Studies of the motion of a medium in the neighbourhood of a point of junction of shock waves in the linear and non-linear cases

Abstract

WE consider the problem of determining the solution of an arbitrary linear system of equations of hyperbolic type with three ( t, x, y) and four ( t, x, y, z) independent variables in the neighbourhood of a point (or line) of junction of a wave of arbitrary profile from a point of diffraction wave. A solution is found in terms of hypergeometric functions. We also obtain simplified non-linear equations describing the motion of the medium in the wave region indicated, and a solution of them is given for the plane problem. As an example the non-linear equations close to a wave in magnetogasdynamics are obtained. In this paper we consider the problem of determining the parameters of the motion of a medium described by a system of quasi-linear hyperbolic differential equations, in the neighbourhood of the front of a weak shock wave. Simplified, non-linear equations, one-dimensional along a ray, and their solution for a compressible ideal fluid will be found in [1–4], the two-dimensional short-wave equations for a homogeneous, initially immobile fluid will be found in [5]. Particular solutions of the short-wave equation enabling the conditions at a weak shock wave to be satisfied approximately will be found in [6, 8]. In [9, 10] the two-dimensional equations close to the wave are derived for an inhomogeneous, initially immobile fluid, and a solution of them is obtained which satisfies the conditions at the shock wave approximately and ensures jointing with the linear solution at the exit from the wave region. In [11] the three-dimensional equations close to the wave were obtained for an inhomogeneous, initially moving fluid, and these were obtained by a somewhat different route in [12] where a particular solution of them satisfying the conditions on the shock wave and the passage to the linear solution were indicated. In [13] the equations in the neighbourhood of the wave were found for a homogeneous, initially immobile electrically conducting fluid in a magnetic field, and in dimensionless variables these are reduced to the equations of [5]. In [12, 14] the two-dimensional and three-dimensional non-linear equations close to the wave are found for an inhomogeneous moving electrically conducting fluid, and these are generalized to the motion of an arbitrary medium. Some problems of the attenuation of magnetogasdynamic waves are considered in [15]. In sections 1 and 3 of the present paper the determination of the structure of the solution of a linear hyperbolic system of equations in the neighbourhood of a characteristic or wave surface is considered for the plane and three-dimensional problem; in sections 2 and 4 the results of sections 1 and 3 are used to determine simplified non-linear equations of motion of the medium close to the wave. In the case of the linear solution, one-dimensional along the ray (the radial approximation), this problem was considered in [16, 17] for an arbitrary linear hyperbolic system of equations. Deviations from one-dimensionality are observed, for example, in the neighbourhood of a point (or line) B of junction of the wave AB (see Fig. 1) with a point or diffraction BB 1 [12, 18–20].