MOTION OF THE GAS SPHERE 285 



ratio of shock Avave energies, because the pertinent vakies are of course 

 those which represent total wastage as heat after the shock wave has 

 passed to infinity. Experimental values give neither this value nor the 

 total of transmitted energy before any wastage has occurred, as they 

 are obtained at a finite distance from the charge. This uncertainty 

 considered, the comparisons of such energy ratios give rather satis- 

 factory indications that no serious discrepancies exist in the total ac- 

 counting of energies, even though this accounting is not yet as directly 

 based on experiment as might be desired. More explicit illustration is 

 unfortunately precluded by security restrictions applying to the more 

 significant examples. 



8.4. General Equations of Noncompressive Motion 



When the effect of gravity is included, the motion of the gas sphere 

 is symmetrical around a vertical axis through the initial position of the 

 charge, but no longer has radial symmetry. To take account of the 

 external force and lower symmetry, it is desirable to use the more power- 

 ful mathematical methods of potential theory rather than solve the 

 dynamical equations by direct integration. In this section, these 

 methods are therefore outlined as they apply to noncompressive flow of 

 an ideal fluid incapable of supporting shearing stress (i.e., neglecting 

 viscosity). No attempt will be made to examine their full possibilities 

 and limitations ; for such discussions, reference should be made to stand- 

 ard treatises. 



For noncompressive motion in which the density is assumed con- 

 stant, the equation of continuity expressing conservation of mass (cf. 

 section 2.1) reduces to 



(8.14) divv = 



where v is the vector particle velocity. For the problems of interest to 

 us, it is possible to define a velocity potential (p at all points in the fluid, 

 from which the components of v are obtained by space differentiation : 



•J 



(8.15) V = — grad ip, i.e., Ux = > etc. 



dx 



With this definition, the potential ip must satisfy Laplace's equation, 

 obtained by substituting (8.15) in (8.14), 



(8.16) div(grad <^) = AV = 



where, in Cartesian coordinates, A^ = d'^/dx^ + d'^/dy'^ + d'^/dz^. In 

 addition to satisfying this equation, the potential ip must also be so 



