Rayleigh waves on a superlattice stratified normal to the surface

We present a theory of Rayleigh waves on a superlattice when the surface is perpendicular to the alternating layers. The method consists of first Fourier analyzing the equations of motion in the direction perpendicular to the layers so as to include automatically the boundary conditions at the different interfaces. For each value of the wave vector $k$ perpendicular to the layers, this gives a set of wave vectors parallel to the layers, i.e., perpendicular to the surface. Then a surface wave is constructed from a superposition of these solutions. The frequencies of the surface Rayleigh waves are obtained by using the Fourier-analyzed conditions of vanishing stresses at the free surface. Numerical calculations are performed by limiting the number of Fourier components taken into account. For two relatively similar layers in the superlattice there are two different Rayleigh branches near the Brillouin-zone boundary which result from the folding of the dispersion curve for a Rayleigh wave on a homogeneous medium. The higher branch can disappear when the two layers differ significantly in their elastic properties. We present a few examples of the dispersion curves and of the associated displacement fields. We also present examples showing the variation of the gap between the two Rayleigh branches versus the parameters of the layers and their thicknesses.