A new finite difference method is employed in obtaining numerical solutions for the complete Navier-Stokes equations. The physical problem treated is the time-dependent propagation of a one-dimensional disturbance, which is produced by the forward motion of a piston through a compressible, viscous, thermally conducting, perfect diatomic gas and its subsequent reflection at a material surface. Two values of disturbance strength are considered, including weak and strong shock waves. The quantities determined include the variation of gas velocity, pressure, density, and temperature as a function of time and distance. In addition, the heat transfer rate and the pressure, which are of considerable interest in the interpretation of shock tube phenomena, are determined at the surface during the period of reflection of the shock wave.