453 
as to the physical features accompanying the decay of the shock wave, the 
theory of this section requires fewer assumptions than does the former 
theory. Since these assumptions are quite different in nature from those 
of the former theory, the comparison of the two theories is of considerable 
interest. 
The partial differential equations of hydrodynamics and the 
Hugoniot relation between pressure and particle velocity may be used to 
provide three relations between the four partial derivatives of pressure 
and particle velocity, evaluated at the shock front, with respect to time 
and distance from the source. If a fourth such relation could be formulated, 
it would then be possible to formulate an ordinary differential equation for 
the peak pressure as a function of the distance from the source. 
dp. F(p,R) , (8.1) 
AR 
On mathematical grounds, it is, of course, futile to seek a fourth relation 
between the partial derivatives which does not involve an integral of the 
fundamental equations of hydrodynamics, However, we shallshow that an ap- 
proximate relation of the desired form can be formulated by imposirg a sim- 
ilarity restraint on the shape of the energy-time curve of the shock wave 
and by utilizing the secemd law of thermodynamics to determine at an arbi- 
trary distance the distribution of the initial energy input between dissi- 
pated energy residual in the fluid already traversed by the shock wave and 
energy available for further propagation. 
The Eulerian equations of hydrodynamics, Eqs. 3,1, for spherical 
| Sacave trebebeaier | Au Dh, ou 2 aoe 
Poy ar | ope’ DE «0 r ies 
