The finite-difference method which Peter D. Lax developed for treating unsteady inviscid flow fields is used to study the transient flow in the shock layer of a sphere that has been struck by a normal shock wave. Transient flow of this sort is encountered when a shock tube is used as a supersonic wind tunnel. Time histories of the shock detachment distance and the stagnation-point pressure and tangential velocity gradient are presented for ranges of the incident-shock Mach number and the perfect-gas specific-heat ratio. These results show that the stagnation-point pressure approaches the steady value much more rapidly than the shock detachment distance. In general, the stagnation-point pressure but more rapidly than the shock detachment distance. As the specific-heat ratio is decreased and the incident-shock Mach number is increased, the variation of the velocity gradient with respect to the shock detachment distance becomes more nearly linear.