Page:Threeshockwaves.pdf/1

Ballistic Research
Laboratory Report No. 367


Chandrasekhar/emh
Aberdeen Proving Ground, Md.
June 5, 1943

ON THE CONDITIONS FOR THE EXISTENCE OF THREE SHOCK WAVES

 

1. Statement of the problem. J. von Neumann has studied the problem of the reflection of shock waves by a rigid plane surface and finds that for a given intensity of the incident shock (as specified for instance by the ratio of the pressures $\zeta$ on the two sides of the shock front) regular refection can take place only for angles of incidence $\alpha$ less than a certain critical value $\alpha _{\mathrm {extreme} }(\zeta )$.

Fig. 1

Fig. 1

Moreover, for a value $\alpha <\alpha _{\mathrm {extreme} }(\zeta )$ two solutions are possible. It is, however, generally supposed that the solution with the smaller of the two possible values $\alpha '$ represents physically realizable solution. The reason for this supposition is that in the acoustic limit $\zeta =1$ the solution with the smaller $\alpha '$ passes continuously into what is believed to be true for the sound waves (namely $\alpha =\alpha '$).

The question arises as to what happens when $\alpha >\alpha _{\mathrm {extreme} }(\zeta )$. Von Neumann suggests that for angles of incidence of a shock wave greater than the critical value the Mach effect takes place (Fig. 2),