MOTION OF THE GAS SPHERE 291 



eral features of the motion, to neglect this term in the energy equation. 

 With this approximation, the only force acting on the water is gravity 

 and the only resistance to flow is that offered by the inertia of the 

 water. 



If the linear scale of the phenomenon is changed, the corresponding 

 change in time scale which preserves similarity of the two motions must 

 be such that the effect of gravity (hydrostatic pressure) is the same in 

 either system. If this is to be true, the invariance of the acceleration 

 of gravity g which has dimensions length/ (time) ^ requires that the time 

 scale be increased by the square root of the scaling factor for length. It 

 is convenient to express this correspondence by using a characteristic 

 length L in terms of which the equations can be expressed in non- 

 dimensional form. The dimensionless variable i' replacing the time t 

 and scaling properly can therefore be written 



^ Q 



provided dimensionless variables a', z' replacing a, z are expressed by 

 a = La\ z = Lz'. Using these definitions, Eqs. (8.23) and (8.25) be- 

 come 



dt' a'^ J 



if the characteristic length L is chosen to have the value {Y /gpoY'^. 

 With this choice of L as made by G. I. Taylor, the equations are ex- 

 pressed in a nondimensional form suitable for numerical integration 

 with assigned initial values of z' , a' , provided the term E{a)/Y is neg- 

 lected. Corresponding values of measured lengths and times are then 

 determined by the relations 



a' z' \pogJ t' \pof) 



where Y is the total energy available after emission of the shock wave. 

 Usually, the energy Y can be expected to be proportional to the 

 weight of the charge and hence to the cube of its linear dimensions. The 

 scaling laws expressed by Eqs. (8.27) thus require that, for proper 

 scaling of bubble phenomena from one size of charge to another, the 



