210 """ 



It may be noted that the second term in the expression for 2k given In equation (iS) Is the 

 ratio of the weight of displaced water to the weight of the sphere. In most underwater explosion 

 experiments with submerged targets this ratio will be less than or equal to unity so that 2k will 

 lie between the values 2 and 3 corresponding to the physical conditions. 



2 k = 2, Infinitely heavy or fixed sphere 



2 k = 3, buoyant sphere just waterOorne' (18) 



(• For brevity this case will Be referred to as "buoyant sphere" in the remainder 

 of the paper). 



For k $ 2, which Includes the above range, the complementary function of the differential 

 equation (lu) is periodic and the formal solution of this equation subject to the Initial 

 conditions (17) is 



' , - M e-'-T sinmT 



t IT a p m 



e 







'''^■^' Slnm(TA)J f W ♦ ^l d\ (19) 



••"^''^ m = / 2k-k' (20) 



For the particular cases of fixed and buoyant sphere the valuss of k and m are thus 



k = 1, m «^ 1, fixed sphere 1 



k = 3/2, m = /J/2 buoyant sphere r '^^' 



Solution for exponential pressure Pulse . 



When the incident pressure pulse is exponential in form we have 



f (T) = e-"^ (2 2) 



and the general solution (19) is easily integrated to give 



. 1 - B (q - k) - kT ,. ^ 



1 = ' e Sin m T 



m 



+ B (e" ''^ Cos m T - e" ''''') (23) 



«77a p^ 



where 



; q"- ^ (2U) 



X (q" - 2kq + 2k) 



From equation (13) the velocity of the sphere is then given by 

 6 (k-1) v„ 



2 kq X 

 where, if p denotes the mass density of the medium. 



(25) 



(26) 



The 



