A perforated metal plate covered with an adhered thin elastic sheet has been studied as a possible anechoic surface for underwater sound. Assuming that those portions of the sheet covering holes in the plate behave as driven damped circular membranes clamped at their edges, expressions for the acoustic impedance of a single membrane and the specific acoustic impedance of the surface have been derived in terms of the tension in the membrane, the number of holes per unit area, the hole size, and damping constant. The results of calculations showing how the impedance varies as these parameters are changed and the possibility of obtaining an impedance match with water, using available materials, are discussed. [This work supported by the U. S. Office of Naval Research.]