# Ultra-Broadband Bending Beam and Bottle Beam Based on Acoustic Metamaterials
## 4 Discussion  
In summary, we have designed a reflected metastructure, simply composed of an array of straight tubes with different spatially modulated depths, to generate a bending beam for sound within an ultra-broad band. The simulated fields via the acoustic metastructure agree well with the theoretical predictions as well as the fields generated by the ideal source. The design has also been extended for the generation of a three-dimensional bottle beam. The transverse trapping behaviours on small rigid objects by the bottle beam have been analyzed with the aid of the force potential. Our work will help the design and the development of broadband acoustic devices involving sound field manipulations, and also suggest the possibility to realize the precise particle manipulations along a curved trajectory.  
The metastructure proposed here has exhibited the performance of the generation of bending beams with the broadband feature in the audible frequency range. It is worth  
Figure 5: Particle manipulation based on the bottle sound beam. (**a**) Simulated three-dimensional bottle sound beam. (**b**) Normalized sound field \(|p|^{2}\) profile along the central beam axis. Normalized sound fields \(|p|^{2}\) and the force potential \(U\) in the transverse direction at (**c**) \(z\approx 5\lambda\) and (**d**) \(z\approx 25.5\lambda\). Insets: corresponding two-dimensional sound fields of \(x-y\) cross section.  
noting that the upper frequency limit could still be extended by reducing the width of each straight tube \(w\) so as to increase the cut-off frequency \(f_{0}=c_{0}/(2w)\) with \(w\) and \(c_{0}\) being the tube width and the background sound speed, respectively. The computation cost will increase for higher frequencies due to the smaller wavelengths and the smaller mesh size required for the simulations based on the finite element methods. When the frequency is too high, for example in the ultrasound frequency range, the thermoviscous effect caused by the structure needs to be considered by including the acoustic boundary layers during the calculation using the thermoviscous acoustic module. However, it might be difficult for the fabrication of the structure since the width of each individual tube is much smaller than the wavelength, and the wavelength is on the scale of millimeters. On the other hand, the applications on low frequencies would be easier because the lower frequency limit could be simply extended by increasing the whole length of the structure along the interface direction (\(y\) direction) so as to make sure the whole size of the metastructure is comparable to the working wavelength.  
Conceptualization, X.F.; methodology, X.F.; software, X.F.; validation, X.F., N.L.; investigation, X.F., X.H., Y.K. and C.L.; writing-original draft preparation, X.F.; writing-review and editing, X.F., N.L., X.H., Y.K., C.L., C.W.; visualization, X.F., N.L., C.W.; supervision, N.L., C.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Funds of the National Key Laboratory of Transient Physics (Grant No. 6142604200202), the Young Scientists Found of the Natural Science Foundation of Jiangsu Province (Grant No. BK20190439) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. 30919011258).  
Not applicable.  
The support from [PERSON] is acknowledged.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.