292 MOTION OF THE GAS SPHERE 



initial values of "hydrostatic" depth z and the scale of length be in- 

 creased as the three-fourths power of the linear dimensions of the charge; 

 correspondingly, the time scale must be increased as the three-eighths 

 power of charge dimensions. Thus neither length nor time scale di- 

 rectly with charge dimensions, and it is not possible to preserve exact 

 similarity for both shock wave and bubble phenomena. A further 

 difficulty in scaling bubble phenomena lies in the fact that the depth 

 variable z is not the depth d of the bubble center below the water sur- 

 face, but is rather a measure of hydrostatic pressure at this depth. If 

 the pressure is atmospheric at the surface and lengths are measured in 

 feet, then z = d -\- 3d feet, and it is evident that z can be less than 33 

 feet only if the pressure at the surface is reduced below atmospheric. 

 Hence model scale experiments on bubble phenomena can in many 

 cases be made to simulate those for large charges only by reducing the 

 pressure at the surface. 



It should be emphasized that the nondimensional form (8.26) of the 

 equations of motion and the scaling laws (8.27) are only approximations 

 obtained by neglecting the internal energy of the gas sphere, and hence 

 are increasingly in error as the bubble radius becomes smaller. Strictly, 

 complete similarity of bubble motions for two different charge sizes is 

 not possible because the scaling laws for the internal energy and hydro- 

 static pressure are not compatible. Any conclusions obtained on the 

 basis of the scaling laws as expressed by Eq. (8.27) are therefore sub- 

 ject to some uncertainty and are particularly unreliable in considering 

 phenomena in the region of greatest contraction. These inaccuracies 

 may, of course, be removed at the expense of generality by restoring 

 the term -Eia)/Y to the right side of Eq. (8.26). For TNT, E(a)/Y 

 varies as {W/a^y^ from section 8.2, and expressed in Taylor's variables 

 has the form: constant- (Tf^^^'^a'-^^''). This term thus varies as W^'^^ 

 and its inclusion requires separate solutions of the equation of motion 

 for each charge weight. 



C. Results obtained neglecting the internal energy. G. I. Taylor (107) 

 has given the results of numerical integration for the initial condition 

 that Zo' = 2.0, where Zo' is the initial value of the depth variable at time 

 t = 0. This integration was carried out by approximate solution of 

 Eqs. (8.26) for the initial stages of the motion, based on the fact that 

 initially a' and the migration velocity dz'/dt' are both small. Neglect- 

 ing small terms in a'^ and dz'/dt', the first of Eqs. (8.26) gives 



da' a'-'i'' „ 2 ,— 

 dt' V27r 5 



assuming a' = for t' = 0. Using this value of t' in the second of 

 Eqs. (8.26) and integrating gives 



