A two-dimensional nonhydrostatic version of the NCEP regional Eta Model together with analytic theory are used to examine flow over isolated mountains in numerical simulations using a step-terrain vertical coordinate. Linear theory indicates that a singularity arises in the steady flow over the step corners for hydrostatic waves and that this discontinuity is independent of height. Analytic solutions for both hydrostatic and nonhydrostatic waves reveal a complex behavior that varies with both horizontal and vertical resolution. Witch of Agnessi experiments are performed with a 2D version of the Eta Model over a range of mountain half-widths. The simulations reveal that for inviscid flow over a mountain using the step-terrain coordinate, flow will not properly descend along the lee slope. Rather, the flow separates downstream of the mountain and creates a zone of artificially weak flow along the lee slope. This behavior arises due to artificial vorticity production at the corner of each step and can be remedied by altering the finite differencing adjacent to the step to minimize spurious vorticity production. In numerical simulations with the step-terrain coordinate for narrow mountains where nonhydrostatic effects are important, the disturbances that arise at step corners may be of the same horizontal scale as those produced by the overall mountain, and the superposition of these disturbances may reasonably approximate the structure of the continuous mountain wave. For wider mountains, where perturbations are nearly hydrostatic, the disturbances above the step corners have horizontal scales that are much smaller than the overall scale of the mountain and appear as sharp spikes in the flow field. The deviations from the “classic” Witch of Agnesi solution are significant unless the vertical resolution is very small compared to the height of the mountain. In contrast, simulations with the terrain-following vertical coordinate produce accurate solutions provided the vertical grid interval is small compared to the vertical wavelength of the mountain waves (typically at least an order of magnitude larger than the mountain height).