417 
solution of the compound rarefaction wave involves the spherical analog of 
the Lagrange ballistic problem and is very complicated. We believe that 
the complication introduced by successive reflections of the rarefaction 
wave between the center and surface of the gas sphere is not of great prac- 
tical importance for two reasons. First, the major part of the water shock 
wave is emitted in an interval of the order of & a/e*, so that the 
first reflected wave would at most disturb its tail at all distances of 
practical interest. Second, due to the attenuating effect of the factor l/r 
in a spherical wave and the fact that the gas sphere will have expanded to 
several times its initial radius before the first reflected wave arrives at 
the surface, the pressure will already be so low when it arrives that it 
will have little accelerating effect on the surface. Nevertheless, it is 
possible that the successive reflected waves in the gas sphere will produce 
small pulsations in the tail of the shock wave, which we shall ignore in the 
present theory. 
At the surface a(t) » we have continuity of pressure and mater- 
ial velocity and therefore of the total time derivatives of these quantities. 
Making use of the continuity condition, we may write the equations of conti- 
nuity exterior and interior to «@(¢} in the form, 
SD ps S% due 2 eu 
ep Dt or aed ace 
aed BE ale ss : 
eres = = Hie 2u, 
cp Dt P Qa 
where € and Pp refer to the water just exterior to the surface; ¢* and pe to 
the gas just interior to the surface, The partial derivatives (dus dr) and 
(du/or)* are in general different. To eliminate them from Eqs. 
30 
