On the behavior of solutions of equations for double waves in the neighborhood of the quiescent region : PMM vol. 39, n 6, 1975, pp. 1043–1050

The structure of solutions of gasdynamic equations is investigated in the case of unsteady double waves in the neighborhood of the quiescent region. A general concept of double waves is presented in the form of special series with logarithmic terms. Results of numerical computations are given.The problem of determining the flow of plane and three-dimensional waves separated from the quiescent region by a weak discontinuity was considered in [1–3], where approximate solutions were derived for that neighborhood, and the formulation of boundary value problems required for solving the equation for the analog of the velocity potential in the hodograph plane was investigated.The more general problem (without the assumption of the degeneration of motion) of arbitrary potential flows of polytropic gas adjacent to the quiescent region and separated by a weak discontinuity was considerd in [4–8]. Solution of that problem was obtained in the form of special series in powers of the mo dulus of the velocity vector r in the space of the time hodograph. The value r = 0 corresponds to the surface of weak discontinuity that separates the perturbed motion region from that at rest. Some applications of derived solutions to problems such as the motion of a convex piston and the propagation of weak shock waves were also investigated in those papers. Convergence in the small of obtained series was proved in [9]. However the attempts of constructing series in powers of r, which were used in [4–8] for the presentation of equations of double waves in the neighborhood of the quiescent region, proved to be unsuccessful.Although parts of expansions in series in powers of r (accurate to within 0 (r²)), were constructed in [1–3], it was found that the coefficient at r⁸ in equations for double waves cannot be determined owing to the insolvability of its equation. This is related to the fact that the surface r = 0in the case of equations for double waves is simultaneously a line of parabolic degeneration and a characteristic.The object of the present note is the formulation of solutions of equations for plane unsteady double waves in the neighborhood of the quiescent region. Parts of the derived series, which generally are nonanalytic functions of r, can be used for defining flows at small r in particular those downstream of two-dimensional normal detonation waves [10] or in problems of angular pistons [11]. The method used for the derivation of series can be also applied in investigations of threedimensional self-similar flows with variables x₁/x₃ and x₂/x₃ (steady flows) or x₁/t, x₂/t and x₃/t (unsteady flows). However it was not possible to obtain in such cases regular series in powers of r.