  We design, simulate, and experimentally characterize an acoustic metasurface
comprising of a 1D array of open, sound-hard, cavities, modulated with
beyond-nearest-neighbor (BNN) couplings in the form of additional connecting
cavities embedded beneath the surface. The hidden complex structure is realized
readily with additive manufacturing techniques (3D printing). The dispersive
properties of the supported localized acoustic surface waves are influenced by
competing power-flow channels provided by the BNN couplings, that generate
extrema in the dispersion spectra within the first Brillouin zone. The
structure supports negatively dispersing 'backwards' waves that we
experimentally verify. Such structures thereby provide a route to enhanced
acoustic sensing by acoustic metasurfaces.
