THEORY OF THE SHOCK WAVE 127 



cjdinder detonated simultaneous!}^ at all points on its axis. This is a 

 mathematical idealization not realized experimentally because no charge 

 is strictly of infinite length and the detonation condition even for a finite 

 cylinder is not easily realized. The best approximation to the relatively 

 simple theoretical assumption is obtained for cylindrical sticks det- 

 onated at one end. In this case the detonation velocity, although not 

 infinite, is sufficiently high that the shock front makes an angle of less 

 than thirty degrees with the surface of the charge. For points not too 

 far from the charge it is reasonable to expect that a theory based on 

 cylindrical rather than axial symmetry would have a rough correspond- 

 ence to observations under such conditions. 



The basic differences between spherical and cylindrical symmetry 

 are in the propagation equations for the water and explosion products, 

 the equations of state and the shock front conditions remaining un- 

 changed. Rice and GinelP have extended the Kirkwood-Bethe propa- 

 gation theory to the cylindrical case, and as in the application of the 

 Kirkwood-Bethe theory the initial conditions are approximated by the 

 pressure in adiabatic conversion of the explosive to its products at 

 the same volume. This condition is unaffected by the symmetry and 

 the essential differences are then in the disturbances propagated away 

 from the discontinuity. 



A basic difficulty in analysis of the propagation lies in the fact that 

 cylindrical waves even in the acoustic approximation necessarily undergo 

 a change of type as they are propagated.'^ There is no exact compari- 

 son with propagation as a function F{t — r/c) as in plane or spherical 

 waves, and only asymptotically is it found that pressure, for example, 

 varies as r~'^''^ F{t — r/co) where F is an undetermined function (as com- 

 pared with r-i F{t — r/co), valid at any distance for acoustic spherical 

 waves). The development of a finite amplitude theory based on this 

 dependence will not therefore be as simply related to the actual state of 

 affairs, and errors incurred in approximations suggested by the relation 

 will be larger than for spherical waves. 



Rice and Ginell develop a propagation theory for the kinetic en- 

 thalpy 12 assuming a propagation function G, level values of which ad- 

 vance outward with a velocity c + c, where c is the local sound velocity 

 and 0- the Riemann function. The discussion of the preceding para- 

 graphs suggests as a first approximation, analogous to that for spherical 

 symmetry, taking G = r^'^il, where H = E + P/p + ^u^. Detailed cal- 

 culations showed, however, that the assumption failed rather badly near 

 the charge, despite its asymptotic validity. A better approximation, 



« Rice and Ginell's treatment is presented in two reports (59, VI and VIII) , the 

 second being a revision and extension of the first. 



' For a discussion of the propagation of cylindrical acoustic waves, see for ex- 

 ample the book by Morse (75). 



