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The source volume and the near field

Assuming that the displacement field is generated by a spherical source of radius a, the source volume V(t) is obtained from the near-field term of eq. 4 as

 \begin{displaymath}V(t)=4 \pi a^2 u_n(a,t)=4 \pi f(t-\frac{a}{c})
\end{displaymath} (9)

Thus, the source function f(t) is simply the source volume, divided by $4 \pi$. If we know the source volume as a function of time then we know the whole wavefield:

 \begin{displaymath}u_r= \frac{1}{4 \pi r^2} \, V(t-\frac{r-a}{c})+\frac{1}{4 \pi r c} \, \dot V(t-\frac{r-a}{c})
\end{displaymath} (10)

The remarkable thing is that we do not need to know the radius of the source, its pressure, or the elastic moduli of the medium (except for their implicit presence in the P-wave velocity c). Further, we see that the source volume is the same for each choice of a, except for the delay a/c which is in most cases negligible. The source volume can be measured at any distance from the source as long as we stay in the near field; the border between the source and the medium is arbitrary. Finally, we observe that the material behaves like an incompressible fluid in the near-field. Although our solution (4) was derived from a compressional potential, in the quasistatic limit it is not associated with a volume compression but with pure shear.


next up previous contents
Next: The source volume and Up: Basics of the volume-source Previous: Near-field and far-field
Erhard Wielandt
2001-09-21