CHAPTER 17 For Deep bursts at increasing scaled depths approaching the Very Deep category, the transition between the two corresponding values of ¢ will have to be determined by additional theoretical or experimental work. Complete derivations of the mathematical forms of N for both deep and shallow bursts are presented in Reference 48. Summaries of the derivations are presented in the following paragraphs, along with Equations 17-10 to 17-12, which are explicit expressions for N. The dose rate, d, due to transit radiation from underwater bursts, can then be calculated by the substitution into Equation 17-5 of Equations 17-6 to 17-8, the appropriate form of Equation 17-9, and the suitable value of N as expressed by Equations 17-10, 17-ll or 17-12. Such calculations have been machine programmed at USNRDL. (1) "Deep" Geometry Consider a solid truncated cone, (Fig.17-10) of radius Ry, height Z, interior angle & between face and base, with the base centered at 0; and a receiver at point P in the plane of the base at a distance S from the axis of the cone. Then, Wr2 + 22 N= l/ln Ife mT? pareedz re + 22 where cylindrical coordinates are used with center at P, z-axis parallel to axis of cone and polar axis PO, and the integration is over the volume of the truncated cone. (All distances are expressed in mean-free-path units. ) To facilitate computation, the z-integration is replaced by 4 finite summation over n increments Az where n 4z = Z. let 2, be the midpoint of the i*® increment: z; = (21 + 1) Z. In effect, the en truncated cone is replaced by a set of n circular disks of thicknese Z/n and of radius R, - z; cot a, i= 1, 2, ....n. N= l/dx n z _ z/n ; e- . Then, 24 eae rdodr. r The value n = 10 was used in computing base surge dose rates. ‘here are 2 forms for the integral depending whether S <R, - z, cot a, or S 2R) - 2; cot a. entire volume: O Best There are thus 3 cases fur the summation over the PY COP a Ls2= C o “" AIL .Ao 17-53 terme reece ee te