Interaction of small perturbations with shock waves

Abstract

Interaction of hydrodynamic waves with plane shock waves under the linear approximation is studied. Two classical problems, namely, wave transformation and a rippling instability on a shock wave, are examined. Starting from 1945, these problems have been analyzed repeatedly. A number of enigmas have been revealed during more than half a century. We argue that some of them can be obviated by a revision of the existing approach to description of shock oscillations. These oscillations originate either due to incident perturbations or spontaneously (the latter case is relevant to the problem of shock stability) and are a first order of smallness. It is well known that the linear approximation enables solution of the problem on the unperturbed discontinuity surface, with the surface oscillations taken as a linearly independent mode. This mode signifies a transition from a local reference frame connected with the perturbed shock surface to a laboratory frame related to the unperturbed plane shock. It is traditionally supposed that two following kinematic effects are essential in this transition. One is a deformation of the surface, leading to small oscillations of the shock normal and shock tangent; the other is an additional velocity of the shock. We state that the conventional approach is incomplete and leads to certain methodical difficulties. In particular, within the conventional framework the shock oscillations do not satisfy the normal component of the Euler equation. Thus, this mode is not a partial solution of the hydrodynamic equations; therefore, it is not a linearly independent one. In order to avoid this difficulty and accomplish the description of the above transition, it is necessary to take into account the effect associated with noninertiality of the local reference frame. The transition into the noninertial frame corresponds to emergence of an inertial force field and additional pressure. The additional pressure is the field potential. It is of the first order of smallness and it influences the perturbed shock surface in the local frame, counterbalancing the effect of dynamic momentum flux oscillations. The physical ground of inertial force appearance is nonideality of the medium inside the thin front of a real shock wave. The role of this additional effect in the interaction of a small-amplitude wave with a plane shock is examined.