# Directivities of linear arrays and linear sources

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.