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The source volume and the far field

The far-field term of eq. (10)

 \begin{displaymath}u_f= \frac{1}{4 \pi r c} \, \dot V(t-\frac{r-a}{c})
\end{displaymath} (11)

permits an interesting generalization. Wielandt (1975) has derived the following formula for the P-wave displacement radiated vertically downward from an explosion at the seabottom:

 \begin{displaymath}u_f(z,t)=\frac{1}{2 \pi z c} \, \frac{Z_1}{Z_1+Z_2} \, \dot{V}(t- \frac{z}{c})
\end{displaymath} (12)

Z1 is the acoustic impedance of the water and Z2 that of the solid; z is the depth below the seabottom. The delay time associated with the radius of the source is neglected. It can be seen that (12) is essentially the same formula as (11), except for the transmission coefficient Z1/(Z1+Z2) which equals 1/2 when the two impedances are the same. The two formulae then become identical for vertical radiation. This is remarkable because eq. (12) was derived under the assumption of cylindrical rather than spherical symmetry. Also, eq. (12) remains valid even when the dynamics of the source are highly nonlinear, such as for explosions; so the same must be true for eq. (11). This simple equation can thus be used to estimate the seismic far-field of explosive sources. Of course absorption in the medium and additional transmission coefficients or spreading factors may have to be taken into account.


next up previous contents
Next: Relation with other source Up: Basics of the volume-source Previous: The source volume and
Erhard Wielandt
2001-09-21