The study of the transverse vibration of thin rectangular plates has been mainly confined to plates of uniform thickness. There are few publications available on the vibration of plates with nonuniform thickness. These plates have application in various fields such as civil engineering and aerospace structures and are often found in high frequency acoustic transducers. Furthermore, it is sometimes possible to achieve minimum weight design of plates by having suitable variations in thickness. For the special case of a plate with two opposite edges simply supported and thickness discontinuities perpendicular to the simply supported edges, an analytical solution is possible using a dynamic flexibility method. Several examples are discussed.