Special case of the density distribution behind a nonstationary shock wave (Q1149334)

The density distribution behind a nonstationary shock wave of arbitrary shape is discussed for a definite value of the Mach number. The jump conditions of the physical quantities at the shock wave and the kinematic condition of consistency of second order are set up in a Lagrangian coordinate system. For a definite value of the shock wave Mach number the determinant of the resulting system of two equations vanishes. This leads to a connection between the second and first derivatives of the density along the normal behind the shock wave at that point of the wave sphere the special Mach number, mentioned above, exists. Therefore the density in the neighbourhood of the shock wave at this point can be expressed by a series expansion up to third order quantities. The calculation of the special Mach number belonging to it is described. Investigations are carried out for plane shock waves moving into a homogeneous medium or into a medium with a linear density distribution as well as for cylindrical and spherical shock waves.