Directivities of linear arrays and linear sources

**Problems in Exploration Seismology and their Solutions**
Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	8
Pages	253 - 294
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	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 $n\to \infty$ while $\Delta x\to 0$ in such a way that $n\Delta x\to a\lambda$ , $a$ 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 $\sin(\pi a\sin \alpha )$ . To get the limit of the denominator, we replace the sine by its argument (because $\Delta x\to 0$ ) and get $(n\pi \Delta x/\lambda )\sin \alpha \to \pi a\sin \alpha$ . Substituting these values in equation (8.6d), we obtain