Magnetohydrodynamic Simple Waves

Abstract

The simple wave solutions, which in ordinary gas dynamics correspond lo expansion flows or Prandtl-Meyer flows are generalized here to ideal magnetohydrodynamic flows. The one-dimensional unsteady (x, t) case is considered. Due to magnetic effects more than one component of field and velocity must be considered, To carry out the simple wave formalism the equations of motion (continuity, momentum, induction) are written in terms of flow velocities (u_1, u_2), Alfvén velocities (b_1, b_2) and sound speed (a), These velocities are then functions only of the phase ξ = x_1 - U(ξ)t; each phase line can be thought of as an infinitesimal wave propagating with a speed c = U - u_1 related to the flow. By elimination of (u_1, u_2) the system of five first-order ordinary differential equations can be reduced to three (homogeneous) equations. The vanishing of the determinant of coefficients provides a famous relation for wave speed c and reduces the problem to integration of two first-order equations, The further introduction of dimensionless variables, ratios of wave speeds, reduces the problem to integration of a single first-order equation, By studying the trajectories of this differential equation an overall view of all possible solutions is obtained; numerical integration is also carried out in the case of slow waves. As applications of this theory various physical problems are studied, the receding piston and waves produced by a current sheet.

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