Macroseismic observations during various historical and recent earthquakes consistently show an increase of damage on narrow stripes located along strong lateral discontinuities, i.e., along contacts between two materials with very different rigidities. This paper presents a series of numerical investigations into this issue. The model consists of a semi-infinite planar, soft layer embedded in a stiffer bedrock, and impinged upon by vertically incident plane SH waves. Computations are based on a finite-difference scheme including anelastic attenuation. Two basic phenomena are observed on such simple lateral discontinuities: the well-known 1D (vertical) resonance, and an efficient wave diffraction from the discontinuity towards the softer side. They induce frequency-dependent amplification and a significant differential motion. A parameter study considering various impedance contrast and damping values shows that the amplification is primarily controlled by the impedance contrast, and to a lesser degree by sediment damping. The amplitude exhibits a slight maximum near the discontinuity, but always remains comparable (within 30%) to the 1D value. Differential motion is also controlled by the impedance contrast. It always exhibits a sharp peak in the immediate vicinity of the discontinuity, the level of which does not depend on sediment damping, and reaches significant values even for moderate incident motion. It is concluded that reported observations of increased damage near such geological structures are very likely connected with effects of differential motion on structures.