The Rankine–Hugoniot Equations: Their Extensions and Inversions Related to Blast Waves

Abstract

The Rankine-Hugoniot equations were developed in their original forms independently by Rankine (Philosophical Transactions of the Royal Society of London, 160, 277–286, 287–288, 1870) and Hugoniot (Journal de l’École Polytechnique (Paris), 57, 3–97; Journal de l’École Polytechnique (Paris), 58, 1–125). The equations describe the relationships between the physical properties in the two possible states of a moving compressible gas for which mass, momentum and energy are conserved. The Rankine-Hugoniot relationships have been in use for more than a century, and subject, therefore, to intense evaluation. All such evaluations have demonstrated the absolute validity of the relationships, assuming the correct value of the ratio of specific heats has been used. The most commonly used forms of the equations relate the hydrostatic overpressure, density and particle velocity to the shock Mach number.

In this chapter, these equations are extended to give eleven different physical properties in terms of the ratio of specific heats, γ, and for γ = 1.4. In the latter case, the equations are also inverted to provide the shock Mach number, and thus the other ten properties, in terms of a specified physical property.