  We investigate the propagation of Rayleigh waves in a half-space coupled to a
nonlinear metasurface. The metasurface consists of an array of nonlinear
oscillators attached to the free surface of a homogeneous substrate. We
describe, analytically and numerically, the effects of nonlinear interaction
force and energy loss on the dispersion of Rayleigh waves. We develop
closed-form expressions to predict the dispersive characteristics of nonlinear
Rayleigh waves by adopting a leading-order effective medium description. In
particular, we demonstrate how hardening nonlinearity reduces and eventually
eliminates the linear filtering bandwidth of the metasurface. Softening
nonlinearity, in contrast, induces lower and broader spectral gaps for weak to
moderate strengths of nonlinearity, and narrows and eventually closes the gaps
at high strengths of nonlinearity. We also observe the emergence of a spatial
gap (in wavenumber) in the in-phase branch of the dispersion curves for
softening nonlinearity. Finally, we investigate the interplay between
nonlinearity and energy loss and discuss their combined effects on the
dispersive properties of the metasurface. Our analytical results, supported by
finite element simulations, demonstrate the mechanisms for achieving tunable
dispersion characteristics in nonlinear metasurfaces.
