In the study of surface waves we make certain drastic assumptions. Although the assumption of isotropy is often approximately satisfied in practice, certain disagreements exist between theory and observation that indicate the necessity of discussing problems of dispersion under less restrictive and possibly more realistic assumptions. In recent years the propagation of elastic waves in anisotropic media has begun to receive some attention. The assumption that the physical properties of an elastic medium vary with repect direction at a fixed point in a medium characterises the medium as being anisotropic. There obvious are reasonable consideration grounds for the assumption that anisotropy may exist in the continents. An is that materials to deposited in water may settle in preferred orientation. Stoneley (1926), Biot (1952) and Tolstoy (1954) have considered the propagation of elastic waves in a system consisting of a liquid layer of finite depth overlying an isotropic half space. Stoneley (1957) considered the case of ocean layering. Abubaker and Hudson (1961) studied the dispersive properties of liquid overlying a semi-infinite homogeneous anisotropic (transversely isotropic) half space.