298 MOTION OF THE GAS SPHERE 



slight contraction in the case Zo' = 1. (It is to be observed, however, 

 that this is the least reaUstic of the cases calculated, as the neglected 

 repulsion of the free surface has a considerable effect in actual cases.) 



Calculations are also given in the report of Comrie and Hartley in 

 which the internal energy of the gas is included. These are for values 

 c = 0.08, 0.09, 0.10, 0.11 in the expression E(a')/Y = ca'-^'\ where 

 c = 0.075Tf ^^^^ and hence correspond to weights of TNT from 3 to 500 

 pounds. The curves are very similar to those of Fig. 8.8 for c = 0, the 

 principal differences being that the maximum radius and migration are 

 about 10 per cent smaller; the period and minimum radius are very 

 nearly the same. 



A number of other detailed numerical solutions have been made by 

 H. M. Nautical Almanac Office (76), primarily for initial values corre- 

 sponding to very small charges (1 gram and 1 ounce TNT); one set of 

 results corresponds to the case zj = 1.0 first computed by Taylor (dis- 

 cussed in section 8.5), and shows reasonably good agreement with this 

 more approximate calculation. In another (77), results are given for 

 formulas developed by Temperley (111) to include an approximation to 

 the nonsphericity of the bubble (see section 8.7 for details of this effect). 



B. Approximate formulas. The detailed numerical calculations 

 cited above apply only to specific relations between charge weight and 

 depth of water, but reports from the Road Research Laboratory (Eng- 

 land) (93) give a number of more general, approximate formulas for 

 principal features of the motion. These results are in some cases based 

 on approximations in the equation of motion found to be justified by 

 the calculations, and in other cases are empirical expressions fitted to 

 calculated points. 



Examination of the calculations shows first that the period of oscil- 

 lation differs only slightly from the value predicted neglecting vertical 

 migration resulting from gravity, and in fact is influenced only slightly 

 by including the internal energy of the gas. The effect of migration on 

 bubble radius is also small up to the time of the first maximum. Changes 

 in the vertical position can therefore be neglected in the equation of 

 motion: z' = Zo and dz'/dt' = in Eq. (8.26). If the internal energy 

 is included in the form E{a')/Y = ca'-^'"^, the maximum radius a'm 

 (for which da' /dt' = 0) is given by 



(8.28) a' J = -^, (1 - ca'm-'i') 



4:TrZo 



This equation is readily solved by graphical methods and is found to 

 agree with exact calculations to within 1 to 5 per cent. 



A second fact observed about the calculations is the near-constancy 



