A source with the isotropic moment M displaces the volume

(5) |

through any spherical surface that encloses the source. This result is obtained in four steps.

1 - The displacement field of a linear vector dipole (LVD, a point source representing two equal colinear forces of opposite sign) can be calculated analytically. The radial component of the displacement can then be integrated over the surface of a sphere centered at the source. The result is given by eq. (5), independent of the radius of the sphere. The derivation is based on formulae for the displacement field of a point force taken from Gerhard MÃ¼ller's lecture notes on the theory of elastic waves (1986), and was doublechecked with the computer-algebra system MuPAD Pro 2.5 (2002).

2 - The result can be generalized to spherical surfaces that are not centered around the point source but enclose it. The result is the same as for centered spheres. A mathematical proof is given in the appendix. The result was also confirmed by numerical integration with random positions of the source within the sphere.

3 - Eq. (5) being true for any position of a point source within a given sphere, it remains valid for distributed sources.

4 - An arbitrary moment tensor is the combination of three LVDs along its principal axes, so (5) is valid for any moment-tensor source.

Equation (5) appears to be the most general moment-to-volume relationship that can be derived without a detailed knowledge of the source geometry. It retains its physical meaning for point sources, for which a "volume change" is otherwise difficult to define.

2003-05-30