Non-monotonic 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 non-monotonic, 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. 
Rolf-Juergen Merath and Udo Seifert