Copyright by Matthew David Guild 2012 The Dissertation Committee for Matthew David Guild certifies that this is the approved version of the following dissertation: Acoustic Cloaking of Spherical Objects Using Thin Elastic Coatings Committee: Andrea Al`u,Supervisor Michael R. Haberman, Supervisor Mark F. Hamilton Preston S. Wilson Loukas F. Kallivokas Acoustic Cloaking of Spherical Objects Using Thin Elastic Coatings by Matthew David Guild, B.S.; M.S.E. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN May 2012 Dedicated to Melissa Acoustic Cloaking of Spherical Objects Using Thin Elastic Coatings Publication No. Matthew David Guild, Ph.D. The University of Texas at Austin, 2012 Supervisors: Andrea Al`u Michael R. Haberman In this thesis, a detailed description of acoustic cloaking is put forth using a coating consisting of discrete layers, enabling the cancellation of the scattered field around the object. This particular approach has previously only been applied to electromagnetic waves, for which it was observed that cloaking could be achieved using isotropic materials over a finite bandwidth. The analysis begins with a presen- tation of the theoretical formulation, which is developed using classical scattering theory for the scattered acoustic field of an isotropic sphere coated with multiple layers. Unlike previous works on acoustic scattering from spherical bodies, the cri- teria for acoustic cloaking is that the scattered field in the surrounding medium be equal to zero, and seeking a solution for the layer properties which achieve this condition. To effectively investigate this situation, approximate solutions are obtained by assuming either quasi-static limits or thin shells, which provide valuable insight into the fundamental nature of the scattering cancellation. In addition, using these approximate solutions as a guide, exact numerical solutions can be obtained, en- abling the full dynamics of the parameter space to be evaluated. Based on this v analysis, two distinct types of acoustic cloaking were found: a plasmonic cloak and an anti-resonance cloak. The plasmonic cloak is a non-resonant type of cloak, named plasmonic be- cause of its analogous behavior to the non-resonant cloak observed in electromag- netic waves which utilizes plasmonic materials to achieve the necessary properties. Due to the non-resonant behavior, this type of cloak offers the possibility of a much broader range of cloaking. To expand this design beyond wavelengths on the order of the uncloaked scatterer, multilayered cloak designs are investigated. The anti-resonance cloak, as the name suggests, uses the anti-resonances of the modes within the cloaking layer to supplement the non-resonant plasmonic cloaking of the scattered field. Although somewhat more limited in bandwidth due to the presence of anti-resonances (and the accompanying resonances), this type of cloak enables a larger reduction in the scattering strength, compared with using a single elastic layer utilizing only non-resonant cloaking. A thorough investigation of the design space for a single isotropic elastic cloaking layer is performed, and the necessary elastic properties are discussed. The work in this thesis describes the investigation of the theoretical for- mulation for acoustic cloaking, expanding upon the use of scattering cancellation previously developed for the cloaking of electromagnetic waves. This work includes a detailed look at the different physical phenomena, including both resonant and non-resonant mechanisms, that can be used to achieve the necessary scattering can- cellation and which can be applied to a wide range of scattering configurations for which cloaking would be desirable. In addition to laying out a broad theoretical foundation, the use of limiting cases and practical examples demonstrates the effec- tiveness and feasibility of such an approach to the acoustic cloaking of a spherical object. vi Table of Contents Abstract v List of Tables x List of Figures xi Chapter 1. Introduction 1 1.1 Thesis objectives . 1 1.2 Thesis overview . 3 Chapter 2. Background and fundamentals of acoustic cloaking 6 2.1 Transformation cloaks for acoustic waves . 7 2.1.1 Inertial cloak . 8 2.1.1.1 Basic formulation . 9 2.1.1.2 Use of acoustic metamaterials . 14 2.1.2 Pentamode materials and acoustic metafluids . 20 2.2 Anomalous resonance cloaking . 22 2.3 EM plasmonic cloaking . 24 2.3.1 Basic formulation . 25 2.3.2 Use of plasmonic materials . 28 2.3.3 Physical analogy between EM and acoustic waves . 30 Chapter 3. Theoretical formulation of acoustic scattering cancella- tion 33 3.1 Method of potentials . 34 3.2 Solution for scattering coefficients and scatter cancellation . 37 3.3 Relation to scattering cross-section . 41 3.4 Numerical implementation . 45 3.4.1 Minimization of scattered field . 45 3.4.2 Parameter space . 48 vii Chapter 4. Investigation of acoustic plasmonic cloaking using a single layer 50 4.1 Analytic expressions for a single fluid cloaking layer . 51 4.1.1 Low frequency approximation . 56 4.1.2 Thin shell approximation . 61 4.2 Cloaking of a rigid sphere . 65 4.3 Cloaking of a fluid sphere . 75 4.4 Cloaking of an isotropic elastic sphere . 80 4.5 Comparison with a single elastic cloaking layer . 89 Chapter 5. Investigation of acoustic anti-resonance cloaking 96 5.1 Formulation for a single elastic layer . 97 5.2 Anti-resonance cloaking of a rigid sphere . 102 5.2.1 Determining anti-resonance behavior . 102 5.2.2 Effects of elastic shear within the cloaking layer . 107 5.2.3 Limiting case of a fluid cloaking layer . 112 5.3 Anti-resonance cloaking of an elastic sphere . 112 5.4 Anti-resonance cloaking of a hollow sphere . 118 Chapter 6. Investigation of acoustic plasmonic cloaking using multi- ple layers 126 6.1 Exact analytic expressions . 126 6.2 Two fluid cloaking layers . 131 6.2.1 Region 1: ρc1 ρc2 . 135 6.2.2 Region 2: ρc1 ρc2 . 140 6.3 Cloaking of a rigid sphere using two fluid layers . 141 6.3.1 Region 1: ρc1 ρc2 . 142 6.3.2 Region 2: ρc1 ρc2 . 145 6.4 Elastic effects . 149 6.4.1 Penetrable elastic core . 149 6.4.2 Scattering strength reduction using two fluid layers . 151 6.4.3 Practical considerations for implementation . 154 6.5 Alternating fluid-fluid and fluid-elastic layers . 159 viii Chapter 7. Theoretical formulation for spherically isotropic elastic layers 166 7.1 Spherical isotropy . 168 7.2 Historical development of solution techniques . 172 7.3 State-space formulation . 174 7.4 Relation to the scattered field in an isotropic medium . 181 Chapter 8. Conclusions 191 8.1 General conclusions and contributions . 191 8.1.1 How the acoustic field scattered from a spherical object can be significantly reduced or cancelled . 191 8.1.2 How a realizable cloak is achieved . 192 8.2 Suggestions for future work . 196 Appendices 199 Appendix A. Linear system of equations derivation for scattering from isotropic sphere coated with isotropic shells 200 A.1 Basic formulation . 200 A.2 Solution for a single layer coated sphere . 201 A.3 Solution for an elastic sphere coated with two fluid layers . 205 A.4 Solution for a submerged sphere covered with a multilayer coating . 207 A.4.1 Surrounding fluid medium . 207 A.4.2 Multilayer coating . 207 A.4.3 Core material . 209 Appendix B. Derivation of expressions containing products of spher- ical Bessel functions of the first and second kind 211 Appendix C. Verification of results using COMSOL multiphysics 214 Index 217 Bibliography 219 ix List of Tables 4.1 Properties of elastic spheres to be cloaked for the three examples considered in this section. 80 4.2 Properties of single fluid layer plasmonic cloak of an elastic core for ka = 0:5, 0:75, and 1:00. 81 4.3 Properties of single fluid layer plasmonic cloak of a fluid core for ka = 0:5, 0:75, and 1:00. 82 5.1 Material properties of a single elastic cloaking layer designed at kd;0a= b 1:0 with =1:30 for a rigid, immovable sphere. 104 a 5.2 Material properties of a single elastic cloaking layer designed at kd;0a= b 1:0 with = 1:30 for a rigid, immovable sphere, based on the first a anti-resonance of the n=1 mode. 108 5.3 Material properties of spheres to be cloaked in Sections 5.3 and 5.4. 115 b 5.4 Properties of a single elastic cloaking layer with = 1:30 for the a different core materials examined in Sections 5.3 and 5.4. 116 6.1 Cloaking layer properties for a steel sphere in water, coated by an acoustic plasmonic cloak consisting of two layers with shell thicknesses δ1 =δ2 =0:04 and a design frequency of kd;0a=2:0. Solutions are given based on analytic thin-shell expressions, exact solutions for the case of two fluid layers, and exact solutions for the case of a fluid inner layer and isotropic elastic outer layer with νc1 =0:3. 152 6.2 Cloaking layer properties for a four layer acoustic plasmonic cloak for a steel sphere in water at a design frequency of kd;0a = 2:0. The shell thicknesses δ of the four layers, in order of the outermost to innermost layer, are 0:0135, 0:0642, 0:0037 and 0:0046, respectively. Exact solutions obtained numerically for the case of four fluid layers, and the case of alternating fluid and isotropic elastic layers with νc1 = 0:3...................................... 161 x List of Figures 2.1 Illustration of the coordinate transformation between an undeformed space (left) and a symmetric deformed space (right).