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HETEROGENEOUS SYSTEMS IN $d$ DIMENSIONS: LOWER SPECTRUM

Published online by Cambridge University Press:  03 November 2015

PAOLO AMORE
PAOLO AMORE*
Affiliation:
Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, Mexico email [email protected]
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Abstract

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The lower part of the spectrum of the Helmholtz equation for a heterogeneous system in a finite region in $d$ dimensions, where the solutions to the corresponding homogeneous system are known, can be systematically approximated by means of iterative methods. These methods only require the specification of an arbitrary ansatz and converge to the desired solution, regardless of the strength of the inhomogeneities, provided the ansatz has a finite overlap with it. In this paper, different boundary conditions at the borders of the domain are assumed, and some applications are used to illustrate the methods.

Keywords

heterogeneous systemsHelmholtz equation
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 2 , October 2015 , pp. 150 - 165
Copyright
© 2015 Australian Mathematical Society 

References

Adams, S. D. M., Craster, R. V. and Guenneau, S., “Bloch waves in periodic multi-layered acoustic waveguides”, Proc. R. Soc. Lond. A 464 (2008) 2669–2692; doi:10.1098/rspa.2008.0065.Google Scholar
Amore, P., “The string of variable density: perturbative and non-perturbative results”, Ann. Physics 325 (2010) 2679–2696; doi:10.1016/j.aop.2010.06.007.CrossRefGoogle Scholar
Amore, P., “The string of variable density: further results”, Ann. Physics 326 (2011) 2315–2355; doi:10.1016/j.aop.2011.04.016.CrossRefGoogle Scholar
Amore, P., “Exact sum rules for inhomogeneous drums”, Ann. Physics 336 (2013) 223–244; doi:10.1016/j.aop.2013.05.010.CrossRefGoogle Scholar
Amore, P., “Exact sum rules for inhomogeneous strings”, Ann. Physics 338 (2013) 341–360; doi:10.1016/j.aop.2013.05.011.CrossRefGoogle Scholar
Amore, P., “Exact sum rules for inhomogeneous systems containing a zero mode”, Ann. Physics 349 (2014) 253–267; doi:10.1016/j.aop.2014.06.019.CrossRefGoogle Scholar
Capdeville, Y., Guillot, L. and Marigo, J. J., “1D non-periodic homogenization for the seismic wave equation”, Geophys. J. Int. 181 (2010) 897–910; doi:10.1111/j.1365-246X.2010.04529.x.Google Scholar
Capdeville, Y. and Marigo, J. J., “Second order homogenization of the elastic wave equation for non-periodic layered media”, Geophys. J. Int. 170 (2007) 823–838; doi:10.1111/j.1365-246X.2007.03462.x.CrossRefGoogle Scholar
Castro, C. and Zuazua, E., “Low frequency asymptotic analysis of a string with rapidly oscillating density”, SIAM J. Appl. Math. 60 (2000) 1205–1233; doi:10.1137/S0036139997330635.CrossRefGoogle Scholar
Cioranescu, D. and Donato, P., An introduction to homogenization (Oxford University Press, Oxford, 1999).CrossRefGoogle Scholar
Ho, S. H. and Chen, C. K., “Free vibration analysis of non-homogeneous rectangular membranes using a hybrid method”, J. Sound Vib. 233 (2000) 547–555; doi:10.1006/jsvi.1999.2808.CrossRefGoogle Scholar
Horgan, C. O. and Chan, A. M., “Vibration of inhomogeneous strings, rods and membranes”, J. Sound Vib. 225 (1999) 503–513; doi:10.1006/jsvi.1999.2185.CrossRefGoogle Scholar
Jackson, J. D., Classical electrodynamics, 3rd edn (Wiley, New York, 1998).Google Scholar
Joannopoulos, J. D., Villeneuve, P. R. and Fan, S., “Photonic crystals: putting a new twist on light”, Nature 386 (1997) 143–149; doi:10.1038/386143a0.CrossRefGoogle Scholar
Kang, S. W. and Lee, J. M., “Free vibration analysis of composite rectangular membranes with an oblique interface”, J. Sound Vib. 251 (2002) 505–517; doi:10.1006/jsvi.2001.4015.CrossRefGoogle Scholar
Masad, J. A., “Free vibrations of a non-homogeneous rectangular membrane”, J. Sound Vib. 195 (1996) 674–678; doi:10.1006/jsvi.1996.0454.CrossRefGoogle Scholar
Nemat-Nasser, S., Willis, J. R., Srivastava, A. and Amirkhizi, A. V., “Homogenization of periodic elastic composites and locally resonant sonic materials”, Phys. Rev. B 83 (2011) 104103 ; doi:10.1103/PhysRevB.83.104103.CrossRefGoogle Scholar
Panasenko, G. P., “Homogenization for periodic media: from microscale to macroscale”, Phys. Atomic Nuclei 71 (2008) 681–694; doi:10.1134/S106377880804008X.CrossRefGoogle Scholar
Parnell, W. J. and Abrahams, I. D., “Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves”, Wave Motion 43 (2006) 474–498; doi:10.1016/j.wavemoti.2006.03.003.CrossRefGoogle Scholar
Rohan, E., “Homogenization of acoustic waves in strongly heterogeneous porous structures”, Wave Motion 50 (2013) 1073–1089; doi:10.1016/j.wavemoti.2013.04.005.CrossRefGoogle Scholar
Smyshlyaev, V. P., “Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization”, Mech. Mater. 41 (2009) 434–447; doi:10.1016/j.mechmat.2009.01.009.CrossRefGoogle Scholar
Srivastava, A. and Nemat-Nasser, S., “Overall dynamic properties of three-dimensional periodic elastic composites”, Proc. R. Soc. Lond. A 468 (2012) 269–287; doi:10.1098/rspa.2011.0440.Google Scholar
Tantau, T., “The TikZ and PGF packages”, manual for version 3.0.0, http://sourceforge.net/projects/pgf/, 20 December 2013.Google Scholar
Tartar, L., The general theory of homogenization (Springer, Berlin–Heidelberg, 2009).Google Scholar
Wang, C. Y., “Some exact solutions of the vibration of non-homogeneous membranes”, J. Sound Vib. 210 (1998) 555–558; doi:10.1006/jsvi.1997.1270.CrossRefGoogle Scholar