\documentclass{article}
\usepackage{amssymb, amsmath, geometry, multicol, cancel}
\usepackage[table, svgnames, dvipsnames]{xcolor}
\usepackage{makecell, cellspace, caption}
\usepackage[europeanresistors]{circuitikz}
\usepackage{empheq}
\usepackage[a]{esvect} % voir la doc pour les différents styles
\let\overrightarrow\vv

\usetikzlibrary{arrows}

\newcommand{\parspace}{\vspace{0.1cm}}
\linespread{1.25}
\geometry{tmargin=.7in, bmargin=.7in, lmargin=.7in, rmargin = .7in}

\begin{document}
\setlength{\parindent}{0mm}

\title{Absorption et Dispersion des Ondes}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\centering

\section*{Câble coaxial avec pertes}

\subsection*{Circuit}
    \begin{circuitikz}
        \draw (-2,0) to [short,] (-2, 0);

        \draw (0,3) to [short,*-] (0.7,3)
        to [R, l_=$r\mathrm{d}x$] (2, 3)

        to [short, i_=${i(x,t)}$] (3.5, 3)
        to [L, l=$\Lambda\mathrm{d}x$] (4.7, 3)
        to [short] (6, 3)
        to [short, -*, i=${i(x+\mathrm{d}x, t)}$] (8, 3);
        \draw (0,0) to [short, *-*,] (8, 0);
        \draw (6,3) to [short] (6, 2.7)
        to [short] (5.5, 2.7)
        to [C, l_=$\Gamma\mathrm{d}x$] (5.5, 0.3)
        to [short] (6, 0.3)
        to [short] (6, 0);
        \draw (6, 2.7) to [short] (6.5, 2.7)
        to [R, l=$g\mathrm{d}x$] (6.5, 0.3)
        to [short] (6, 0.3)
        to [short] (6, 0);
        \draw [->] (0, 0.2) to (0, 1.5) node[right] {$u(x,t)$} to (0, 2.8);
        \draw [->] (8, 0.2) to (8, 1.5) node[right] {$u(x+\mathrm{d}x,t)$} to (8, 2.8);
    \end{circuitikz}
    
\vspace*{0.3cm}

\begin{multicols}{2}
    \subsection*{Equation des télégraphistes}
    Équation vérifié par $u$ dans le circuit ci-dessus /
    \[\boxed{\frac{\partial^2 u}{\partial x^2} - \Lambda\Gamma\frac{\partial^2 u}{\partial t^2} - (g\Lambda + r\Gamma)\frac{\partial u}{\partial t} - gru = 0}\]

    Avec $\displaystyle \boxed{\tau = \frac{\Lambda\Gamma}{\Lambda g + \Gamma r}}, \quad \boxed{\omega_0 = \sqrt{\frac{rg}{\Lambda\Gamma}}} \quad \text{et} \quad \boxed{c_0 = \frac{1}{\sqrt{\Lambda\Gamma}}}$ :
    \[\boxed{\underset{\text{D'Alembert}}{\underbrace{\frac{\partial^2 u}{\partial t^2} - c_0^2\frac{\partial^2 u}{\partial x^2}}}\ \ \underset{\text{Pertes}}{\underbrace{-\frac{1}{\tau}\frac{\partial u}{\partial t} - \omega_0^2u}}} = 0\]
\end{multicols}

\vspace*{0.3cm}

\subsection*{Relation de dispersion du câble coaxial avec pertes}
\[\boxed{k^2 = \frac{\omega^2 - \omega_0^2}{c_0^2} - j\frac{\omega}{\tau c_0^2}}\]

\vspace*{0.3cm}

\begin{multicols}{3}
    \subsection*{Vecteur d'onde}
    \[\boxed{\underline{k}(\omega) = k_r(\omega) + jk_i(\omega)}\]

    \subsection*{Vitesse de phase}
    \[\boxed{v_\varphi \triangleq \frac{\omega}{k_r}}\]

    \subsection*{Dispersion ?}
    Un milieu est \underline{dispersif} si $v_\varphi$ dépend de la fréquence
\end{multicols}

\begin{multicols}{3}
    \subsection*{Amortissement}
    ($k_i < 0$ pour propagation selon $+\overrightarrow{e_x}$)
    \[\boxed{\delta = \frac{1}{|k_i|}}\]

    \subsection*{Absorption?}
    Si $k_i \neq 0$ alors il y a absorption\\
    Plus rarement, il peut y avoir amplification

    \subsection*{Forme complète de l'onde}
    \[\boxed{\begin{array}{ll}
        \psi(x,t) &= \psi_0e^{-\frac{x}{\delta}}\cos(\omega t - k_r x + \varphi)\\
        &= \psi_0e^{-\frac{x}{\delta}}\cos\bigg[\omega\bigg(t - \frac{x}{v_\varphi}\bigg) + \varphi\bigg]
    \end{array}}\]
\end{multicols}

\end{document}