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A mathematical source model

For a given isotropic moment M, we can easily construct a mathematical source model with any desired positive or negative volume change. We simply let a linear vector dipole (LVD) with the moment 3M act in a small sphere, either in its center or evenly distributed. (If the isotropic moment equals zero, we may use a suitable scaled CLVD - a compensated linear vector dipole - instead.) The displacement field outside the source sphere has the divergence

\begin{displaymath}\nabla \vec{u} = 3 M \frac{1-3 \cos^2(\theta)}{4 \pi (\lambda + 2 \mu)
\,r^3}\
\end{displaymath} (6)

The volume integral of this divergence is zero over any spherical shell enclosing the source, but since the divergence decays as r-3 in any given direction from the source, the integral diverges to plus or minus infinity if taken over a suitable cone in the polar or equatorial direction. Our definition of the source permits to consider any part of such a cone as part of the source. We can thus define the boundary of the source in such a way that its total volume change has any desired value.


next up previous contents
Next: Acknowledgment Up: Draft On the relationship Previous: Source volume and moment
Erhard Wielandt
2003-05-30