Analytical solution of the problem of a convergent shock in gas for one-dimensional case

The analytical solution of the problem of a convergent shock in the vessel with an impermeable wall is constructed for the cases of planar, cylindrical and spherical symmetry. The negative velocity is set at the vessel boundary. The velocity of cold ideal gas is zero. At the initial time the shock spreads from this point into the center of symmetry. The boundary moves under the particular law which conforms to the movement of the shock. In Euler variables it moves but in Lagrange variables its trajectory is a vertical line. Equations that determine the structure of the gas flow between the shock front and the boundary as a function of time and the Lagrange coordinate as well as the dependence of the entropy on the shock wave velocity are obtained. Self-similar coefficients and corresponding critical values of self-similar coordinates were found for a wide range of adiabatic index. Thus, the problem is solved for Lagrange coordinates. It is fundamentally different from previously known formulations of the problem of the self-convergence of the self-similar shock to the center of symmetry and its reflection from the center which has been constructed for the infinite area in Euler coordinates.