Elastic wave propagation in a medium consisting of two finite layers is considered. Two types of solutions are treated. The first is a Rayleigh train of waves. It is seen that for this case, when the wavelength becomes short, the waves approach two Rayleigh waves plus a possible Stoneley wave. When the wavelength becomes large, there are two waves; i.e., a flexural wave and an axial wave. Calculations are presented for this case. The propagation of SH waves is treated, but no calculations are presented.