THEORY OF THE SHOCK WAVE 125 



tances.^ In order to carry out this derivation, we note that the quantity 

 /3(7 evahiated at the shock front decreases monotonically toward zero as 

 a hmit, as the shock front progresses outward, and the derived quantity 

 Z (which equals 1 + (/3o-)~0 therefore increases with increasing R/go. 

 For large values of R/ao, the behavior of the dissipation factor x and 

 time spread factor 7 will be determined primarily by the terms involving 

 log Z/Za in Eqs. (4.13) and (4.14) for x and Eq. (4.16) for 7, the other 

 terms varying only slowly or becoming insignificant. 



From Eq. (4.10), the limiting behavior of Z with increasing R/aoX is 

 seen to be given by 



rr Co R R 



Z = — ; — ■ as > cc 



/312(ao) aoX ao 



and Za{0) is, from the basic equation of propagation, determined by 



Za{0) ^ (3Q(ao) 



[ZaiO) - lY Co 



Hence for large R/ao we have 



^^ ZaiO) °^ \aox) IZaiO) - 1 J 



and the dominant term is log (R/aox) as {R/ao) -^ ^ . The quantity 

 Ko, in terms of which Eq. (4.14) for x is expressed, becomes increasingly 

 great compared to unity at large distances, and the asymptotic behavior 

 of X is therefore given by 



(4.19) '"^ w.. ,Co Co 



/lo 

 Gia,t)dt . p\-i/s 



— ^ (log^l 



ao ^^{ao) G{ao) — G{a, To) \ aoxj 



The quantity Ki defined by Eq. (4.13), which depends on R/ao only 

 through G{ao) — G{a, To) and the integral over G{a, t),^ is a slowly vary- 

 ing quantity as To increases. The dissipation factor x itself therefore 

 varies slowly for large R/ao, the dominant factor being the logarithmic 

 decay. Strictly, of course, Eq. (4.19) is transcendental, but the vari- 

 ation of X at large R/ao is so gradual that its effect in the logarithmic 

 term can be neglected. As a result, the factor x decreases slowly as 

 (log R/ao)~'^'^ for large values of R/ao, and even in this limit the peak 

 pressure at the shock front decays more rapidly than in the acoustical 



* These limiting laws are derived in the original report of Kirkwood and Bethe 

 (59, I), which the discussion given here follows closely. 



^ The function G{a,t) is a monotonically decreasing function, approaching zero 

 for large t, and the integral accordingly varies slowly for sufficiently large t. 



