Numerical modeling and experiments on sound propagation through the sonic crystal and design of radial sonic crystal

NUMERICAL MODELING AND EXPERIMENTS ON SOUND PROPAGATION THROUGH THE SONIC CRYSTAL AND DESIGN OF RADIAL SONIC CRYSTAL ARPAN GUPTA NATIONAL UNIVERSITY OF SINGAPORE 2012 NUMERICAL MODELING AND EXPERIMENTS ON SOUND PROPAGATION THROUGH THE SONIC CRYSTAL AND DESIGN OF RADIAL SONIC CRYSTAL ARPAN GUPTA (B-Tech Indian Institute of Technology Delhi) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgements Firstly, I would like to thank the Supreme Lord, for giving me the intelligence and the ability to this work. Research work requires inspiration, knowledge, hard work, success in endeavor and many other resources. Therefore, I would like to acknowledge the mercy of the Supreme Lord to carry out this work. I hope I can use this gift for the service of mankind. I would like to express my gratitude to my supervisor Prof. Lim Kian Meng for his very helpful suggestions and feedbacks during my PhD. He spent lot of time with me teaching me various aspects in doing research. I have benefited in various aspects, such as in computational methods, being professional in research, writing technically, etc. I am also very grateful to my supervisor Prof. Chew Chye Heng for teaching me various aspects in experimental acoustics. Both of my supervisors gave me ample opportunity to be creative and to pursue my thoughts. They also gave valuable and timely suggestions to improve my work. I am also grateful to Prof. S.P. Lim and Prof. H. P. Lee for their comments during my oral qualifying exam. Their comments helped me to be more focused in my work. I would also like to express my deepest gratitude to my professor of numerical methods at IIT Delhi, whom I lovingly call as ‘Sir’. His contribution in my life is much more than numerical methods. He is the person who has brought some good qualities and good character in my life. He is an ideal example of a truly selfless person and a genuine i well wisher for others. I am very grateful for his wonderful teachings which have significantly transformed my life. I would also like to thank Dr. Sujoy Roy, for being a senior friend and mentor to me during my stay here at Singapore. He gave me lot of inspiration and shared with me his valuable experiences. I would also like to thank my friends Karthik, Ruchir, Dhawal etc. for their help, friendship and happiness they shared with me during my stay here in Singapore. I am also grateful to NUS to provide me with full research scholarship. Thanks to Dynamics lab to provide me with all the facilities to my work. I am also thankful to Mr. Cheng for his prompt help during my experiments. Thanks to HPC (High Performance Computing) for the computational resources to carry out the numerical modeling. I am also very thankful to my lab mates Tse Kwong Ming, Guo Shieffeng, Zhu Jianghua, Liu Yang, Thein etc, for their friendship and valuable discussions. Lastly, I would like to thank my parents, grandmother and brother for their support and patience during this work. Arpan Gupta ii Table of Contents Acknowledgements i Summary vi List of figures . viii List of symbols . xiv Chapter 1. Introduction . 1.1 Periodic structures and band gaps . 1.2 Motivation . 1.3 Objective of the thesis . 1.4 Organization of the thesis . 1.5 Original contribution of the thesis 10 1.6 Acoustic wave propagation . 11 Chapter 2. Literature Review 17 2.1 Sound insulation 18 2.2 Frequency filters and acoustic waveguides . 20 2.3 Metamaterials and radial wave crystal 21 2.4 Other applications . 23 2.5 Numerical Methods for calculation and optimization of the band gap . 24 2.6 Evanescent wave . 26 iii 2.7 Webster horn equation 27 Chapter 3. One dimensional model for sound propagation through the sonic crystal.29 3.1 Computation of band structure 29 3.2 Complex frequency band structure and decay constant 35 3.3 Sound attenuation by the sonic crystal using the Webster horn equation . 39 3.4 Conclusion 48 Chapter 4. Validation of 1-D model by experiment and finite element simulation . 50 4.1 Experiment 50 4.2 Finite element simulation 55 4.2.1 Validation of finite element simulation with published work 59 4.2.2 Mesh convergence study . 61 4.3 Parametric study on rectangular sonic crystal 63 4.4 Conclusion 68 Chapter 5. Quasi 2-D model for sound attenuation through the sonic crystal 69 5.1 Quasi 2-D model for sound propagation through the sonic crystal 70 5.2 Decay constant for the sonic crystal using the quasi 2-D model 75 5.3 Conclusion 78 Chapter 6. 6.1 Radial sonic crystal . 80 Sound propagation in two dimensional waveguide with circular wavefront 81 iv 6.1.1 Problem Definition 83 6.1.2 Numerical Formulation . 84 6.1.3 Validation of the 1-D model with the finite element simulation 88 6.2 Analysis of an intuitive radial sonic crystal 93 6.3 Design of periodic structure in cylindrical coordinates 95 6.4 Sound attenuation by the radial sonic crystal . 99 6.5 Conclusion 101 Chapter 7. Experiment and finite element simulation on the radial sonic crystal 104 7.1 Experiment 104 7.2 Finite element simulation 108 7.2.1 Mesh convergence test 109 7.3 Results . 111 7.4 Conclusion 117 Chapter 8. Conclusion and future direction of work 118 8.1 Conclusion 118 8.2 Future direction of work . 123 References . 126 Publications . 140 v Summary Sound propagation through a rectangular sonic crystal with sound hard scatterers is modeled by sound propagation through a waveguide. A 1-D numerical model based on the Webster horn equation is proposed to obtain the band structure for sound propagation in the symmetry direction of the rectangular sonic crystal. The model is further modified to obtain the complex dispersion relation, which gives the additional information of decay constant of the evanescent wave. The decay constant is used to predict the sound attenuation over a finite length of the sonic crystal in the band gap region. Alternatively, sound transmission over the finite length of sonic crystal can be directly obtained using the Webster horn equation. Theoretical results from the model are compared with the finite element simulation and experiment. The model developed is used to perform a parametric study on the various geometrical parameters of the rectangular sonic crystal to find optimal design guidelines for high sound attenuation. It is found that a particular kind of rectangular structure is better suited for sound attenuation than the normal square arrangement of scatterers. The 1-D numerical model is further extended to a quasi 2-D model for sound propagation in a waveguide. The assumption in the 1-D model was due to the Webster horn equation, which assumes a uniform pressure across the cross-section of a waveguide. Quasi 2-D model is derived from the weighted residual method and Helmholtz equation, to include a parabolic pressure profile across the cross-sectional area of the waveguide. This quasi 2-D model for sound propagation in a waveguide is used to vi obtain band structure of the sonic crystal and to obtain sound attenuation over a finite length. The results match well with the 2-D finite element simulation and experimental results. The quasi 2-D model also shows significant improvement over the 1-D model based on the Webster horn equation. It is also shown that Webster horn equation is a special case of the quasi 2-D model. Lastly, radial sonic crystal is envisioned and a numerical model is proposed to obtain its design parameters. Most of the sound sources generate pressure waves which are non-planar in nature. Instead of scatterers arranged in square lattice with a plane wave propagating through it, scatterers are arranged in radial coordinates to attenuate sound wave with circular wavefront. Sound propagation through such sonic crystal is modeled by an equation for sound propagation through radial waveguide. Although such a structure may not be physically periodic (i.e. a unit cell by simple translation can form the whole structure), but such a structure is mathematically periodic by implementing the property of invariance in translation on the governing equation. Such periodic structure in radial coordinates, are termed as radial sonic crystal. Based on the design from the numerical model, finite element simulations and experiments are performed to obtain sound attenuation for radial sonic crystal. The results are in good agreement and it shows a significant sound attenuation by radial sonic crystal in the band gap region. vii List of figures Figure 1-1 Different types of sonic crystals. (a) 1-D sonic crystal consisting of plates arranged periodically (b) 2-D sonic crystal with cylinders arranged on a square lattice (c) 3-D sonic crystal consisting of periodic arrangement of sphere in simple cubic arrangement . Figure 1-2 An example of band gaps for sonic crystal represented by the shaded region. Figure 1-3 First experimental revelation of the sonic crystal was found by an artistic structure designed by Eusebio Sempere in Madrid . Figure 3-1(a) A two dimensional periodic structure made of circular scatterers arranged on a square lattice. On the left side there is plane wave sound source. The dotted square shows a unit cell. (b) Magnified view of a unit cell with various geometric parameters. 30 Figure 3-2 Band gap for an infinite sonic crystal corresponding to Fig. 3.1 with a = 4.25 cm and d = cm, along the symmetry direction ΓΧ . . 34 Figure 3-3 Complex band structure for an infinite sonic crystal. (a) normal band structure. (b) Decay constant as a function of frequency. The decay constant is non-zero in the band gap regions. . 38 Figure 3-4 Sound attenuation predicted by the decay constant. . 39 Figure 3-5 (a) Sound propagating over a sonic crystal consisting of five layers of scatterers. 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Pierce, A.D., Acoustics : an introduction to its physical principles and applications / Allan D. Pierce. McGraw-Hill series in mechanical engineering. 1981, New York: McGraw-Hill Book Co. 642. 139 Publications 1. A. Gupta, K.M. Lim, C.H. Chew, Analysis of frequency band structure in onedimensional sonic crystal using Webster horn equation, Applied Physics Letters, 98 (2011). 2. A. Gupta, K.M. Lim, C.H. Chew, Parametric Study on Rectangular Sonic Crystal, Applied Mechanics and Materials, 152 – 154 (2012) 3. A. Gupta, C.H. Chew, K.M. Lim, Effect of periodic structure on sound propagation, Fourth International Conference on Experimental Mechanics, 7522 (2010). 4. A. Gupta, K.M. Lim, C.H. Chew, A quasi two dimensional model for sound attenuation by the sonic crystals, Journal of the Acoustical Society of America (2012) (accepted). 5. A. Gupta, K.M. Lim, C.H. Chew, Radial sonic crystal – periodic structures in polar coordinates (under review). 140 [...]... intuitive design of radial sonic crystal The mathematical background for the design of radial sonic crystal is presented Chapter 7 presents the experiments and simulations performed to test the design of radial sonic crystal 9 1.5 Original contribution of the thesis In this thesis, firstly a numerical model based on the Webster horn equation is presented for sound propagating through the symmetry direction... directions of wave 3 propagation is known as a complete band gap However, in the present work, wave propagation is considered along one of the symmetry directions The details of the band gap are discussed in chapter 3 Figure 1-2 An example of band gaps for sonic crystal represented by the shaded region One major difference between the periodic structure in the photonic crystal and in the sonic crystal. .. direction in a sonic crystal For an infinite periodic structure, band gaps are obtained The method is further modified to obtain the complex dispersion relation, which gives additional information of the decay constant of the evanescent wave in the band gap region The decay constant can be used to predict sound attenuation over a finite length of the sonic crystal The sound transmission by the sonic crystal. .. within the band gap by introducing a defect in the sonic crystal Thus such structure can be used as a frequency filter 2.1 Sound insulation Since the sound wave is inhibited to propagate through the sonic crystal in the band gap region, one of the direct applications of sonic crystal is in selective sound reduction Multilayer partition based on periodic arrangement of layers showed the properties of one... lattice 2 The solution of the Bloch wave for periodic potential leads to the formation of bands of allowed and forbidden energy regions The allowed energy region is known as conduction and valence band, whereas, the forbidden band of energies where there is no solution for the Bloch wave is known as the band gap These band gaps are quite common in semiconductor materials and they form the basis of all... a new concept, and such kind of sonic crystals can help in sound attenuation from a point or a line source 7 1.3 Objective of the thesis The main objective of this thesis is to develop numerical models for obtaining sound attenuation through the sonic crystal and validate them with the experiment and finite element simulations A one dimensional numerical model based on the Webster horn equation is presented... both, the complex band gap and sound attenuation by the sonic crystal The results predicted by the quasi 2-D model are in good agreement with the finite element simulations and experiments Lastly, a radial sonic crystal is envisioned and designed based on the mathematical principle of invariance in translation of a unit cell A governing equation for sound propagation in waveguide with circular wavefront,... introduction and motivation of studying the wave propagation through the periodic structures It also discusses about the basic equation for acoustic wave propagation in free space Chapter 2 describes some of the recent and past developments in the field of sonic crystals and some of its applications Chapter 3 presents a one dimensional model based on the Webster horn equation to predict the band gap and sound. .. wave propagating equation 97 Figure 6-10 A radial sonic crystal 99 Figure 6-11 (a) Sonic crystal made of circular scatterers based on intuitive design (b) Radial sonic crystal designed based on periodic condition 100 Figure 6-12 Sound attenuation as a function of frequency for radial sonic crystal and the intuitive structure made of circular scatterers of constant diameter ... situations The benefit of sonic crystal is that it can attenuate sound significantly (~ 30 dB) in a particular frequency band Also the property of selective sound attenuation by the sonic crystal can be useful in designing frequency filters The conventional method uses a partition or solid barrier But in sonic crystal, the sound attenuation is due to interaction of wave with the periodic structure The . NUMERICAL MODELING AND EXPERIMENTS ON SOUND PROPAGATION THROUGH THE SONIC CRYSTAL AND DESIGN OF RADIAL SONIC CRYSTAL ARPAN GUPTA NATIONAL UNIVERSITY OF SINGAPORE 2012 NUMERICAL. NUMERICAL MODELING AND EXPERIMENTS ON SOUND PROPAGATION THROUGH THE SONIC CRYSTAL AND DESIGN OF RADIAL SONIC CRYSTAL ARPAN GUPTA (B-Tech Indian Institute of Technology Delhi) A THESIS. through the sonic crystal 69 5.1 Quasi 2-D model for sound propagation through the sonic crystal 70 5.2 Decay constant for the sonic crystal using the quasi 2-D model 75 5.3 Conclusion 78 Chapter