The dynamics of elastic m~dia has been studied in the past mainly from the wave propagation point of view (Ewing et ai, 1957). This is probably because of the importance of such problems in fields like geophysics, wherein one mostly considers layers of infinite length or layers of infinite depth. However, while considering finite elastic media the approach of the theory of vibrations would be more informative and useful. As an example the inplane vibrations of a gravity dam idealized as a thin rectangular medium can be cited. Also in dynamic soil-structure interaction studies the foundation may sometimes be modelled as a two-dimensional elastic medium of finite depth and width, resting on a rock bed. In such problems probably the most desired results are the natural frequncies and 1)10de shapes for the free vibration of the two·dimensional medium. These can be used invariably in forced vibration studies. These modelling of soil deposits as elastic continua for purposes of dynamic analysis is not new. Previously Idriss (1968) and Idriss and Seed (1967, 1968, 1970) while studying the seismic response of soil layers, have suggested an one-dimensional analysis for semi-infinite layers. Clough and Chopra (1966) have ana’ lysed an earth dam section as a two-dimensional elastic medium by the finite element method. Eernisse (1966) has considered two-dimensional problems in dynamic elasticity for which closed form solutions are generally not available. Exact solutions may not be possible because of the complicated boundary conditions tbat are to be satisfied at the edges. Nevertheless, general expressions for the eigenfunctions would be most desirable since the forced response can be expanded as a series in terms of these functions.