Fig. 2. 3D channel-modulated acoustic crystal and its dispersion property.

a Schematic of the 3D channel-modulated acoustic crystal and its unit cell. b Photograph of the printed 3D channel-modulated sample. c Dispersion diagram of a supercell composed of $9\times 9$ coupled cavities, terminated by a hard-wall boundary along x and y directions, and a periodic boundary condition along the z-direction. Bulk modes are shown in gray, topological pumping of edge and corner modes are in blue and green, respectively. d Wave number $k_z$ as a function of the pumping parameter for fixed frequencies $f=7498\mathrm{Hz}$ (upper panel) and $f=6175\mathrm{Hz}$ (lower panel), corresponding to the cross-section illustrated by the red and purple planes in c. The inset in the upper panel shows the diagonal orbit with the phason value from $\left(-0.2\pi ,-0.2\pi \right)$ to $\left(0.2\pi ,0.2\pi \right)$. The insets below c and d show representative mode shapes for each type of mode. The color scale corresponds to the pressure amplitude.