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\section{Slip on a Fault} 
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\label{Slip CHAP} 
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In this next example we want to calculate the displacement field $u\hackscore{i}$ for any time $t>0$ by solving the wave equation: 
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\index{wave equation} 
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\begin{eqnarray}\label{WAVE general problem fault} 
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\rho u\hackscore{i,tt}  \sigma\hackscore{ij,j}=0 
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\end{eqnarray} 
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in a three dimensional block of length $L$ in $x\hackscore{0}$ 
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and $x\hackscore{1}$ direction and height $H$ 
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in $x\hackscore{2}$ direction. $\rho$ is the known density which may be a function of its location. 
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$\sigma\hackscore{ij}$ is the stress field \index{stress} which in case of an isotropic, linear elastic material is given by 
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\begin{eqnarray} \label{WAVE stress fault} 
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\sigma\hackscore{ij} & = & \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i}) 
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\end{eqnarray} 
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where $\lambda$ and $\mu$ are the Lame coefficients 
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\index{Lame coefficients} and $\delta\hackscore{ij}$ denotes the Kronecker symbol\index{Kronecker symbol}. 
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On the boundary the normal stress is given by 
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\begin{eqnarray} \label{WAVE natural fault} 
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\sigma\hackscore{ij}n\hackscore{j}=0 
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\end{eqnarray} 
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for all time $t>0$. 