Directivities of linear arrays and linear sources

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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 while in such a way that , 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 . To get the limit of the denominator, we replace the sine by its argument (because ) and get . Substituting these values in equation (8.6d), we obtain

Because the angles and are equivalent, equations (8.6d) and (7.5d) are equivalents.

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