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% If you are using CVS use this line to give version information
% $Header$
\documentclass{article}
% Use the Cactus ThornGuide style file
% (Automatically used from Cactus distribution, if you have a
% thorn without the Cactus Flesh download this from the Cactus
% homepage at www.cactuscode.org)
\usepackage{../../../../doc/latex/cactus}
\begin{document}
\title{IDScalarWave}
\author{Gabrielle Allen \\ Horst Beyer}
\date{$ $Date$ $}
\maketitle
% Do not delete next line
% START CACTUS THORNGUIDE
\begin{abstract}
Initial Data for the 3D Scalar Wave Equation
\end{abstract}
\section{Purpose}
This thorn provides different initial data for the 3D scalar wave
equation.
\section{Spherically Symmetric Solutions}
The general spherically symmetric solution can be written
\begin{equation}
\Psi(r,t) = \frac{1}{r}\left(f(r+t)+g(r-t)\right)
\end{equation}
where the functions $f$ and $g$ can be freely chosen.
Making the additional requirement of time symmetry at $t=0$, forces
\begin{equation}
f(r)=g(r)
\end{equation}
Thus if the solution at t=0 is given by $\phi(r)$, the general solution
will be
\begin{equation}
\Psi(r,t) = \frac{1}{2r}\left( (r+t)\phi(r+t)+(r-t)\phi(r-t) \right)
\end{equation}
\section{Gaussian}
The gaussian solution is {\it spherically symmetric} about the
origin of the Cartesian coordinate system, and is {\it time symmetric}.
The initial profile is
\begin{equation}
\phi(r) = A \exp (- r^2/\sigma)
\end{equation}
with the solution at the origin being
\begin{equation}
\Psi(r=0,t) = \left(1-2\frac{t^2}{\sigma}\right)\exp(-t^2/\sigma)
\end{equation}
The Gaussian solution is set with the parameters
\begin{itemize}
\item {\tt amplitude} = $A$
\item {\tt sigma} = $\sigma$
\end{itemize}
% Do not delete next line
% END CACTUS THORNGUIDE
\end{document}
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