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. 2021 Aug 19;12:5028. doi: 10.1038/s41467-021-25305-z

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 3 — a, b Experimental a and numerical b observation of edge-bulk-edge pumping. The sound is injected at the bottom-left edge with the frequency at $f = 7498 Hz$ . c, d Experimental c and numerical d observation of corner-bulk-corner pumping. The sound is injected at the bottom-left corner with the frequency at $f = 6175 Hz$ . Note that the pumping parameter $ϕ (z)$ for each horizontal layer of these samples is evenly distributed from $- 0.2 π$ to $0.2 π$ along the z-direction. The color scale corresponds to the pressure amplitude.