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revision 624 by gross, Fri Mar 17 05:48:59 2006 UTC revision 625 by gross, Thu Mar 23 00:41:25 2006 UTC
# Line 1  Line 1 
1  % $Id$  % $Id$
2    %
3    %           Copyright © 2006 by ACcESS MNRF
4    %               \url{http://www.access.edu.au
5    %         Primary Business: Queensland, Australia.
6    %   Licensed under the Open Software License version 3.0
7    %      http://www.opensource.org/license/osl-3.0.php
8    %
9  \section{3D Wave Propagation}  \section{3D Wave Propagation}
10  \label{WAVE CHAP}  \label{WAVE CHAP}
11    
# Line 63  on the solution at the previous two time Line 70  on the solution at the previous two time
70    
71  In each time step we have to solve this problem to get the acceleration $a^{(n)}$ and we will  In each time step we have to solve this problem to get the acceleration $a^{(n)}$ and we will
72  use the \LinearPDE class of the \linearPDEs to do so. The general form of the PDE defined through  use the \LinearPDE class of the \linearPDEs to do so. The general form of the PDE defined through
73  the \LinearPDE class is discussed in \Sec{SEC LinearPDE}. The forms which are relevant here are    the \LinearPDE class is discussed in \Sec{SEC LinearPDE}. The form which is relevant here are  
74  \begin{equation}\label{WAVE dyn 100}  \begin{equation}\label{WAVE dyn 100}
75  D\hackscore{ij} a^{(n)}\hackscore{j} = - X\hackscore{ij,j}\; .  D\hackscore{ij} a^{(n)}\hackscore{j} = - X\hackscore{ij,j}\; .
76  \end{equation}  \end{equation}
# Line 71  Implicitly, the natural boundary conditi Line 78  Implicitly, the natural boundary conditi
78  \begin{equation}\label{WAVE dyn 101}  \begin{equation}\label{WAVE dyn 101}
79  n\hackscore{j}X\hackscore{ij} =0  n\hackscore{j}X\hackscore{ij} =0
80  \end{equation}  \end{equation}
81  is assumed. The \LinearPDE object allows defining constraints of the form  is assumed.
 \begin{equation}\label{WAVE dyn 102}  
 u\hackscore{i}(x)=r\hackscore{i}(x) \mbox{ for } x \mbox{ with } q\hackscore{i}>0  
 \end{equation}  
 where $r$ and $q$ are given functions. However in this problem we don't need to define  
 any constraints for our problem.  
82    
83  With $u=a^{(n)}$ we can identify the values to be assigned to $D$, $X$:  With $u=a^{(n)}$ we can identify the values to be assigned to $D$, $X$:
84  \begin{equation}\label{WAVE  dyn 6}  \begin{equation}\label{WAVE  dyn 6}

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