Special case of the density distribution behind a nonstationary shock wave

Abstract

The density distribution behind a nonstationary shock wave for a definite value of the Mach number M_*, which depends on γ = c_p/c_v, is considered. Use is made of the previously established fact [1] that for M = M_*(γ) there exists a connection between the first and second derivatives of the density along the normal behind the wave. An investigation is made into the density profile in dimensionless variables behind plane, cylindrical, and spherical shock waves in the neighborhood of the shock front. In the first case, if the gas in front of the wave is homogeneous, only two types of density profile are possible (up to small quantities of third order in the coordinate). In the second and third cases, the form of the density distribution also depends on a parameter, the ratio of the first derivative along the normal of the density behind the wave to the radius of curvature of the wave.