Update cheatsheet.fr.md

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

@@ -556,6 +556,8 @@ $`\quad\boldsymbol{\mathbf{=\color{brown}{2\,A\cdot cos\Big(\dfrac{\varphi_1-\va
 
 ----------------------------
 
+<br>
+
 **Calcul de l'onde résultante** *en notation complexe*
 
 * Une **onde harmonique réelle $`U_1`$** s'écrit comme la *partie réelle de l'onde harmonique complexe $`\underline{U_1}`$*.
@@ -593,6 +595,24 @@ $`\quad\boldsymbol{\mathbf{=\color{brown}{2\,A\cdot cos\Big(\dfrac{\varphi_1-\va
   <br>
   $`\color{brown}{\mathbf{U(x,t)}}\; = U_1(x,t) + U_2(x,t)`$   
 
+$`\begin{align} \quad &=A\;\big[\,e^{\,i\;(\underbrace{(kx - \omega t}_{\color{blue}{\text{  posons }\\ kx - \omega t \,=\, \alpha}} + \varphi_1) + e^{\,i\;(\underbrace{(kx - \omega t}_{\color{blue}{=\; \alpha}} + \varphi_1)\,\big]
+&\\
+&=A\;\big[\,cos\Big(\alpha + \dfrac{\varphi_1+\varphi_1}{2} + \dfrac{\varphi_2-\varphi_2}{2}\Big) \\
+&\quad\quad\quad\quad + \,cos\Big(\alpha + \dfrac{\varphi_2+\varphi_2}{2} + \dfrac{\varphi_1-\varphi_1}{2}\Big)\,\Big]\\
+&\\
+&=A\;\big[\,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\color{blue}{=\;\alpha '}} + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
+&\quad\quad\quad\quad + \,cos\Big(\underbrace{\alpha + \dfrac{\varphi_1+\varphi_2}{2}}_{\color{blue}{\text{nous avons posé }\\ \alpha + (\varphi_1+\varphi_2)/2\; = \;\alpha '}} - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
+&\\
+&=A\;\big[\,cos\Big(\alpha ' + \dfrac{\varphi_1-\varphi_2}{2}\Big) \\
+&\quad\quad\quad\quad + \,cos\Big(\alpha ' - \dfrac{\varphi_1-\varphi_2}{2}\Big)\,\Big]\\
+&\\
+&=A\;\big[\,\underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,-\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1-\varphi_2}{2}}_{\color{blue}{\text{car   }cos(a+b)\;=\;cos\,a\,cos\,b\;-\;sin\,a\,sin\,b}}\big)\\
+&\quad + \underbrace{cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)\,+\,sin(\alpha ')\,sin\Big(\dfrac{\varphi_1-\varphi_2}{2}}_{\color{blue}{\text{car   }cos(a-b)\;=\;cos\,a\,cos\,b\;+\;sin\,a\,sin\,b}}\big)\,\Big]\\
+&\\
+&=2\,A\cdot cos(\alpha ')\,cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big)
+\end{align}`$   
+<br>
+$`\quad\boldsymbol{\mathbf{=\color{brown}{2\,A\cdot cos\Big(\dfrac{\varphi_1-\varphi_2}{2}\Big) \cdot cos\Big(}\color{blue}{\underbrace{\color{brown}{kx - \omega t + \dfrac{\varphi_1+\varphi_2}{2}}}_{\text{pulsation }\omega\text{ inchangée}}}\color{brown}{\Big)}}}`$