Fig. 1. 2D channel-modulated acoustic crystal and its dispersion property.

a Schematic of the 2D channel-modulated acoustic crystal and its unit-cell. b Photograph of the printed 2D channel-modulated sample. c Dispersion diagram of a supercell composed of 15 coupled cavities, terminated by a hard-wall boundary along the x-direction, and a periodic boundary condition along the z-direction. The topological pumping of edge states is represented by blue surfaces, whereas the bulk bands are indicated by gray surfaces. d Wave number $k_z$ as a function of the pumping parameter for frequency f = 4960 Hz, corresponding to the cross-section illustrated by the red horizontal sheet in c. The insets show the topological mode from left to right localization with the change of the pumping parameter $\varphi$ from $-0.2\pi$ to $0.2\pi$ along the z-direction. The color scale corresponds to the pressure amplitude. e, f Experimental e and numerical f demonstrations of topological edge pumping in the 2D crystal with the frequency at f = 4960 Hz. The color scale corresponds to the pressure amplitude.