A theory is developed for the propagation of two-dimensional unattenuated waves in a system consisting of a liquid layer overlying an infinitely thick solid. Special attention is given to the interaction between the Stoneley type of wave and the Rayleigh wave. It is shown that the type of wave discussed corresponds to a dispersion branch for which the velocity varies continuously from a value lower than the velocity of sound in the liquid to that of the Rayleigh waves. The possible importance of this fact is pointed out in connection with the interpretation of the T phase of shallow-focus submarine earthquakes. The physical nature of these waves is illustrated by showing that they exist at the interface of a massless solid and an incompressible fluid.