462 
where the coefficients are determined as functions of p mn by the Hugoniot 
relations, Eqs. 8.6. By differentiation of the second of Eqs. 8.19 we 
obtain a total differential equation relating D to [ee andl, 
AD | _ pe? 
dk Aa (8.21) 
Equations 8.20 and 8.2] are exact, although in using them we shall employ 
the similarity restraint in estimating the slowly varying function v . We 
remark that Eqs. 8.20 are valid for plane and cylindrical shock waves when 
the terms 2 u/ R are replaced by zero and Uy / R » respectively. 
Solution of the equations 8.20 for (dp/at) and (Jp far). 
and the application of Eq. 8.7 yields the desired total differential equa- 
tion for ap /AR along the shock front. This equation is to be solved 
simultaneously with Eq. 8.21. The final differential equations can be 
written in the form, 28/ 
28/ We remark that if these methods are applied to the one-dimensional 
cases of the plane shock wave and the cylindrical shock wave result- 
ing from adiabatic constant volume conversion of an infinite cylinder 
of explosive to its products, the resulting propagation equations can 
be written in the form, 
ol b a Ri f CG 
p Bey ae as) 
as ae M(p,) Der) [g0* ip ~S ) 
Pi uise KR hth..) 
£0. A Rhl ba), 
where & = O for the plane wave and & = | for the cylindrical wave, 
and where So e 
Dir) 2 | ha’. hl p(rddr, ' 
R 
The suockewave energy per unit area of initial generating surface is 
equal to ag*D¢<F . Eqs. 8.22 for the spherical wave are obtained 
withA=2,. It may also be noted that the theory has not required 
the approximation of adiabatic flow employed in the kinetic enthalpy. 
propagation theory. The exact Hugoniot curves of the fluid may be em- 
ployed in the numerical integration of the propagation equations. As 
a result, the present theory is applicable to blast waves in air as 
well as to explosion waves in water. 
73 
