28 HYDRODYNAMICAL RELATIONS 



Taking the changes dr for a given dt to be (c + u)dt and — (c — u)dt for 

 the two functions, the equations become 



(2.18) dN = -—-dt, dr = (c + u) dt 



r 



cu 



dQ = dt, dr = —{c — u) dt 



r 



If at a time t, values of c, u, and a are known as a function of r, incre- 

 ments in N, Q may be calculated for a sufficiently small interval dt and 

 corresponding values of dr. Carrying out this process gives new values 

 of N and Q at distances r -\- dr and time t + dt. From these new 

 values of N and Q as a function of r, at time t -\- dt, c and t^ can be 

 determined if a is known from the equation of state for the fluid and 

 the process can be repeated. 



Penney (83) and later Penney and Dasgupta (85) have carried out 

 calculations of this kind for spherical TNT charges. The details of the 

 process, such as the largest permissible values of dt for sufficient ac- 

 curacy, and insurance that the iteration accounts for all points in the 

 fluid from which a disturbance can be propagated to a given (r, t), evi- 

 dently depends on the initial conditions. In the case of high explosives, 

 these are evidently the distribution of pressure and particle velocity 

 in the products when the charge is completely detonated. Penney and 

 Dasgupta base their calculations for TNT on results of H. Jones and 

 G. I. Taylor described in Chapter 3; their method of computation from 

 these results is given in more detail in section 4.4. A further consid- 

 eration is the fact that the Riemann equations apply only where dis- 

 sipation effects can be neglected. The conditions realized for a dis- 

 continuity of compression advancing in the fluid must therefore be 

 determined by other factors, as described in section 2.5. Although 

 Penney's method of integrating the Riemann equations is simple and 

 straightforward in principle, the numerical calculations are sufficiently 

 tedious and complex that they have been carried out only to six charge 

 radii for one explosive, TNT. It is therefore highly desirable to have 

 a more flexible analytic theory not involving excessive approximation 

 which can readily be applied to a number of explosives and a wide range 

 of distances. A theoretical development of this kind is the subject of 

 the next section. 



2.4. Kirkwood-Bethe Propagation Theory 



A more analytical approach to shock wave propagation in water, 

 than that of Penney, has been developed by Kirkwood and Bethe and 

 extended by Kirkwood and co-workers to detailed calculations of shock 



