126 THEORY OF THE SHOCK WAVE 



approximation (for which x would be independent of (R/ao)). Neg- 

 lecting G{a, To) in comparison to G{ao), the asymptotic peak pressure 

 variation is, from Eq. (4.18), 



and is therefore determined by the time integral of Ga(t) on the gas 

 sphere rather than its initial value. 



The asymptotic behavior of the time scale factor y is similarly found 

 from Eq. (4.16) to be 



(4.21) ,^1-'1J-^U^) 



Co^ clT \ao/ 



If the peak approximation G(a, t) = G{ao)e~^/^^ is an adequate descrip- 

 tion we find that the time constant 6 = ydi for the initial part of the 

 pulse (r = To) is given by 



(4.22) e = 2'^^^^\oJ^\ 



Co Co \(J'o/ 



and therefore increases gradually with R/ao. This result is much more 

 approximate than the expression for peak pressure. In the later por- 

 tions of the pulse, the derivative dGa{T)/dT is not accurately determined 

 by the peak approximation and furthermore its value will become in- 

 creasingly small. As a result, the second term in Eq. (4.21) for y will 

 become increasingly smaller compared to unity. At sufficiently late 

 portions of the pulse, the broadening of the time scale represented by 

 this term will become unimportant and the later profile of the shock 

 wave is represented by a time scale increasingly similar to that on the 

 gas sphere (except of course for the time required for the wave to be 

 propagated from the gas sphere to the point considered). 



4.4. The Shock Wave for Cylindrical Symmetry 



The theories of the shock wave so far described have dealt with tlie 

 case of spherical symmetry, which is the simplest symmetry corre- 

 sponding to an experimentally realizable situation. In many practical 

 problems it is neither necessary nor desirable to have the explosive 

 charge in the form of a sphere detonated at its center. Any tractable 

 theory for some other shape of charge is therefore desirable, if only to 

 reveal the nature of the resulting differences. The simplest geometry 

 for this purpose is evidently the one-dimensional case of an infinite 



