Droplet impacts with rough surfaces described by Fourier series are investigated assuming gas cushioning is negligible. For impacts leading to a contiguous contact patch, a mixed boundary value problem for the displacement potential is formulated by extending models of inertially dominated droplet impacts with a flat plate. For large times after impact, the contact line evolution for impacts with periodic rough substrates is found to tend to the contact line evolution obtained for a droplet impact with a flat plate vertically positioned at the average height of the rough substrate. For symmetric impacts with even substrate geometries represented by Fourier cosine series, the contact line evolution is given by a Schlömilch series in which the coefficients are related to the coefficients of the corresponding Fourier series. A method for determining whether secondary impacts occur for particular geometries is described and regime diagrams, which show the boundary of the region of substrate parameters associated with single contiguous impacts, are obtained. The loads associated with droplet impacts with periodic rough substrates are calculated and compared with the loads associated with impacts with a flat plate. As the height of the roughness increases, the load associated with an impact with a rough substrate may initially differ significantly from the flat-plate case, although the load on a flat plate is recovered in the limit of large time. The implications of the results for more general droplet impacts with roughness are discussed from both a theoretical and experimental standpoint.
