% beamer version of chapter 1 "Introduction" in "Computational Seismology, A Practical Introduction" (H. Igel)

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\usepackage{multimedia}

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\title[Computational Seismology] % (optional, use only with long paper titles)
{Computational Seismology: Wave equations}

%\subtitle{Introduction} 

\author[Heiner Igel]{Heiner Igel}


\institute[] 
  {Department of Earth and Environmental Sciences \\ Ludwig-Maximilians-University Munich} 



\date[] % (optional)
{}

\subject{Talks}


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\begin{document}

% ----------------------------------------------------------

\begin{frame}
  \titlepage
\end{frame}



% ---------------------------------------------------------

	\begin{frame}{Goals of this lecture}
	\footnotesize
		\begin{itemize}
			\item Presenting the basic \textbf{wave equations} used in this course
			\item Demonstrating the differences between \textbf{scalar} and \textbf{vectorial} wave equations
			\item Showing \textbf{first order} and \textbf{second order} wave equations
			\item Presenting \textbf{analytical solutions} for acoustic and elastic wave equations
			\item Introducing some \textbf{fundamental solutions} of elastic waves in homogeneous media
		\end{itemize}
	\end{frame}



% ---------------------------------------------------------

\section{Scalar wave equation: analytical solutions}


	\begin{frame}{Scalar wave equation}
	
	\footnotesize
		\begin{minipage}{0.49\textwidth}
		
		
\begin{block}{Acoustic wave equation}
\begin{center}
$$
\partial^2_t p \ = \ c^2 \Delta p  + s 
$$
\end{center}
\end{block}

\hskip2truecm $p \rightarrow p(\textbf{x},t) $, pressure

\hskip2truecm $c \rightarrow c(\textbf{x}) $, velocity

\hskip2truecm $s \rightarrow s(\textbf{x},t) $, source term

\begin{block}{Initial conditions}

$$
\hskip1truecm p(\textbf{x},t=0) \ = \ p_0(\textbf{x},t)
$$
$$
\hskip.7truecm \partial_t p(\textbf{x},t=0) \ = \ 0
$$

\end{block}
		\end{minipage}
	\mbox{}		
	\hfill	
		\begin{minipage}{0.49\textwidth}
			\includegraphics[width=0.8\textwidth]{\figpath/fig_waves_init.png}
			
			Snapshot of $p(\textbf{x},t)$ (solid line) after some time for initial condition $p_0(\textbf{x},t)$ (Gaussian, dashed line), 1D case.
		\end{minipage}
		
	\end{frame}
	
% -----------------------------------------------------------------------

	\begin{frame}{Analytical solution: source-free case}
	
\begin{minipage}{0.49\textwidth}	
\begin{center}

\vskip1truecm
\begin{block}{Solution}
\begin{center}
$$
p(x,t) \ = \ \frac{1}{2} p_0(ct-x)\ + \frac{1}{2} p_0(ct+x)
$$
\end{center}
\end{block}
\end{center}
	\end{minipage}
		
	\mbox{}
	\hfill
		\begin{minipage}{0.49\textwidth}	
		\vskip-4truecm
\begin{center}
\movie[height = 0.6 \textwidth,width = .8 \textwidth,poster,loop]{}{\figpath/movie.avi}
\end{center}
	\end{minipage}
	
	The solution shown in the movie has been obtained with a finite-difference approximation.
	
	\end{frame}		
	
% ---------------------------------------------------------


	\begin{frame}{Acoustic wave equation: external source}
	
	\footnotesize
		\begin{minipage}{0.65\textwidth}
		
		
	
					\begin{block}{Green's Function G}
					\begin{center}
$$
\partial^2_t G(\textbf{x},t;\textbf{x}_0,t_0) - c^2 \Delta G(\textbf{x},t;\textbf{x}_0,t_0) = \delta(\textbf{x}-\textbf{x}_0)\delta(t-t_0)
$$
\end{center}
\end{block}
\begin{block}{Delta function $\delta$}
					\begin{center}
$$
\delta(x) = \left\{
\begin{array}{ll}
	\infty &x=0 \\
	0 &x\neq 0 	
\end{array}
\right.
$$

$$
\int_{-\infty}^{\infty}\delta(x)dx\ = \ 1 \ , \   \int_{-\infty}^{\infty}f(x)\delta(x)dx\ = \ f(0) 
$$
\end{center}
\end{block}



		\end{minipage}
	\mbox{}		
	\hfill	
		\begin{minipage}{0.33\textwidth}
		\begin{center}
			\includegraphics[width=0.7\textwidth]{\figpath/fig_waves_boxcar.png}
			
			$\delta$-generating function using boxcars.
			\end{center}
		\end{minipage}
		
		
		
	\end{frame}

% ---------------------------------------------------------

\begin{frame}{Acoustic wave equation: analytical solutions}
	
	
Green's functions for the inhomogeneous acoustic wave equation for all dimensions. $H(t)$ is the Heaviside function. 
	
\begin{table}[h]

\normalsize
\begin{tabular}{ccc}
\\
1D & 2D & 3D \\
\hline
$\frac{1}{2c}H(t-\frac{|r|}{c})$ &  $\frac{1}{2\pi c^2}\frac{H(t-\frac{|r|}{c})}{\sqrt{t^2-\frac{r^2}{c^2}}}$ & $\frac{1}{4 \pi c^2 r}\delta(t-r/c)$ \\
\\
$r=x$ & $r = \sqrt{x^2+y^2}$ & $r = \sqrt{x^2+y^2+z^2}$ \\
\hline
\end{tabular}

\end{table}
		
		
\end{frame}

% ---------------------------------------------------------

\begin{frame}{Acoustic wave equation: analytical solutions}
	
	
\begin{center}
			\includegraphics[width=0.75\textwidth]{\figpath/fig_waves_green_ana.png}
			\end{center}
		
\end{frame}

% ---------------------------------------------------------


\end{document}
