348 



{Z.2) 2TrpA^^lA'^(l + Af^) - 2a>2a'b' + ^ b'^{1 + A^g)] 



where (p~., y?p saia cp,^ are fTxnctlons of A, B axid. li which 

 represent the efrect of the surface and the hot ton. Note 

 that 9?,, <p^, <p„ are of the dimensions (length)"", 



(length)" , (length)"""' respectively. 



The potential energy of the displaced water is equal 

 to the volijine of the bubble multiplied by the hydrostatic 

 pressure at the center of the bubble, that is, to 

 4TrA"^pgZ/3 . 



Assuiae the pressure and volume of one unit mass of 

 the gas in the bubiale under an adlabatic change are con- 

 nected by the relation 



(3.3) PV*'' = K. 



If M is the mass of the explosive, then for the actual 

 bubble 



PV*^ = YJ.1^ 



and the internal energy of the gas when the bubble has 

 radius A is 



G(A) = /pdv = iai*''v-'-"V(Y - 1) 

 = ^1*^(3 TTA^)-'-'V(r - 1). 



By our assumption the total energy, E, left in the 

 bubble after the shock wave has passed is equal to the 

 sum of the kinetic energy of the water and the potential 

 energy of the displaced water and the internal energy of 

 the gas . V/e have , then 



2iTpA^[A'^(l + kf-^) - SA^jPg^'B* + i B'^d + A^^^)] 



(3.5) , 3 



+ V TTA pgZ + G(A) = E. 



