A concise and efficient scattering matrix formalism for stable analysis of elastic wave propagation in multilayered anisotropic solids

This paper presents a concise and efficient scattering matrix formalism for stable analysis of elastic wave propagation in multilayered anisotropic solids. The formalism is capable of resolving completely the numerical instability problems associated with transfer matrix method, thereby obviating the extensive reformulation in its modified versions based on delta operator technique. In contrast to the earlier reflection matrix formalisms, all scattering matrices are obtained in a direct manner without invoking wave-propagator or scatterer operator concepts. Both local and global reflection and transmission matrices corresponding to scatterings in two and more layers are derived. The derivation of global scattering matrices in terms of the local ones is carried out concisely based on physical arguments to provide better insights into scattering mechanism. Another formulation which is even more succinct is also devised for obtaining the global scattering matrices directly from eigensolutions. The resultant expressions and algorithm are terse, efficient and convenient for implementation.     

Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media.

The numerical instability problem in the standard transfer matrix method has been resolved by introducing the layer stiffness matrix and using an efficient recursive algorithm to calculate the global stiffness matrix for an arbitrary anisotropic layered structure. For general anisotropy the computational algorithm is formulated in matrix form. In the plane of symmetry of an orthotropic layer the layer stiffness matrix is represented analytically. It is shown that the elements of the stiffness matrix are as simple as those of the transfer matrix and only six of them are independent. Reflection and transmission coefficients for layered media bounded by liquid or solid semi-spaces are formulated as functions of the total stiffness matrix elements. It has been demonstrated that this algorithm is unconditionally stable and more efficient than the standard transfer matrix method. The stiffness matrix formulation is convenient in satisfying boundary conditions for different layered media cases and in obtaining modal solutions. Based on this method characteristic equations for Lamb and surface waves in multilayered orthotropic media have been obtained. Due to the stability of the stiffness matrix method, the solutions of the characteristic equations are numerically stable and efficient. Numerical examples are given.     

Investigation of Lamb elastic waves in anisotropic multilayered composites applying the Green’s matrix

This article presents a numerical study of dispersion characteristics of some symmetric and antisymmetric composites modelled as multilayered packets of layers with arbitrary anisotropy of each layer. The authors introduce a subsidiary boundary problem of three-dimensional elasticity theory for the system of partial differential equations describing the harmonic oscillations of the composite caused by a surface load. The problem reduces to a boundary problem for ordinary differential equations by employing the Fourier transform. An algorithm of constructing the Fourier transform of the Green’s matrix of the given boundary problem is presented. The wave numbers of Lamb waves propagating in composites, their phase velocity surfaces and group wave surfaces are presented through the poles of the transform of the Green’s matrix. The authors obtain the dispersion curves for different directions and frequencies and investigate the dispersion curves and surfaces of wave numbers, phase velocities and group wave surfaces for various composites. The numerical results are then compared with the results obtained by applying other methods.     

Bending wave propagation of carbon nanotubes in a bi-parameter elastic matrix

This article studies transverse waves propagating in carbon nanotubes (CNTs) embedded in a surrounding medium. The CNTs are modeled as a nonlocal elastic beam, whereas the surrounding medium is modeled as a bi-parameter elastic medium. When taking into account the effect of rotary inertia of cross-section, a governing equation is acquired. A comparison of wave speeds using the Rayleigh and Euler-Bernoulli theories of beams with the results of molecular dynamics simulation indicates that the nonlocal Rayleigh beam model is more adequate to describe flexural waves in CNTs than the nonlocal Euler-Bernoulli model. The influences of the surrounding medium and rotary inertia on the phase speed for single-walled and double-walled CNTs are analyzed. Obtained results turn out that the surrounding medium plays a dominant role for lower wave numbers, while rotary inertia strongly affects the phase speed for higher wave numbers.     

On beam propagation in anisotropic media: one-dimensional analysis

We theoretically investigate light beam propagation in (1+1)D homogeneous anisotropic uniaxials where ordinary and extraordinary waves are decoupled, accounting for the vectorial character of the electromagnetic field and addressing the nonparaxial limit.     

Guided wave propagation in multilayered piezoelectric structures

A general formulation of the method of the reverberation-ray matrix (MRRM) based on the state space formalism and plane wave expansion technique is presented for the analysis of guided waves in multilayered piezoelectric structures. Each layer of the structure is made of an arbitrarily anisotropic piezoelectric material. Since the state equation of each layer is derived from the three-dimensional theory of linear piezoelectricity, all wave modes are included in the formulation. Within the framework of the MRRM, the phase relation is properly established by excluding exponentially growing functions, while the scattering relation is also appropriately set up by avoiding matrix inversion operation. Consequently, the present MRRM is unconditionally numerically stable and free from computational limitations to the total number of layers, the thickness of individual layers, and the frequency range. Numerical examples are given to illustrate the good performance of the proposed formulation for the analysis of the dispersion characteristic of waves in layered piezoelectric structures. Supported by the National Natural Science Foundation of China (Grant Nos. 10725210 and 10832009), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060335107), the National Basic Research Program of China (Grant No. 2009CB623204), and the Scientific Research Foundation for Tsuiying Talents of Lanzhou University     

A stationary phase argument for the modal wave beam deviation in the time-space domain for anisotropic multilayered media

The aim of the paper is to describe the physical phenomenon of the excitation of modal waves, such as Lamb waves, in anisotropic multilayered media by a monochromatic incident beam and then by a time depending signal. A modal beam is generated in the structure and, due to the anisotropy of the media constituting the structure, is deviated with respect to the sagittal plane of the incident bounded beam. Using a stationary phase approach, it is possible to determine the deviation direction of the modal beam in the far field at a given frequency. This direction is normal to the modal curve, at the point corresponding to the main modal wave vector. Using Lagrange multipliers, it is possible to obtain the equation of an oblique plane in which the modal beam reradiates in the external fluid. As the modal waves are dispersive, the group velocity and the direction of propagation of the principal modal wave vary with the frequency. So, in the far field, for a time depending signal, the different monochromatic components of the main modal wave are found in different directions. In general, the main crest line of this modal wave packet is not a straight line.     

Light propagation in stratified anisotropic media

Light propagation in stratified anisotropic media with arbitrary orientation of optic axes smoothly varying from layer to layer is considered. In the WKB approximation, a general expression for the field is obtained. For the case of a uniaxial medium, the normal waves are found and specific features of the light propagation are analyzed. General conditions are obtained that determine the turning points and forbidden zones. It is shown that the developed approach allows one to find trajectories of rays in anisotropic media with arbitrary layered structure.     

Accurate calculation of reflectance spectra for thick one-dimensional inhomogeneous optical structures and media: stable propagation matrix method

Numerical instability is usually observed when the propagation matrix method is used to calculate the reflectance and transmittance spectra for the thick one-dimensional inhomogeneous optical structures and media. To remove this numerical instability we applied two procedures, the normalization and the singular-value decomposition, for the propagation matrix and the matrix involved in calculating the matrix of reflection coefficients, respectively. Examples of a cholesteric liquid crystal and a helical structure of ferroelectric liquid crystals with a twist defect show that the modified propagation matrix method is able to accurately calculate the reflectance spectra for thick structures.     

各向异性超常材料中倒退波的传播研究

研究了完全各向异性超常材料中的倒退波传播现象,得到了在材料本征轴和传输轴成任意角度情形下倒退波形成的条件,分析了超常材料的介电张量和磁导率张量、电磁波的偏振方式对倒退波形成和传播的影响. 在此基础上,进一步分析了几种不同色散曲线关系的各向异性超常材料中倒退波的产生情况,获得了电磁波波矢和坡印亭矢量(能流)夹角的具体表达式和倒退波传播的一般性结论. 此外,还研究了近零介电常数超常材料中倒退波的传播特性,发现在此类超常材料中倒退波只能是完美倒退波. 关键词： 超常材料 负折射 倒退波 各向异性     

Pseudo-Hermiticity and electromagnetic wave propagation: The case of anisotropic and lossy media

Pseudo-Hermitian operators can be used in modeling electromagnetic wave propagation in stationary lossless media. We extend this method to a class of non-dispersive anisotropic media that may display loss or gain. We explore three concrete models to demonstrate the utility of our general results and reveal the physical meaning of pseudo-Hermiticity and quasi-Hermiticity of the relevant wave operator. In particular, we consider a uniaxial model where this operator is not diagonalizable. This implies left-handedness of the medium in the sense that only clockwise circularly polarized plane-wave solutions are bounded functions of time.     

Sound-transparent anisotropic media for backscattering-immune wave manipulation

The scattering behavior of an anisotropic acoustic medium is analyzed to reveal the possibility of routing acoustic signals through the anisotropic layers with no backscattering loss. The sound-transparent effect of such a medium is achieved by independently modulating the anisotropic effective acoustic parameters in a specific order, and is experimentally observed in a bending waveguide by arranging the subwavelength structures in the bending part according to transformation acoustics. With the properly designed filling structures, the original distorted acoustic field in the bending waveguide is restored as if the wave travels along a straight path. The transmitted acoustic signal is maintained nearly the same as the incident modulated Gaussian pulse. The proposed schemes and the supporting results could be instructive for further acoustic manipulations such as wave steering, cloaking and beam splitting.     

Differential matrix formalism for depolarizing anisotropic media

Azzam's differential matrix formalism [J. Opt. Soc. Am. 68, 1756 (1978)], originally developed for longitudinally inhomogeneous anisotropic nondepolarizing media, is extended to include depolarizing media. The generalization is physically interpreted in terms of means and uncertainties of the elementary optical properties of the medium, as well as of three anisotropy absorption parameters introduced to describe the depolarization. The formalism results in a particularly simple mathematical procedure for the retrieval of the elementary properties of a generally depolarizing anisotropic medium, assumed to be globally homogeneous, from its experimental Mueller matrix. The approach is illustrated on literature data and the conditions of its validity are identified and discussed.     

A computational method for wave propagation from a point load in an anisotropic material

One approach which is employed to solve dynamic point load problems in plates and laminates is to take integral transforms to reduce the governing equations to a system of ordinary differential equations with respect to the depth variable. The solution of this system leads to expressions for the transforms of the displacement and stress components at any level in the plate and the transient response at any location may then be recovered by inversion of the multiple transforms. The formal transform inversion involves a double infinite integral but by making a change of variable this may be replaced by an infinite integral associated with a line source and a finite integral with respect to the orientation of the line. A first attempt at applying this approach to obtain the point load response of quasi-isotropic fibre composite laminate led to a non-causal predicted signal. This paper deals with an investigation of this proposed method applied to the simpler model problem of wave propagation in a two-dimensional anisotropic medium. Results are obtained for two different time histories of point loads, namely: a delta function; and a single period of a sine function. In the case of the delta function source a comparison is made with the analytic solution and the errors arising from the numerical approach are discussed. Graphs are also presented showing the non-causal contributions to the overall response which arise at individual angles of orientation of the line source.     

Matrix method for obtaining Rayleigh wave equations for anisotropic media with hexagonal syngony

Equations for the existence of Rayleigh waves ona free boundary of anisotropic media of hexagonal syngony are explicitly derived by the matriciant method. The propagation of waves along the X and Y axes and the velocity of wave propagation are considered.     

A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.     

Theory of mixed-polarization propagation in anisotropic cubic media

Polarization dynamics are considered in the presence of an anisotropic Kerr non-linearity in the most common semiconductor waveguide geometry. The equations of motion are formulated in terms of Stokes polarization parameters and their Hamiltonian form is derived. Stationary solutions and their stability are found for plane-wave propagation. It is found that the non-integrable problem of mixed-polarization spatial soliton dynamics can be largely explained in terms of the equivalent plane-wave solutions.     

Electromagnetic wave propagation in random media

The propagation of a narrow frequency band beam of electromagnetic waves in a medium with randomly varying index of refraction is considered. A novel formulation of the governing equation is proposed. An equation for the average Green function (or transition probability) can then be derived. A Fokker-Planck type equation is contained as a limiting case. The results are readily generalized to include the features of the random coupling model and it is argued that the present problem is particularly suited for an analysis of this type.     

Anomalous wave propagation in quasiisotropic media

Based on boundary conditions and dispersion relations, the anomalous propagation of waves incident from regular isotropic media into quasiisotropic media is investigated. It is found that the anomalous negative refraction, anomalous total reflection and oblique total transmission can occur in the interface associated with quasiisotropic media. The Brewster angles of E- and H-polarized waves in quasiisotropic media are also discussed. It is shown that the propagation properties of waves in quasiisotropic media are significantly different from those in isotropic and anisotropic media.