Abstract
Transformation method provides an efficient way to control wave propagation by materials. The transformed relations for field and material during a transformation are essential to fulfill this method. We propose a systematic method to derive the transformed relations for a general physic process, the constraint conditions are obtained by considering geometrical and physical constraint during a mapping. The proposed method is applied to Navier’s equation for elastodynamics, Helmholtz’s equation for acoustic wave and Maxwell’s equation for electromagnetic wave, the corresponding transformed relations are derived, which can be used in the framework of transformation method for wave control. We show that contrary to electromagnetic wave, the transformed relations are not uniquely determined for elastic wave and acoustic wave, so we have a freedom to choose them differently. Using the obtained transformed relations, we also provide some examples for device design, a concentrator for elastic wave, devices for illusion acoustic and illusion optics are conceived and validated by numerical simulations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Smith D.R., Pendry J.B., Wiltshire M.C.K.: Metamaterials and negative refractive index. Science 305, 788 (2004)
Cai X.B., Zhu R., Hu G.K.: Experimental study for metamaterials based on dielectric resonators and wire frame. Metamaterials 2, 220–226 (2008)
Liu Z.Y., Zhang X.X., Mao Y. et al.: Locally resonant sonic materials. Science 289, 1734 (2000)
Fang N., Xi D., Xu J. et al.: Ultrasonic metamaterials with negative modulus. Nat. Mater. 5, 452–456 (2006)
Lee S.H., Park C.M., Seo Y.M. et al.: Acoustic metamaterial with negative density. Phys. Lett. A 373, 4464–4469 (2009)
Yao S.S., Zhou X.M., Hu G.K.: Experimental study on negative effective mass in a 1D mass-spring system. New J. Phys. 10, 043020 (2008)
Bui H.D.: Fracture Mechanics: Inverse Problems and Solutions. Springer, Dordrecht (2006)
Greenleaf A., Lassas M., Uhlmann G.: On nonuniqueness for Calderon’s inverse problem. Math. Res. Lett. 10, 685–693 (2003)
Pendry J.B., Schurig D., Smith D.R.: Controlling electromagnetic fields. Science 312, 1780 (2006)
Leonhardt U.: Optical conformal mapping. Science 312, 1777 (2006)
Schurig D., Pendry J.B., Smith D.R.: Calculation of material properties and ray tracing in transformation media. Opt. Express 14, 9794–9804 (2006)
Schurig D., Mock J.J., Justice B.J. et al.: Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977 (2006)
Lai Y., Chen H.Y., Zhang Z.Q. et al.: Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell. Phys. Rev. Lett. 102, 093901 (2009)
Lai Y., Ng J., Chen H. et al.: Illusion optics: the optical transformation of an object into another object. Phys. Rev. Lett. 102(25), 253902 (2009)
Leonhardt U., Philbin T.G.: Transformation optics and the geometry of light. Prog. Opt. 53, 69–152 (2009)
Genov D.A., Zhang S., Zhang X.: Mimicking celestial mechanics in metamaterials. Nat. Phys. 5(9), 687–692 (2009)
Greenleaf A., Kurylev Y., Lassas M. et al.: Full-wave invisibility of active devices at all frequencies. Comm. Math. Phys. 275(3), 749 (2007)
Chen H., Chan C.T.: Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett. 91, 183518 (2007)
Zhang X., Liu Z.: Negative refraction of acoustic waves in two-dimensional phononic crystals. Appl. Phys. Lett. 85(2), 341 (2004)
Chen T., Weng C.N., Chen J.S.: Cloak for curvilinearly anisotropic media in conduction. Appl. Phys. Lett. 93, 114103 (2008)
Zhang S., Genov D.A., Sun C. et al.: Cloaking of matter waves. Phys. Rev. Lett. 100, 123002 (2008)
Milton G.W., Briane M., Willis J.R.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006)
Cummer S.A., Rahm M., Schurig D.: Material parameters and vector scaling in transformation acoustics. New J. Phys. 10, 115025 (2008)
Norris A.: Acoustic cloaking theory. Proc. R. Soc. A 464, 2411 (2008)
Hu, J., Liu, X.N., Hu, G.K.: Constraint condition on transformed relation for generalized-acoustics and optics (arXiv:0912.5462)
Zhou X.M., Hu G.K., Lu T.J.: Elastic wave transparency of a solid sphere coated with metamaterials. Phys. Rev. B. 77, 024101 (2008)
Farhat M., Guenneau S., Enoch S.: Ultrabroadband elastic cloaking in thin plates. Phys. Rev. Lett. 103, 024301 (2009)
Hu, J., Chang, Z., Hu, G.K.: Controlling elastic waves by transformation media (arXiv.1008.1641)
Lai W.M., Rubin D., Krempl E.: Introduction to Continuum Mechanics. 3rd edn. Butterworth–Heinemann, Burlington (1995)
Hu, J.: Coordinate transformations method based on deformation view and its application to invisibility design. PhD Thesis, Beijing Institute of Technology, Beijing (2009) (in Chinese)
Milton G.W., Willis J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. A 463, 855–880 (2007)
Brun M., Guenneau S., Movchan A.B.: Achieving control of in-plane elastic waves. Appl. Phys. Lett. 94, 061903 (2009)
Zheng, Y., Huang, X.: Earth Resources Laboratory 2002 Industry Consortium Meeting 1–18 (2002)
Milton G.W., Cherkaev A.V.: Which elasticity tensors are realizable?. J. Eng. Mater. Technol. 117, 483–493 (1995)
Hu J., Zhou X.M., Hu G.K.: Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation. Opt. Express 17, 1308 (2009)
Yan W., Yan M., Ruan Z. et al.: Coordinate transformations make perfect invisibility cloaks with arbitrary shape. New J. Phys. 10, 043040 (2006)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The project was supported by the National Natural Science Foundation of China (10832002), and the National Basic Research Program of China (2006CB601204).
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Chang, Z., Hu, J. & Hu, GK. Transformation method and wave control. Acta Mech Sin 26, 889–898 (2010). https://doi.org/10.1007/s10409-010-0386-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-010-0386-8