460 
The function f( AR ' v7, ) is the energy-time integrand, normalized by its peak 
value RH bat the shock front, expressed as a function of A and a reduced 
time P vinich normalizes its initial slope to -l if i does not vanish. Thus 
f(KR? ) is a function having the properties, 
PER Oral =, 
[otf] y. 0 a al 
We also assume Fo be a monotone decreasing function of +, Elimination of 
(8.16) 
Ins between the first two of Eqs, 8.15 yields the desired fourth relation be- 
tween the partial derivatives at the shock front supplementingEqs. 8.8. It, 
of course, involves integrals of Eqs. 8.5 for the knowledge of the reduced 
energy-time function ¥( R,) . However, if f( Qo, WY) is initially a mono- 
tone decreasing function of Y, F(R, Y) will remain so, and in fact will at 
large R become asymptotically a quadratic function of a> corresponding to 
the linear form of the pressure-time curve shown in section 6 to be asymptot- 
ically stable. This means that y is a very slowly varying function of R 5 
for which sufficiently accurate estimates for many purposes can be made without 
explicit integration of the hydrodynamic equations, Eqs. &.5. 
The assignment of a constant value, independent of R » to y is 
equivalent to imposing a similarity restraint on the energy-time curve of the 
shock wave, This type of approximation is equivalent in principle to that 
underlying the Rayleigh-Ritz method for solving vibration problems, although 
we have not, of course, developed a variational procedure designed to carry 
the result to any desired degree of approximation. 
The initial pressure-time, and also energy-time curve, of an ex— 
plosion wave is rapidly decreasing. An expansion of the logarithm of the 
(a 
