388 



9x' 9y' 9z ^^^ components in the X, Y , Z directions of the gravita- 

 tional acceleration g, 

 C is a constant depending on the initial conditions, and 



M, N^, Ny, N^ stand for simple functions of X, Y, and Z, depending 



upon the choice of axes and upon the nature and location 

 of the boundaries. 



An accurate solution of these equations can be effected only by 

 numerical Integration. This method has the disadvantage that many repeti- 

 tions of the entire calculation are required to obtain results covering a 

 wide variety of conditions. 



For the first oscillation of the globe, on the other hand, formulas 

 can be obtained which, although less accurate, are widely applicable. These 

 formulas give the position of the gas globe at the peak of the first recom- 

 pression, which is particularly important because it is the point of origin 

 of the first secondary pulse of pressure. 



The method of approximation is based upon the observation that, as 

 the time t advances, the Integrals in Equations [2], [3], and [4] grow chief- 

 ly while R is large, whereas, because of the factor l/fl* preceding the inte- 

 grals, they are effective in causing displacement of the gas globe chiefly 

 while R is small. Approximate values for the displacement during the first 

 recompression can be obtained, therefore, by substituting for these integrals 

 in Equations [2], [3], and [U] constants equal to the values of the integrals 

 at the Instant of greatest compression. In calculating these values, on the 

 other hand, an approximate value of dR/dt, obtained by neglecting certain 

 terms in Equation [1], is sufficiently accurate. The same expression for 

 dR/dt leads to a corrected value of the period. 



The first period is thus found to be 



r, = T,o(l + 0.20 Mi?2) [5] 



where T^^ is the period when no bounding surfaces are near. Here M Is the 

 coefficient that occurs in Equation [1], and R2 is the maximum radius of the 

 gas globe during the first expansion; see Equation [18]. 



The approximate formulas obtained for the displacement of the cen- 

 ter of the gas globe from the position of detonation X^, Y^, Z„ to the point 

 Xj, Yi, Zi at which the next minimum radius occurs, may be written 



X^- Xo = Ve B^R^U [6a] 



y, - y^ = /6 ByR^U [6b] 



Z^- Zq = VQ B2R2U [6c] 



