\,2\,\Delta A_{1-2}\,\,sin\,\theta_{moy}\;sin\,\Delta\theta_{12}\end{array}`$   
   <br>
   **$`\begin{array}\quad = &+\,2\,A_{moy}\,cos\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
-    &\quad\times\,cos\big(\Delta \omega_{1-2} t + \Delta \overrightarrow{k}_{1-2}\cdot\overrightarrow{r}+\Delta\varphi_{1-2}\big)\\
+    &\quad\times\,cos\big(\Delta \omega_{12} t + \Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}\big)\\
    &-\,2\,\Delta A_{1-2}\,sin\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
-    &\quad\times\,sin\big(\Delta \omega_{1-2} t + \Delta \overrightarrow{k}_{1-2}\cdot\overrightarrow{r}+\Delta\varphi_{1-2}\big)
+    &\quad\times\,sin\big(\Delta \omega_{12} t + \Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}\big)
     \end{array}`$**
 
 
@@ -1628,12 +1628,12 @@ $`\quad\quad \;\; = \cdots`$
    <br>
   **$`\mathbf{\boldsymbol{U(\overrightarrow{r},t)}}`$**  
   $`\begin{array}\quad = &+\,2\,\color{blue}{\underbrace{A_{moy}}_{=\;A}}\,cos\big(\omega_{moy} t + \color{blue}{\underbrace{\overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}}_{=\;\varphi_A}} \big)\\
-    &\quad\times\,cos\big(\Delta \omega_{1-2} t + \color{blue}{\underbrace{\Delta \overrightarrow{k}_{1-2}\cdot\overrightarrow{r}+\Delta\varphi_{1-2}}_{=\;\varphi_B}}\big)\\
-   &-\,2\,\color{blue}{\underbrace{\Delta A_{1-2}}_{=\;0}}\,sin\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
-    &\quad\times\,sin\big(\Delta \omega_{1-2} t + \Delta \overrightarrow{k}_{1-2}\cdot\overrightarrow{r}+\Delta\varphi_{1-2}\big)
+    &\quad\times\,cos\big(\Delta \omega_{12} t + \color{blue}{\underbrace{\Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}}_{=\;\varphi_B}}\big)\\
+   &-\,2\,\color{blue}{\underbrace{\Delta A_{12}}_{=\;0}}\,sin\big(\omega_{moy} t + \overrightarrow{k}_{moy}\cdot\overrightarrow{r}+ \varphi_{moy}\big)\\
+    &\quad\times\,sin\big(\Delta \omega_{12} t + \Delta \overrightarrow{k}_{12}\cdot\overrightarrow{r}+\Delta\varphi_{12}\big)
     \end{array}`$    
    <br>
-   **$`\begin{array}\quad = \underbrace{2\,A\,\,cos\big(\Delta \omega_{1-2} t + \varphi_B)}_{\color{blue}{\text{lentement variable} \\ \text{fonction enveloppe}}} \times cos\,(\omega_{moy} t + \varphi_A)
+   **$`\begin{array}\quad = \underbrace{2\,A\,\,cos\big(\Delta \omega_{12} t + \varphi_B)}_{\color{blue}{\text{lentement variable} \\ \text{fonction enveloppe}}} \times cos\,(\omega_{moy} t + \varphi_A)
    \end{array}`$**