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revision 2647 by gross, Fri Sep 4 05:25:25 2009 UTC revision 2651 by jfenwick, Mon Sep 7 03:39:45 2009 UTC
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4  In this next example we want to calculate the displacement field $u\hackscore{i}$ for any time $t>0$ by solving the wave equation:  In this next example we want to calculate the displacement field $u\hackscore{i}$ for any time $t>0$ by solving the wave equation:
5  \index{wave equation}  \index{wave equation}
6  \begin{eqnarray}\label{WAVE general problem}  \begin{eqnarray}\label{WAVE general problem fault}
7  \rho u\hackscore{i,tt} - \sigma\hackscore{ij,j}=0  \rho u\hackscore{i,tt} - \sigma\hackscore{ij,j}=0
8  \end{eqnarray}  \end{eqnarray}
9  in a three dimensional block of length $L$ in $x\hackscore{0}$  in a three dimensional block of length $L$ in $x\hackscore{0}$
10  and $x\hackscore{1}$ direction and height $H$  and $x\hackscore{1}$ direction and height $H$
11  in $x\hackscore{2}$ direction. $\rho$ is the known density which may be a function of its location.  in $x\hackscore{2}$ direction. $\rho$ is the known density which may be a function of its location.
12  $\sigma\hackscore{ij}$ is the stress field \index{stress} which in case of an isotropic, linear elastic material is given by  $\sigma\hackscore{ij}$ is the stress field \index{stress} which in case of an isotropic, linear elastic material is given by
13  \begin{eqnarray} \label{WAVE stress}  \begin{eqnarray} \label{WAVE stress fault}
14  \sigma\hackscore{ij} & = & \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i})  \sigma\hackscore{ij} & = & \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i})
15  \end{eqnarray}  \end{eqnarray}
16  where $\lambda$ and $\mu$ are the Lame coefficients  where $\lambda$ and $\mu$ are the Lame coefficients
17  \index{Lame coefficients} and $\delta\hackscore{ij}$ denotes the Kronecker symbol\index{Kronecker symbol}.  \index{Lame coefficients} and $\delta\hackscore{ij}$ denotes the Kronecker symbol\index{Kronecker symbol}.
18  On the boundary the normal stress is given by  On the boundary the normal stress is given by
19  \begin{eqnarray} \label{WAVE natural}  \begin{eqnarray} \label{WAVE natural fault}
20  \sigma\hackscore{ij}n\hackscore{j}=0  \sigma\hackscore{ij}n\hackscore{j}=0
21  \end{eqnarray}  \end{eqnarray}
22  for all time $t>0$.  for all time $t>0$.

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