  We investigate how Rayleigh waves interact with modulated resonators located
on the free surface of a semi-infinite elastic medium. We begin by studying the
dynamics of a single resonator with time-modulated stiffness. In particular, we
evaluate the accuracy of an analytical approximation of the resonator response
and identify the parameter ranges in which its behavior remains stable. Then,
we develop an analytical model to describe the interaction between surface
waves and an array of resonators with spatio-temporally modulated stiffness. By
combining our analytical models with full-scale numerical simulations, we
demonstrate that spatio-temporal stiffness modulation of this elastic
metasurface leads to the emergence of non-reciprocal features in the Rayleigh
wave spectrum. Specifically, we show how the frequency content of a propagating
signal can be filtered and converted when traveling through the modulated
medium, and illustrate how surface-to-bulk wave conversion plays a role in
these phenomena. Throughout this article, we indicate bounds of modulation
parameters for which our theory is reliable, thus providing guidelines for
future experimental studies on the topic.
