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For the anisotropic elastic case the differential operator
is
a 6x3 matrix of partial spatial derivatives and when operating on
the displacement field results in the symmetric Lagrange strain tensor.
| ![\begin{displaymath}
\nabla^T = {1\over2}({\partial\over{\partial x_l}}+
{\partial\over{\partial x_k}}) \qquad\qquad{\rm with}\qquad k,l=1,2,3\end{displaymath}](img6.gif) |
(2) |
Giving us the following set of first order (in space) equations:
| ![\begin{eqnarray}
\epsilon_{kl} & = & {1\over2}({\partial u_k\over{\partial x_l}}...
...r{\partial x_j}})
\sigma_{ij}
+{{\partial}\over{\partial{t^2}}}u_j\end{eqnarray}](img7.gif) |
(3) |
| (4) |
| (5) |
where a is the density and b represents the stiffness matrix.
In general the stiffness matrix is a sparse matrix, and can be simplified
for different degrees of symmetry.
Next: Acoustic Medium
Up: WAVE EQUATIONS
Previous: WAVE EQUATIONS
Stanford Exploration Project
11/17/1997