. Directivity function of a general receiving array for spherical and plane sound waves. Sound-waves. Figure 12. Element on spherical surface. Also, the direction cosines expressed in spherical coordi- nates are coscc = sin7COsx cos3 = sin/siny cos 7 = cos/ Where x is "the angle measured from the x-axis to projec- tion of the unit vector (coscc, cos(3, COS7) on the x-y plane. Let E{x, y,z) = 1, then substituting the above relationships into the expression for B, and simplifying B = a' sF ipjka Jsin0sin7f cos(9-x)+cos0cos7j 0=0 e=o Then, integrating with respect to 8, sinjZ 7? = 2na2 {expj

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. Directivity function of a general receiving array for spherical and plane sound waves. Sound-waves. Figure 12. Element on spherical surface. Also, the direction cosines expressed in spherical coordi- nates are coscc = sin7COsx cos3 = sin/siny cos 7 = cos/ Where x is "the angle measured from the x-axis to projec- tion of the unit vector (coscc, cos(3, COS7) on the x-y plane. Let E{x, y,z) = 1, then substituting the above relationships into the expression for B, and simplifying B = a' sF ipjka Jsin0sin7f cos(9-x)+cos0cos7j 0=0 e=o Then, integrating with respect to 8, sinjZ 7? = 2na2 {expj7i;acos7cos0} Jb (tozsin7sin0)sin 0 36. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Warner, K. K; U. S. Navy Electronics Laboratory (San Diego, Calif. ). San Diego, Calif. : U. S. Navy Electronics Laboratory

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