  We present a model for wave propagation in a monolayer of spheres on an
elastic substrate. The model, which considers sagittally polarized waves,
includes: horizontal, vertical, and rotational degrees of freedom; normal and
shear coupling between the spheres and substrate, as well as between adjacent
spheres; and the effects of wave propagation in the elastic substrate. For a
monolayer of interacting spheres, we find three contact resonances, whose
frequencies are given by simple closed-form expressions. For a monolayer of
isolated spheres, only two resonances are present. The contact resonances
couple to surface acoustic waves in the substrate, leading to mode
hybridization and "avoided crossing" phenomena. We present dispersion curves
for a monolayer of silica microspheres on a silica substrate, assuming
adhesive, Hertzian interactions, and compare calculations using an effective
medium approximation to a discrete model of a monolayer on a rigid substrate.
While the effective medium model does not account for discrete lattice effects
at short wavelengths, we find that it is well suited for describing the
interaction between the monolayer and substrate in the long wavelength limit.
We suggest that a complete picture of the dynamics of a discrete monolayer
adhered to an elastic substrate can be found using a combination of the results
presented for the discrete and effective medium descriptions. This model is
potentially scalable for use with both micro- and macroscale systems, and
offers the prospect of experimentally extracting contact stiffnesses from
measurements of acoustic dispersion.
