# Directivities of linear arrays and linear sources

Jump to navigation Jump to search
ADVERTISEMENT
Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 8 253 - 294 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem

Show that the directivity equations (8.6d) and (7.5d) are consistent.

### Solution

Since equation (8.6d) applies to a discontinuous array of geophones whereas equation (7.5d) applies to a continuous source, we find the limit of equation (8.6d) as the number of geophones becomes infinite. We require that ${\displaystyle n\to \infty }$ while ${\displaystyle \Delta x\to 0}$ in such a way that ${\displaystyle n\Delta x\to a\lambda }$, ${\displaystyle a}$ being the same constant as in problem 7.5, thus keeping the array length equal to the source length. In the limit the numerator of equation (8.6d) becomes ${\displaystyle \sin(\pi a\sin \alpha )}$. To get the limit of the denominator, we replace the sine by its argument (because ${\displaystyle \Delta x\to 0}$) and get ${\displaystyle (n\pi \Delta x/\lambda )\sin \alpha \to \pi a\sin \alpha }$. Substituting these values in equation (8.6d), we obtain

{\displaystyle {\begin{aligned}F=\sin(\pi a\sin \alpha )/\pi a\sin \alpha =\sin c(\pi a\sin \alpha ).\end{aligned}}}

Because the angles ${\displaystyle \alpha }$ and ${\displaystyle \theta _{0}}$ are equivalent, equations (8.6d) and (7.5d) are equivalents.

## Continue reading

Previous section Next section
Response of a linear array Tapered arrays
Previous chapter Next chapter
Seismic equipment Data processing