Refraction of Plane Shock Waves

It is assumed that when a plane shock wave is incident on an interface between two gases of different densities $\ensuremath{\rho}$ and ${\ensuremath{\rho}}_{1}$, and different ratios of specific heats $\ensuremath{\gamma}$ and ${\ensuremath{\gamma}}_{1}$, respectively, a three shock configuration results, involving an incident shock, a reflected shock, and a transmitted shock. It is further assumed that in the various angular domains the pressure is constant. The Rankine-Hugoniot equations are used to formulate the following conditions: (a) the pressure across the interface is continuous and (b) the deflection of the flow caused by the incident and reflected waves is equal to that caused by the transmitted wave. Rational polynomial equations of the twelfth degree are obtained, the roots of which determine the position and strength of the reflected and transmitted waves as functions of the strength, angle of incidence of the incident wave, and three parameters characterizing the pair of gases involved. The solutions of these equations are studied as multiple branched functions of the five parameters. It is shown that one branch behaves similarly to the acoustic case, and it is suggested that this branch is the only physically realizable one. Relations are obtained between the strength of the incident shock, its angle of incidence, and the three parameters characterizing the pair of gases which determine the ranges of these parameters where real physically realizable solutions may exist. One of these relations shows that the configuration is impossible for angles of incidence corresponding to the angle of total reflection. The cases for which numerical computations were made are listed, and the method of computation is briefly described. These computations were planned and supervised by Mrs. Adele Goldstein and were carried out on the Eniac which was made available through the cooperation of the Army Ordnance department.