B. John, D. Pendry, . Schurig, R. David, and . Smith, Controlling electromagnetic fields, science, vol.312, issue.5781, pp.1780-1782, 2006.

M. J. Zheng, J. J. Xiao, and K. W. Yu, Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains, Journal of Applied Physics, vol.106, issue.11, p.113307, 2009.

M. Lu, L. Feng, and Y. Chen, Phononic crystals and acoustic metamaterials, Materials Today, vol.12, issue.12, pp.34-42, 2009.

D. M. Mead, WAVE PROPAGATION IN CONTINUOUS PERIODIC STRUCTURES: RESEARCH CONTRIBUTIONS FROM SOUTHAMPTON, 1964?1995, Journal of Sound and Vibration, vol.190, issue.3, pp.495-524, 1996.

Y. Xu, X. Zhou, W. Wang, L. Wang, F. Peng et al., On natural frequencies of non-uniform beams modulated by finite periodic cells, Physics Letters A, vol.380, issue.40, pp.3278-3283, 2016.

J. Wen, G. Wang, D. Yu, H. Zhao, and Y. Liu, Theoretical and experimental investigation of flexural wave propagation in straight beams with periodic structures: Application to a vibration isolation structure, Journal of Applied Physics, vol.97, issue.11, p.114907, 2005.

N. Olhoff, B. Niu, and G. Cheng, Optimum design of band-gap beam structures, International Journal of Solids and Structures, vol.49, issue.22, pp.3158-3169, 2012.

S. Vladislav, J. J. Sorokin, and . Thomsen, Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties, Journal of Sound and Vibration, vol.347, pp.14-26, 2015.

Y. Liu, D. Yu, L. Li, H. Zhao, J. Wen et al., Design guidelines for flexural wave attenuation of slender beams with local resonators, Physics Letters A, vol.362, issue.5-6, pp.344-347, 2007.

Y. Xiao, J. Wen, and X. Wen, Broadband locally resonant beams containing multiple periodic arrays of attached resonators, Physics Letters A, vol.376, issue.16, pp.1384-1390, 2012.

Y. Xiao, J. Wen, D. Yu, and X. Wen, Flexural wave propagation in beams with periodically attached vibration absorbers: Band-gap behavior and band formation mechanisms, Journal of Sound and Vibration, vol.332, issue.4, pp.867-893, 2013.

Y. Xiao, J. Wen, D. Yu, and X. Wen, Flexural wave propagation in beams with periodically attached vibration absorbers: Band-gap behavior and band formation mechanisms, Journal of Sound and Vibration, vol.332, issue.4, pp.867-893, 2013.

Y. Xiao, J. Wen, G. Wang, and X. Wen, Theoretical and Experimental Study of Locally Resonant and Bragg Band Gaps in Flexural Beams Carrying Periodic Arrays of Beam-Like Resonators, Journal of Vibration and Acoustics, vol.135, issue.4, p.41006, 2013.

X. Wang and M. Y. Wang, An analysis of flexural wave band gaps of locally resonant beams with continuum beam resonators, Meccanica, vol.51, issue.1, pp.171-178, 2015.

Z. Shuguang, N. Tianxin, W. Xudong, and F. Jialu, Studies of band gaps in flexural vibrations of a locally resonant beam with novel multi-oscillator configuration, Journal of Vibration and Control, vol.23, issue.10, pp.1663-1674, 2015.

C. Yang and L. Cheng, Suppression of bending waves in a beam using resonators with different separation lengths, The Journal of the Acoustical Society of America, vol.139, issue.5, pp.2361-2371, 2016.

D. Yu, Y. Liu, H. Zhao, G. Wang, and J. Qiu, Flexural vibration band gaps in Euler-Bernoulli beams with locally resonant structures with two degrees of freedom, Physical Review B, vol.73, issue.6, 2006.

L. Liu and M. I. Hussein, Wave Motion in Periodic Flexural Beams and Characterization of the Transition Between Bragg Scattering and Local Resonance, Journal of Applied Mechanics, vol.79, issue.1, p.11003, 2011.

W. Yu-dianlong and . Yaozong, Flexural vibrations band gaps in periodic beams including rotary inertia and shear deformation effects, Qiu Jing Wang Gang, vol.19, p.1, 2006.

Y. Zhang, Z. Ni, L. Han, Z. Zhang, and L. Jiang, Flexural vibrations band gaps in phononic crystal Timoshenko beam by plane wave expansion method, Optoelectronics and advanced materials -rapid communications, vol.6, p.6, 2012.

Y. Zhang, Z. Ni, L. Jiang, L. Han, and X. Kang, Study of the bending vibration characteristic of phononic crystals beam-foundation structures by Timoshenko beam theory, International Journal of Modern Physics B, vol.29, issue.20, p.1550136, 2015.

L. Raghavan and A. S. Phani, Local resonance bandgaps in periodic media: Theory and experiment, The Journal of the Acoustical Society of America, vol.134, issue.3, pp.1950-1959, 2013.

B. Sharma and C. T. Sun, Local resonance and Bragg bandgaps in sandwich beams containing periodically inserted resonators, Journal of Sound and Vibration, vol.364, pp.133-146, 2016.

Z. Wang, P. Zhang, and Y. Zhang, Locally Resonant Band Gaps in Flexural Vibrations of a Timoshenko Beam with Periodically Attached Multioscillators, Mathematical Problems in Engineering, vol.2013, pp.1-10, 2013.

A. Søe-knudsen, Design of stop-band filter by use of curved pipe segments and shape optimization, Structural and Multidisciplinary Optimization, vol.44, issue.6, pp.863-874, 2011.

S. V. Sorokin and O. A. Ershova, Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading, Journal of Sound and Vibration, vol.278, issue.3, pp.501-526, 2004.

A. Hvatov and S. Sorokin, Free vibrations of finite periodic structures in pass- and stop-bands of the counterpart infinite waveguides, Journal of Sound and Vibration, vol.347, pp.200-217, 2015.

F. Errico, M. Ichchou, S. De-rosa, O. Bareille, and F. Franco, The modelling of the flow-induced vibrations of periodic flat and axial-symmetric structures with a wave-based method, Journal of Sound and Vibration, vol.424, pp.32-47, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02090712

E. D. Nobrega, F. Gautier, A. Pelat, and J. M. Dos-santos, Vibration band gaps for elastic metamaterial rods using wave finite element method, Mechanical Systems and Signal Processing, vol.79, pp.192-202, 2016.
URL : https://hal.archives-ouvertes.fr/hal-02453194

H. Zhang, Y. Xiao, J. Wen, D. Yu, and X. Wen, Flexural wave band gaps in metamaterial beams with membrane-type resonators: theory and experiment, Journal of Physics D: Applied Physics, vol.48, issue.43, p.435305, 2015.

C. W. Zhou, J. P. Lainé, M. N. Ichchou, and A. M. Zine, Wave Finite Element Method Based on Reduced Model for One-Dimensional Periodic Structures, International Journal of Applied Mechanics, vol.07, issue.02, p.1550018, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01716026

E. Deckers, S. Jonckheere, L. Van-belle, C. Claeys, and W. Desmet, Prediction of transmission, reflection and absorption coefficients of periodic structures using a hybrid Wave Based ? Finite Element unit cell method, Journal of Computational Physics, vol.356, pp.282-302, 2018.

G. Trainiti, J. J. Rimoli, and M. Ruzzene, Wave propagation in periodically undulated beams and plates, International Journal of Solids and Structures, vol.75-76, pp.260-276, 2015.

C. M. Da-fonseca, On the eigenvalues of some tridiagonal matrices, Journal of Computational and Applied Mathematics, vol.200, issue.1, pp.283-286, 2007.

Y. Hsue, A. J. Freeman, and B. Gu, Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals, Physical Review B, vol.72, issue.19, 2005.

A. Geslain, S. Raetz, M. Hiraiwa, M. Abi-ghanem, S. P. Wallen et al., Spatial Laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems, Journal of Applied Physics, vol.120, issue.13, p.135107, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01445395