328 MOTION OF THE GAS SPHERE 



The two differential equations, (8.49) and (8.50), are more con- 

 veniently expressed in nondimensional, or reduced, variables for length 

 and time. The variables emploj^ed by Taylor in his analysis of bubble 

 motion under gravity could be emploj^ed, but these are not as con- 

 venient in the present problem where the position of the bottom is also 

 a factor. Shiffman and Friedman^^ have therefore adopted somewhat 

 different characteristic length and time units L,* C* to effect the trans- 

 formation to dimensionless variables which are defined by the relations 



''■''' ^* = (iS:)"' ^* = ^*(ir 



where Po is the initial hydrostatic pressure at the depth of explosion in 

 the differential equations. Reduced variables (denoted by asterisks) 

 are then expressed in terms of these scaling factors by 



a = L*a*, h = L%*, t = CH* 



and initial values of hydrostatic depth Zo (corresponding to pressure Po) 

 and charge position bo above the bottom by 



Zo = L^Zo"^, bo = L%o* 



The scaling factors L* and C* so chosen are seen on comparison with 

 the results of section 8.2 to be simply the maximum radius and two- 

 thirds the period of oscillation obtained neglecting the internal energy 

 and external influences of gravity and boundaries. They thus should 

 be expected to have a close connection with these simple properties in 

 the less approximate solutions, as proves to be the case. 



Using the reduced variables, Eq. (8.49) and (8.50) become 



'-> «-[<'+«(&7+5"+'«(f)"-'Kf)(f)] 



(8.53) 



(It 



26*2 \_da \dt* J 2da\ dt*J da \dt* J \ dt*j] 



+ 



a^ 



26*2 



" The notalion employed here differs from that of Shiffman and Friedman in 

 order to reduce the danger of confusion with other parts of the text. It is unfortu- 

 nate that the usages and conventions of various authors are conflicting and not al- 

 ways int(!rnally consistcmt, a difficulty which must be kept in mind when comparing 

 different results. 



