Introduction

The theoretical description of non-steady shock propagation is difficult in general as it involves the integration of the non-linear conservation equations. However, if the shock wave is weak, the entropy change across the shock is small and the shock can be treated as an isentropic compression wave. For a weak shock propagating into a uniform or quiescent medium, the flow behind it is a simple wave flow which can be determined by initial or boundary conditions. The shock motion can then be obtained by fitting the shock to the known simple wave flow behind it. The weak shock theories described in this chapter are of this nature. In Chandrasekhar's (1943) method, the characteristic equation is first transformed into the kinematic wave equation which has a particular solution corresponding to a linear distribution of flow variables behind the shock. The decay of the shock is then obtained by fitting the shock to the solution of the flow behind it. In the method of Friedrichs (1948), a simple wave flow is assumed behind the shock. The shock motion is then determined by matching the shock strength to give the same state as that carried by the characteristic that intersects it. The weak shock theory of Whitham (1956) is also very similar in nature to that of Freidrichs' method, except Whitham used a modified acoustic solution for the flow behind the shock. By modified acoustic theory, the linear acoustic solution is “non-linearized” by replacing the acoustic trajectories by the characteristics. In so doing, the distortion of the wave front can now be realized, whereas linear acoustic theory would give a wave front of permanent form. Oswatitsch (1956) also gave a theory for weak shock propagation. He assumed, as in Chandrasekhar's solution, a linear profile behind the shock and obtained a perturbation solution of the flow equations in the neighborhood of the sonic point (u = 0, c = c 0) of the triangular shock profile. Since the profile is linear, the solution at the sonic point can be extended to the shock front. The shock strength is then matched to the solution behind it. The principal approximations used in all these weak shock theories are essentially similar. The difference lies in the description of the flow behind it and the method used to match the shock to the flow.

Introduction

The description of the propagation of non-steady shock waves requires the integration of the partial differential equations for the flow behind the shock subject to the boundary conditions at the shock and at some rear boundary. Since the shock velocity is not known, the numerical solution is not straight forward. There are practical problems where a complete description is not required, and only the variation of the shock strength with position is desired. Thus there is a need for approximate methods of solution for the shock front only. During the early 1940s, Kirkwood and Brinkley (1945) and Brinkley and Kirkwood (1947) developed a simple method to describe blast waves in water and air. They derived a simple “shock front evolution equation” which gave surprisingly good predictions for the blast wave decay in air and water. Brinkley and Kirkwood (henceforth referred to as BK) derived a pair of ordinary differential equations for the variation of the shock strength and the blast energy with shock radius from the conservation equations, the Rankine–Hugoniot conditions at the shock front, and the energy integral. The unique feature of the BK theory is the formulation of the blast energy integral. The blast energy is defined as the sum of the kinetic and internal energy between the shock front and a particle path which acts like a piston. The blast energy is then equated to the work done by the particle path. Since no mass crosses a particle trajectory, it is essentially a solid expanding piston that displaces the fluid ahead of it. When time is taken to infinity where all motion ceases, the total work done is manifested as the residual internal energy in the shocked fluid. The residual internal energy also corresponds to the energy dissipated by the shock wave. The BK theory recognizes the finite entropy increment of the fluid particle that crosses the shock wave. The energy dissipation reduces the available energy of the shock wave and results in its decay. Thus the BK theory addresses explicitly the mechanism of shock decay.

Descriptions of the BK method, described in the wartime NDRC reports and in the Physical Review paper by Brinkley and Kirkwood, are rather brief and difficult to follow.