  The recent exotic topological corner modes (CMs) in photonic higher-order
topological insulators with long-distance interactions have attracted numerous
attentions and enriched the physics than their condensed-matter counterparts.
While the next-nearest-neighbour (NNN) coupling appears between NNN lattice
sites unselectively, and the NNN coupling and their associated dynamics remains
elusive in acoustics due to the waveguide-resonator model, an analogy of
tight-binding model (TBM), drastically hinders its NNN coupling. Here, in
acoustics, we demonstrate selective NNN coupling-induced CMs in split-ring
resonators-based kagome crystal and observe dual-band CMs. Three types of CMs
are demonstrated in the first bulk gap which can be explained by TBM
considering NNN coupling and one type of CMs is observed in the second. All of
these findings are verified theoretically and experimentally which reveals rich
physics in acoustics, opening a new way towards tunable or multi-band
metamaterials design, and offering opportunities for intriguing acoustic
manipulation and energy localization.
