The authors present a second order explicit finite-difference scheme for elastic waves in 2-D nonhomogeneous media. These schemes are based on integrating the equations of motion and the stress-free surface conditions across the discontinuities before discretizing them on a grid. As an alternative for the free-surface treatment, a scheme using zero density above the surface is suggested. This scheme is first order and is shown to be a natural consequence of the integrated equations of motion and is called a vacuum formalism. These schemes remove instabilities encountered in earlier integration schemes. The consistency study reveals a close link between the vacuum formalism and the integrated/ discretized stress-free condition, giving priority to the vacuum formalism when a material discontinuity reaches the free surface. The two presented free-surface treatments coincide in the sense of the limit (grid size --> 0) for lateral homogeneity at or near the free surface.