  Synthetic dimensions can be rendered in the physical space and this has been
achieved with photonics and cold atomic gases, however, little to no work has
been succeeded in acoustics because acoustic wave-guides cannot be weakly
coupled in a continuous fashion. Here, we establish the theoretical principles
and for the first time manufacture acoustic crystals composed of arrays of
acoustic cavities strongly coupled through modulated channels to evidence
one-dimensional (1D) and two-dimensional (2D) dynamic topological pumpings. In
particular, the topological edge-bulkedge and corner-bulk-corner transport are
physically illustrated in finite-sized acoustic structures. We delineate the
generated 2D and four-dimensional (4D) quantum Hall effects by calculating
first and second Chern numbers and demonstrating robustness against the
geometrical imperfections. Synthetic dimensions could provide a powerful way
for acoustic topological wave steering and open up a new platform to explore
higher-order topological matter in dimensions four and higher.
