On the basis of the expanded multi-dimensional Virial equation, the gas equation of state (EOS) is expanded the series to the second term, so that the density effect of real gas can be considered. Using the basic formula of the shock wave and small parameter contained in the Virial equation, the explicit expressions of the gas density, pressure and velocity behind the shock wave changing with the shock wave velocity and the shock wave Mach number are obtained by the perturbation method. The results show that for the same Mach number of the shock waves, compared with the results by the ideal gas model, the pressure, velocity and density behind the waves are all lower, especially the density. With the increase of the Mach number and the density of the gas, the difference in results will be even greater. These relations are the proper corrections to the ideal gas, which reflects the influence of the volume and repulsion effects of the gas molecule, and are fit for the gases whose density is lower than 100 kg·m^-3. The formula of the shock wave for the ideal gas can be regarded as its zero-order approximation. It is very convenient to use these relations to analyze the properties of shock waves.

To reveal shock wave entrainment of particles on a wall, simulation is carried out for raising of single particle on a wall behind a shock wave perpendicular to the wall. The particle is assumed air-born(depart from the wall immediately) at initial time, and is acted by gravity, gas resistance, and Saffman force. The model equation is a combination of boundary layer equations of gases behind shock wave and ordinary differential equations describing movement of the particle. A single parameter method and a 4-order Rung-Kutta method are employed to solve boundary layer equations and particle movement equations, respectively. Calculated particle velocities and their tracks show that the basic dynamic of particle entraining is from Saffman force provided by strong shear flows in boundary layer. The raising height of particle is independent of intensity of the shock wave. It changes with size of the particle. Calculated results agree with experimental results in references. It validates the model and the assumptions.