416 
to the error € in examples to which the theory is to be applied indicate 
that it does not exceed 20% in the most unfavorable cases and probably is 
much less than this. Since € is algebraically negative, the approximate 
theory in which it is neglected yields an upper bound to G e 
4, Motion of Surface of Gas SphereL6/ 
According to the initial conditions described in Section 2, we 
have at the instant t =Oa gas sphere of radius Q, at a uniform pressure 
be » the equilibrium pressure of the product gases arising from adiabatic 
conversion of the solid explosive at constant volume. The water surrounding 
the gas sphere is at a uniform pressure Pe (1 atm. or for practical pur- 
poses zero). The discontinuity in pressure Peo P. at the boundary between 
water and gas is removed instantaneously with the establishment of a pressure 
P, and velocity U, on the sphere |” 7a,- At the same time, an advancing 
shock wave starts in the water and a receding rarefaction wave starts in the 
gas sphere, The motion of the surface a(t) is controlled by these two waves. 
The propagation theory that we have developed for the wave in water can be 
applied with slight modification to the wave in the gas sphere, since in a 
rarefaction wave the requirement of no dissipation is satisfied exactly. 
However, in setting up the equations of motion of the gas-ephere surface, we 
shall restrict our considerations to an initial period of time during which 
the wave just interior to the surface a(t) is a simple recessive rarefaction 
Wave. * 
After a period certainly greater than 2a,/o¥ where C, is 
the initial velocity of sound in the gas, a reflected wave will arrive at 
the sphere surface, after which the rarefaction wave becomes compound. A 
16/ See Reference 7, Section 3. 
29 
