403 



It is particularly unfortunate that calculations do not exist for 

 charges detonated on the bottom. They are made difficult by the Inevitable 

 distortion of the gas globe. As Z Is diminished below 2R^, the attractive 

 effect of the bottom should probably Increase and then decrease again. This 

 conclusion Is based on the following Ideal case. The water flow around a 

 hemispherical charge lying with Its flat face on a rigid bottom and detonated 

 at its center should resemble half of the flow around a spherical charge of 

 the same radius detonated in open water; gravity should, therefore, cause the 

 gas globe from the hemisphere to rise. From the analytical results It may 

 reasonably be surmised that the gas globe from 10 pounds or over, detonated 

 on the bottom under any depth of water of practical interest, will probably 

 rise during the first recompression. 



MIGRATION OF A GAS GLOBE IN SHALLOW WATER 



The combined effect of the free surface and of a parallel rigid 

 bottom can be obtained by extending the method of Images. If Z is taken to 

 stand for the distance of the center of the gas globe above the bottom, it is 

 found that only those changes need to be made in the formulas, as obtained 

 for a rigid bottom alone, which correspond to the assumption. Instead of 

 [29a, b] 



U = T,-^^^, Nz = Sz [40a, b] 



where D is the total depth of the water, 1.39 represents 2 log 2, and Tj and 

 Sj stand for the series 



T, = i ' 1 , , 1 . 1 1 



Z D-Z D+Z 2D-Z 2D+Z 3D - Z 



^ _ ± ^ _J 1 1__ , 1 .... 



^^~ Z^ (D-Z)^ (D + Z)^ T2D^^W Tw+W 



Hence the displacement of the gas globe measured upward, during the first ex- 

 pansion and recompression, is 



2.60 B^ R2 Dili 



Bz = 0.0346 [1 + 0. 23 (t, - ^^) Kg] ^ 



-0.223 [1- 0.18 (r,- ^j/Jg] 52-^2^ ['^2] 



