Abstract: This paper investigates wave propagation and its interaction with discontinuities in media using numerical models which employ a finite difference method. The paper has two parts, the first of which considers wave propagation and scattering by simulating a compressive body source in a 2-D half-space containing a step or a slot on the surface, which represents a cliff or a rift valley that has dimensions of the order of the wavelength in the source. The second part of the paper reports a hybrid technique which combines a perturbation method and the finite difference method for wave propagation again caused by the body source in a 2-D half-space system but with a shallow surface obstacle, such as a smoothed ridge, with elastic constants slightly different from bulk material and the slops of the topography being shallow. This hybrid technique is also reported being applied to a half-space with a horizontal insert or void. Results obtained are of importance in both theoretical studies on wave propagation and interaction and applied researches on the analysis and isolation of the damage caused by seismic waves.