% miko:type: atomic
% miko:title: Wave Equation
% miko:summary: The differential equation describing fields that propagate at fixed speed.
% miko:tags: em
% miko:difficulty: undergrad
% miko:related: maxwell-equations, faraday-law, schrodinger-equation
% miko:keywords: propagation, field, electromagnetic wave

\documentclass{article}
\usepackage{amsmath}
\title{Wave Equation}
\begin{document}
\maketitle

\section{Form}

For a scalar field $\psi(x,t)$,

$$
\frac{\partial^2 \psi}{\partial t^2} = v^2\nabla^2\psi.
$$

\section{Electromagnetic Waves}

In vacuum, Maxwell's equations imply

$$
\nabla^2\mathbf E-\mu_0\varepsilon_0\frac{\partial^2\mathbf E}{\partial t^2}=0.
$$

Thus light propagates at $c=1/\sqrt{\mu_0\varepsilon_0}$.

\end{document}
