The viscosity of the gas in the proximity gap between x-ray mask and silicon wafer causes damping of any membrane motion. As x ray is extended to future integrated circuit generations, this effect becomes increasingly significant, because the gap must be rapidly scaled down to permit resolution of finer lithographic features. Damping is much greater at a reduced gap. Damping can be beneficial in reducing unwanted fluctuations in the gap due to vibration, convection, or ambient sound. The longer settling time following gap adjustment, however, can be detrimental to the throughput of x-ray steppers. This article introduces simple hydrodynamic models, formulas, and numerical algorithms to calculate membrane response to adjustments in gap setting. It applies the analysis to a variety of mask formats, including advanced "pedestal" design. Time constants of several seconds are shown to result from gap setting < 10 mum, which may be needed for 0.12 mum proximity x-ray lithography.