Only the fundamental assumption of a continuum flow with an infinitesimally thin shock is pertinent to this section. A fixed Cartesian coordinate system xi and its corresponding orthonormal basis |ˆ i are introduced. The shock wave surface, which may be in motion, is represented by
F = F(xi,t) = 0 (2.1)
Conditions just upstream and just downstream of the surface are denoted with subscripts 1 and 2, respectively. In a more conventional treatment, the upstream flow is uniform and steady, and the “just upstream” qualification is unnecessary. The velocity, in a laboratory frame, just upstream and downstream of the shock, is written as
V V x t j( , ) |ˆ , 1,2j j i k i,
= = (2.2)
where xk and t satisfy Equation (2.1), and i is summed over. The arbitrary sign of F is chosen so that
1 0⋅∇ ≥ (2.3)
for some region of the shock’s surface. For this region, the flow is primarily in the downstream direction.