In a previous paper [J. Acoust. Soc. Am.53, 719 (1973)] we discussed the calculation of the scattered field for scalar plane‐wave incidence on a type of filled periodic corrugated surface. The surface consisted of parallel half planes, alternate adjacent pairs of which were filled with a material having different density, and supporting waves of different velocity structure from those in the surrounding medium. This paper considers a related problem with plates of finite rather than semi‐infinite depth. Both soft‐ (Dirichlet) and hard‐ (Neumann) boundary conditions are considered. Limiting cases of the problems considered here correspond to the results of Deryugin. The most characteristic effects observed are the Brewster‐angle and resonance anomalies.