The interaction of shock waves with viscosity is one of the central problems in the supersonic regime of compressible fluid flow. The motion of weak shock waves in a highly viscous medium has been investigated by Chester-Chisnell- Whitham method. It is found that the shock velocity and shock strength both decrease as shock advances for the low viscous region of a medium to the high viscous region. The dependence of Mach number on propagation distance, Pressure, Particle velocity, and shock velocity as well as on adiabatic index has also been analyzed for the cases. In the last comparison between the results obtained here are made mutually as well as with those obtained by other methods. The obtained expansions are computed and discussed through tables and graphs.