A non-stationary analogue of Chaplygin's equations in one-dimensional gas dynamics

As an extension of the results obtained in [1], two equivalent uniformly divergent systems of equations are constructed in thespeedograph plane, each of which is the analogue of Chaplygin's equation in the hodograph plane. Each of the systems reduces to a linear second-order equation, in one case for the particle function (the Lagrange coordinate) ψ, and in the other for the time t. These systems possess an infinite set of exact solutions. It is shown that a uniformly divergent system of first-order equations correspond to each of these, and, related to them, the simplest non-linear homogeneous second-order equation in the modified events plane (ψ, t) and the conservation law in the events plant (x, t). Clear relations are obtained between the velocities of the fronts of constant values of the newly constructed dependent variables and the velocity of sound. Examples are given which demonstrate the relation between the exact solutions with the uniformly divergent equations and the conservation laws of one-dimensional non-stationary gas dynamics and, simultaneously, enable one to compare the newly obtained results (the exact solutions, the equations and conservation laws, and the relations for the velocities of the front) with existing results, including those for plane steady flows. The so-called additional conservation laws, to which Godunov drew attention, are considered.