Shock Interaction and the Formation of a Contact Discontinuity

In this work the classical problem of the interaction of two shock waves traveling in opposite directions is analyzed using modern methods of infinitesimal analysis. Two shock waves, a right-running shock and a left-running shock, are modeled as traveling waves in a one-dimensional, inviscid, ideal gas. The shock waves are represented by nonstandard Heaviside jump functions and are defined using infinitesimal analysis. It is shown that the interaction of the two shocks admits a solution that may be expressed using a third nonstandard Heaviside function which combines the two shocks downstream of the interaction and demonstrates that in general a contact discontinuity must exist between the two shock waves. Nonstandard analysis is applied to describe the jump functions and their derivatives. Nonstandard analysis is an area of modern mathematics that studies extensions of the real number system to number systems that contain both infinitesimal numbers and infinitely large numbers and provides a rigorous framework for infinitesimal analysis. It is assumed that the shock wave thicknesses occur on idealized infinitesimal intervals and that the nonstandard jump functions in the thermodynamic and kinematic parameters vary smoothly across these shock layers. The equations of motion are cast in nonconservative form and applied to derive unambiguous relationships between the nonstandard jump functions and their products for the flow parameters in the regions behind the two shock waves.