60 HYDRODYNAMICAL RELATIONS 



If the reflected shock travels with a velocity U' relative to the fluid be- 

 hind it, the tangential velocity ahead of this shock is U'/sin a' less the 

 tangential component of u^, and 



U' , . , {U' -u') +u'cosW 



Ux = ; — u sm a = -^ -. 



sm a sm a 



Equating these values gives 



fo An\ f \ — (^ ~ '^) + ^ cos^q ; _ {W — u^) + u' cos^o:^ 

 sm a sm a 



The pair of equations (2.39) and (2.40) therefore determine the 

 values of {V — u') and a for given values of {U — u) and a (since 

 {V — u') and u' are not independent for any given fluid), and can be 

 solved by numerical methods. There are, in general, two values of a 

 for a given intensity and direction of incidence, and corresponding 

 values of U' and u', these values being identified with two values of 

 pressure P'. As the value of a increases, how^ever, these two solutions 

 merge into one, and real solutions for a do not exist for more oblique 

 incidence and larger a. This failure indicates that the postulated 

 reflection scheme cannot be realized physically and some other system 

 of shock fronts must replace the simple one drawn in Fig. 2.10. Before 

 considering what "irregular" reflections can satisfy the physical condi- 

 tion it is of interest to determine at what angle of incidence aextr the 

 regular reflection breaks down, and to consider the relation between 

 incident and reflected shock fronts in the regular region as the extreme 

 value is approached. 



The first question involved in the regular scheme of reflection is that 

 of which of the two values of a for a given P and a is physically real- 

 ized. General considerations as well as detailed calculations show that 

 the larger value of a' always corresponds to a greater pressure discon- 

 tinuity, and this solution thus represents a shock wave stronger than 

 the one for smaller a\ which moreover is ahead of the weaker shock. 

 Intuitively it is to be expected that the wave realized will be the in- 

 ferior one of smaller a in which energy is degraded to a greater extent, 

 and the existing experimental data all support (or at least do not con- 

 flict with) this conclusion. The evidence therefore excludes the larger 

 of the two values of a. Values of a and P' /Po as functions of a and 

 P/Po, where Po is atmospheric pressure, have been computed for water 

 by Polachek and Seeger (87) from the equivalent of Eqs. (2.39), (2.40). 

 In these calculations an adiabatic law of the form of Eq. (2.30) was 

 used with 7 = 7.15. Some of their values for a' as a function of a are 

 plotted in Fig. 2.11. The dashed parts of the curves correspond to the 

 larger, unrealized values of a. It will be seen that a is always greater 



