Nonmonotonic fluctuation spectra of membranes pinned or tethered discretely to a substrate.

The thermal fluctuation spectrum of a fluid membrane coupled harmonically to a solid support by an array of tethers is calculated. For strong tethers, this spectrum exhibits nonmonotonic, anisotropic behavior with a relative maximum at a wavelength about twice the tether distance. The root-mean-square displacement is evaluated to estimate typical membrane displacements. Possible applications cover pillar-supported or polymer-tethered membranes.