HYDRODYNAMICAL RELATIONS 5§ 



sure front OR can then be thought of as propagating into a fluid under 

 compression which has velocity away from OR. Its velocity U' in 

 fixed coordinates will therefore become increasingly supersonic for both 

 reasons as a approaches 90°. 



Regardless of the direction of OR, it encounters for sufficiently large 

 values of a a fluid in which its displacement parallel to BB' will exceed 

 that of OS. The validity of the conclusion is perhaps seen most easily 

 for a frame of reference in which the incident wave OS is at rest, as 

 shown in Fig. 2.10(b). The fluid entering OS has initially a flow 

 velocity parallel to BB' modified by OS into a flow F which has a com- 

 ponent normal to BB' . The component of F normal to OS is increas- 

 ingly reduced as it, the flow left normal to OS in a fixed frame of refer- 

 ence, increases. As OS becomes increasingly obhque (a -^90°), this 

 normal component represents more nearly the whole of F and the flow 

 velocity F itself will, for sufficiently high compression, be less than 

 sonic, i.e., less than the velocity with which even an acoustic wave is 

 propagated. Any shock wave OR Avill therefore advance into this 

 fluid, as its resultant velocity normal to its front in the moving frame, 

 obtained by compounding the component of F normal to OR and the 

 shock front velocity, is always directed away from BB' . The point of 

 contact of OR must therefore move to the right in Fig. 2.10(b) if OS is 

 fixed. In a fixed frame of reference OR advances on OS, and the point 

 O must move downward from the wall. 



The possibility of a breakdown of regular reflections having been 

 shown, it remains to determine the critical condition for which it fails 

 and to examine what happens on either side of this condition. If 

 regular reflection is to hold it must be true that the resultant velocity 

 left by the two shock waves has no normal component. It is convenient 

 to choose a frame of reference fixed with respect to the two fronts. 

 The condition for no normal component of velocity is then 



(2.39) u co^ a -^ u' cos a = 



where u' is the velocity acquired in passing through the reflected front. 

 It is convenient to formulate the relation between the tangential com- 

 ponents by the condition that the tangential velocity Ux, of a point in 

 the fluid between the two shocks, must be the same whether calculated 

 from the conditions ahead or behind the front. Any such point arrives 

 at the initial shock with velocity [7/sin a if t/ is the velocity of this 

 shock relative to the fluid ahead of it. It acquires a velocity u normal 

 to the front and the tangential velocity Uj, behind the front is therefore 



U . (U — u) -\- u cos^« 



Ux = ~ u sm a = -^ 



sm a sin a 



