  Macroscopic two-dimensional sonic crystals with inversion symmetry are
studied to reveal higher-order topological physics in classical wave systems.
By tuning a single geometry parameter, the band topology of the bulk and the
edges can be controlled simultaneously. The bulk band gap forms an acoustic
analog of topological crystalline insulators with edge states which are gapped
due to symmetry reduction on the edges. In the presence of mirror symmetry, the
band topology of the edge states can be characterized by the Zak phase,
illustrating the band topology in a hierarchy of dimensions, which is at the
heart of higher-order topology. Moreover, the edge band gap can be closed
without closing the bulk band gap, revealing an independent topological
transition on the edges. The rich topological transitions in both bulk and
edges can be well-described by the symmetry eigenvalues at the high-symmetry
points in the bulk and surface Brillouin zones. We further analyze the
higher-order topology in the shrunken sonic crystals where slightly different
physics but richer corner and edge phenomena are revealed. In these systems,
the rich, multidimensional topological transitions can be exploited for
topological transfer among zero-, one- and two- dimensional acoustic modes by
controlling the geometry.
