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Spherical sources

An elementary calculation shows that the actual volume change of an expanding spherical source desribed by the moment tensor (1) is given by eq. (3) while eq. (2) defines the transformational volume change, which is larger. It is instructive to look at the moments required for a given expansion $\Delta V$. The total moment $ M = (\lambda + 2 \mu) \Delta V $ results from a moment $ M = (\lambda + 2 \mu / 3) \Delta V $ that expands the interior of the sphere, and a moment $ M = (4 \mu / 3) \Delta V $ that pushes the surrounding medium away. The outer displacement field is divergence-free both in liquids and in solids, so the volume change of the spherical source can be measured at any distance and over any closed surface.

Erhard Wielandt