% $Id: notes.tex,v 1.4 2016/09/22 20:09:37 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
\usepackage{graphicx,natbib}
\usepackage{bm}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input macros
\title{}
\author{}
\date{\today,~ $ $Revision: 1.4 $ $}
\begin{document}
\maketitle

\section{Setup}

A wave train with
\begin{equation}
\ln\rho=ux=(A/k)\,e^{-x^2/2\sigma^2}\sin(kx)
\end{equation}
is initialized with $A=10^{-4}$, $\sigma=0.5$, and $k=10$
in a non-periodic domain $-\pi\leq x\leq\pi$.

For $u_x$ we use an antisymmetric condition, but for
the density there is no boundary condition.
If we put an antisymmetric condition for $\ln\rho$,
we find reflecting waves with almost unchanged amplitude.
If we put a symmetric condition, the amplitude decreases
significantly with each reflection.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmap}
\end{center}\caption[]{
$xt$ diagram of $u_x(x,t)$
for $\nu=10^{-3}$ and nx=1024 meshpoints.
}\label{pmap}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmom}
\end{center}\caption[]{
Linear momentum evolution using 'a'
for $\nu=10^{-3}$ and nx=1024 meshpoints.
}\label{pmom}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmom_s}
\end{center}\caption[]{
Linear momentum evolution using 's'
for $\nu=10^{-3}$ and nx=1024 meshpoints.
}\label{pmom_s}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmom_d1s}
\end{center}\caption[]{
Linear momentum evolution using 'd1s' for $u_z$
and $1s$ for $\rho$
for $\nu=10^{-3}$ and nx=1024 meshpoints.
}\label{pmom_d1s}\end{figure}

\section{Questions}

What is actually the correct condition for the density in this case?

%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

\begin{verbatim}
\end{verbatim}

%\begin{itemize}
%\item
%\end{itemize}

%r e f
%\begin{thebibliography}{}

%\bibitem[Biskamp \& M\"uller(1999)]{BM99}
%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

%\end{thebibliography}

\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/public_html/teach/PencilCode/material/NonlinearSound/WaveTrain/notes.tex,v 1.4 2016/09/22 20:09:37 brandenb Exp $
\end{verbatim}

\end{document}
