Instability of Shock Waves in Inhomogeneous Gases

Abstract

The instability of an ideal-gas shock which propagates along the direction of density decrease, with respect to deformations of the shape of the shock front, is studied. A linearized ordinary differential equation which governs the disturbance is derived from an equation which gives the variation in shock strength through the density decrease of the gas and the divergence of the shock ray. Solutions of the equation are obtained in the case of plane shocks. The growth of deformation of a spherical shock-front propagating from the center of a star to the surface is discussed by using the solutions. It is shown that the shock can reach very near the surface before the shock front undergoes so remarkable deformation as to break up to small parts. The stability of hydrodynamic shocks in a homogeneous medium with respect to deformations of the shape of the shock front has been studied by a number of authors/'' 2' and it is known that plane shocks are stable in an ideal gas, that is, random deformation of the front decays with time. It is also known that cylindrically converging shocks in a homogeneous gas are unstable. 2' The insta­ bility is due to the increase of shock strength in the propagation. In an inhomogeneous gas, the increase of shock strength occurs even for a plane shock when it propagates toward the density decrease. In these circum­ stances, leading parts of the shock front travel faster than the others and lead more; hence, it can be conjectured that the plane shock is unstable. Skalafuris8l has studied this problem, using Whitham's. kinematic equations2l of a shock front. He has reduced the ordinary differential equations to linear ones by means of a hodograph transformation; the way is erroneous, however, since the equations for a shock in an inhomogeneous medium are not reducible;*' therefore, his results are not correct. *l Whitham's kinematic equations for two-dimensional shock propagation are 88 1 8A ap=Maa • Here, M is the Mach number of the shock; A is a measure of the shock-ray separation; the curves a:=constant denote the shock positions; the curves P=constant denote the shock-rays; 8 denotes the inclination angle of the shock-rays. In a homogeneous medium, M is a function of A alone and the equations can be reduced to linear ones by interchanging the roles of dependent and independent variables. In an inhomogeneous medium, however, M is a function of A and the medium density which depends on a and /1. Hence the equations are not reducible.