Controlling Acoustic Wave With Radially-symmetric Gradient-index System

In recent years, extensive attentions have been paid to the development and application of metamaterials. Particularly, considerable efforts have been dedicated to the gradient index (GRIN) media in which the refractive index varies from point to point. By tailoring the parameters of GRIN media, various interesting optical phenomena can be realized, e.g.,bending, convergence and collimation of light and optical GRIN lens. The wave equation governing the propagation of acoustic wave-another classical wave-exhibits surprisingly-high mathematical similarity with that of its electromagnetic counterpart. Inspired by the theoretical principle and application potential of optical GRIN lenses, the concept of acoustic GRIN lens has been proposed. It has been extensively proven that acoustic GRIN media can be realized by properly selecting the structural parameters of sonic crystals (SCs) in the direction transverse to the acoustic propagation or employing acoustic metamaterials formed by coiling up space. By carefully manipulating structural parameters on the direction of the sound propagation path, one can simply regulate acoustic GRIN to reasonable control the refractive index distribution of the sonic crystal. In the absence of the resonance element, this method can avoid the loss effect of acoustic energy. It is therefore expectable that acoustic GRIN systems should also have the potential to manipulate the acoustic waves in a variety of ways, which would have deep implications for acoustic devices, acoustic applications and the field of acoustics in general.Despite the intensive investigation on the acoustic GRIN media, the primary attentions are only focused on their ability of designing acoustic lenses. Although the possibility of using a radially-symmetric GRIN shell to guide the incident acoustic wave has been studied, only a highly-limited case is considered where the refractive index distribution along the radial directional exactly follows an inverse-square law. So far, the potential of acoustic GRIN system to yield various kinds of acoustic manipulations has not yet been explored fully. Furthermore, a detailed theoretical description to the wave propagation in acoustic GRIN systems still lacks.In this paper, we have proposed an acoustic gradient-index system with radial symmetry and investigated the wave propagation in such a system numerically. Geometric acoustic theory is employed to predict the acoustic trajectory in the system. Then we perform finite-element-method-based simulations to verify the validity of the geometric acoustic theory numerically, and the numerical results are proven to agree well with the geometric ones. The numerical results demonstrate the potential of the proposed system to yield various kinds of manipulations on acoustic waves such as acoustic bending, trapping and absorbing, which can be tuned by tailoring the distribution function of refractive index. The proposed system comprises radially-symmetric positive-index medium only, and thus has an omnidirectional and broadband functionality. With the flexibility of controlling acoustic waves in different ways, the designed system may find applications in various situations where special manipulations of acoustic waves are required.This article is divided into the following four sections.In the first chapter, we review the research background and the latest developments of the phononic crystal, acoustic metamaterial and the acoustic gradient index metamaterial.In the second chapter, we introduce several methods to calculate the acoustic wave propagation in the material. We begin with the solid state theory and then introduce some basic knowledge of the crystal structure. Based on this, the equation governing the propagation of acoustic wave in the artificial structure has been given. Finally, several methods used to calculate the acoustic wave propagation in the material, including the plane wave expansion method, finite element method and multiple scattering methods, have been introduced.In the third chapter we briefly describe the acoustic gradient index materials, esp., the concepts and mechanisms in such materials. Starting from the Einstein field equations, acoustic wave equation describing the propagation trajectory in the gradient index materials has been derived. Then we use the finite element method for the numerical calculation for validation. Next we strictly deduce the equation of the wave propagation in the material gradient index. The acoustic fields in several materials with typical distributions of refractive indices have been calculated via theoretical analysis. The results of the theoretical analysis and the numerical simulation agree with each other perfectly.In the fourth chapter,by solving the equation of the acoustic wave propagation trajectory, we obtained a special solution describing the interesting phenomenon of acoustic trapping, which has been verified by using the finite element method and the multiple-scattering method.In the fifth chapter, we present the main conclusions of this paper as well as prospects for future work.The main innovations of this paper are listed as below:1 We present scheme of designing acoustic gradient lens at will. Compared with the traditional acoustic gradient lens, this structure is more flexible and has more versatile functionality of acoustic manipulation.2 From the analogue between the acoustic propagation in inhomogeneous media and motion of massless particle in curved spacetimes caused by gravitational fields, acoustic GRIN system is proposed to mimic an arbitrary "potential well" for acoustic waves. We depart from the Einstein field equation, and derive strictly the propagation trajectory of acoustic wave in gradient index materials. An interesting special solution has been found on this basis. This particular solution describes the acoustic waves propagate along the border of the system with a particular distribution of gradient index, the center will focus on acoustic gradient refractive index of the material point of symmetry in a circular motion. This phenomenon is then verified via numerical simulation.3 The wave equation for describing the propagation of acoustic wave in graded-index material has been derived. The acoustic field has been calculated by using the finite element method and derived theoretical formula. Excellent agreement has been observed between the theoretical predictions and the numerical results.