Theory of the Propagation of Shock Waves

Abstract

A new theory of propagation of one-dimensional shock waves is described. The partial differential equations of hydrodynamics and the Hugoniot relation between pressure and particle velocity are used to provide three relations between the four partial derivatives of pressure and particle velocity, with respect to time and distance from the source, at the shock front. An approximate fourth relation is set up by imposing a similarity restraint on the shape of the energy-time curve of the shock wave and by utilizing the second law of thermodynamics to determine, at an arbitrary distance, the distribution of the initial energy input between dissipated energy residual in the fluid already traversed by the shock wave and energy available for further propagation. The four relations are used to formulate a pair of ordinary differential equations for peak pressure and shock wave energy as functions of distance from the source. The theory takes proper account of the finite entropy increment of the fluid produced by the passage of the shock and permits the use of the exact Hugoniot curve of the fluid in the numerical integration of the basic equations.