| 3 |
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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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