408 
other case with constant velocity C 9 » we are led to ask whether the spher- 
ical shockewave problem possesses solutions corresponding to an outgoing 
progressive wave in which r{d or G is propagated behind the shock front 
with a variable velocity c depending upon the intensity of the wave. To 
investigate this possibility, we consider the characteristics of G with 
the differential equations, 
(arfot), = G 
shown schematically in Figure 3.1. 
(3.27) 
It is assumed that the characteristics ret ,G) are monotonic 
curves with finite positive slope C » which do not intersect at any point 
behind the shock front. We then define a function P(I,t ) by the relation, 
GT) eo Wine ) , (3.28) 
where for the present G. is an arbitrary function except for the requirement 
that J be a single-valued function of G. Eqe 3.27 may therefore be written 
f= (3), (3.29) 
We now integrate Eq. 3.29 along a path of constant { in the form 
dis’ 
«2 z - FO?) , (3.30) 
a(t) 
as 
where a(t) is the radius of the expanding gas sphere at time t. We remark 
that {7 has been assigned the dimensions of a time and that the integration 
constant in Ea. 3.30 has been so chosen that 
Blatt ty. pee! 
Eqs. 329 and 3.31 then determine the arbitrary function G (t) as 
Git) = Glattx,t] , (3.32) 
