Solution Of Dispersion Equation Research Articles

Formation Model for Acoustic Characteristics of Solid Media with Ordered Fracturing

Introduction. The development of new structural materials and improvement of existing technologies for the production of new products on their basis lead to the emergence of new types of medium discontinuities. Therefore, the development of new models of discontinuities that take the previously ignored parameters into account seems to be relevant for the purposes of nondestructive testing and structural measurements. This concerns, e.g., the roughness of adjacent surfaces of microcrack ordered sets.Aim. Theoretical substantiation for the processes of elastic waves propagation through an elastic medium containing an ordered lattice of microcracks with boundary conditions in the linear slip approximation, modified by taking into account the parameters of micro-convexities of microcrack rough boundaries. Database formation for experimental studies aimed at determining the physical and mechanical characteristics of structural materials.Materials and methods. The acoustic characteristics of materials were determined based on the derivation and solutions of dispersion equations describing the formation and propagation of effective longitudinal, transverse, and Rayleigh surface elastic waves in elastic media with ordered cracking. Their values were also used to determine the effective speed of Rayleigh surface waves.Results. The conducted simulation of elastic wave formation processes showed that an increase in the concentration of microcracks leads to a decrease in the phase velocities of effective longitudinal, transverse, and surface waves, as well as to an increase in the attenuation coefficients at given ultrasound frequencies and material parameters.Conclusion. The radius of the microsphere that replaces the surface micro-convexity and the roughness parameter Rz have a significant impact on the formation of physical and mechanical characteristics of materials, which are determined by the results of ultrasonic measurements. The developed model can be recommended as a basis for interpreting the results of ultrasonic measurements.

Dispersion Equations of Transverse Electromagnetic Waves Narrowly Localized Near the Interface between the Media with Different Graded-Index Profiles

"Combinations of three pairs of contacting media with linear, exponential, and parabolic profiles of dielectric permittivity are described theoretically. Three new types of narrowly localized transverse electric waves propagating along the interfaces between the considered graded-index media are found. The exact dispersion equations for each type of the waves determining the effective refractive index in dependence on optical and geometric characteristics are obtained. The influence of the thicknesses of the graded-index layers on dispersion equation solutions is analyzed. It is found that the effective refractive index increases with an increase in the thickness of the gradedindex layers in all considered combinations of the contacting media. The thickness of the parabolic graded-index profile has the least significant effect on the effective refractive index, compared to the linear and exponential ones. "

The method of simple iterations with correction of convergence and the minimal discrepancy method for plasmonic problems

The problem of searching for complex roots of the dispersion equations of plasmon-polaritons along the boundaries of the layered structure-vacuum interface is considered. Such problems arise when determining proper waves along the interface of structures supporting surface and leakage waves, including plasmons and polaritons along metal, dielectric and other surfaces. For the numerical solution of the problem, we consider a modification of the method of simple iterations with a variable iteration parameter leading to a zero derivative of the right side of the equation at each step, i.e. convergent iterations, as well as a modification of the minimum residuals method. It is shown that the method of minimal residuals with linearization coincides with the method of simple iterations with the specified correction. Convergent methods of higher orders are considered. The results are demonstrated by examples, including complex solutions of dispersion equations for plasmon-polaritons. The advantage of the method over other methods of searching for complex roots in electrodynamics problems is the possibility of ordering the roots and constructing dispersion branches without discontinuities. This allows you to classify modes.

Low frequency limits for Lamb waves in homogeneous, stratified and functionally graded anisotropic plates

The closed form expressions for the low frequency velocity limits of Lamb waves propagating in anisotropic homogeneous, stratified and functionally graded (FG) plates with arbitrary elastic anisotropy and arbitrary (continuous) transverse variation of physical properties are constructed. The applied method is based on Cauchy sextic formalism coupled with the stratification convergence theorem for FG plates. The constructed solutions are obtained in the form of vanishing auxiliary determinants related to derivatives of the closed form solutions of dispersion equations.

The new features of generalized order parameter localization in multi-layered magnetic structures

The new types of nonlinear localized state of generalized order parameter in the three-layered structure of band antiferromagnets with congruent sections of the Fermi surface are obtained. The nonlinear effects inside the narrow plane defects between the layers representing the local magnetization in the direction of polarization of the spin density wave are described by the nonlinear source function. The variation of the Ginsburg–Landau functional near the Lifshitz point leads to nonlinear Schrödinger equation (NSE) with cubic nonlinear terms and self-consistent nonlinear potential modeling the plane defects. The dispersion equations of two type stationary nonlinear localized states are derived. The solutions of dispersion equations in explicit analytical form in dependence on characteristics of plane defects and the distance between them are found and analyzed. The localized states of order parameter existing only near the plane defects with nonlinear response are obtained.

Closed-form approximation of symmetric thin-film multi-layer plasmonic dispersion equation solutions

An original asymptotic method is developed and used to find closed-form approximations to the symmetric thin-film three- and multi-layer plasmonic dispersion equations. Closed-form analysis of three-layer metal-insulator-metal (MIM: "M" is metal and "I" is insulator) and IMI devices shows a complementary physics underpinning their properties. Analysis of multi-layer symmetric devices, considered for a seven-layer MIMIMIM example, uncovers a remarkable departure from the physics governing MIM and IMI features. Multi-layer propagation length and attenuation are determined by proximity, in the space of cladding thickness and wavelength, to singularities that exist in the limit of vanishing imaginary part of the cladding dielectric constant. Exploitation of this phenomenon will expand the development of a broader range of thin-film applications in optoelectronics.

Localization in the model of contacting media with specific nonlinearity and interface interaction

In this paper, we consider the new model of nonlinear contacting media based on nonlinear Schrodinger equation with point potential and term, which is depended stepwise on field amplitude. Such a model theoretically describes a change in properties of the boundary regions along the interface between a Kerr-type crystal with cubic nonlinearity and a nonlinear medium characterized by abruptly change in dielectric constant depending on field amplitude. The short-range local interaction between wave and interface is taken into account by point potential in nonlinear Schrodinger equation. We obtain two new types of localized states characterized by composite structure consisting of three parts of the field distributions. We find exact and approximate solutions of dispersion equations. We described new properties of the spectrum of localized states arising as a result of the interaction of the wave with the interface and the presence of threshold field of the switching between the medium constants. All results are obtained in an analytical form. The proposed theory can be used to describe the propagation features of intense light beams localized along media interfaces in nonlinear optics, and to describe Bose-Einstein condensates with cubic nonlinearity.

Stability of propagation features under time-asymptotic approximations for a class of dispersive equations

We consider solutions of dispersive equations on the line defined by Fourier multipliers with initial data having compactly supported Fourier transforms. In this paper, a refinement of an existing method permitting to expand time-asymptotically the solution formulas is proposed. Here the first term of the expansion is supported in a space-time cone whose origin depends explicitly on the initial datum. As an important consequence of our refined method, the first term inherits the mean position of the solution together with a constant variance error and a shifted time-decay rate is obtained. Hence this refinement, which takes into account both spatial and frequency information of the initial datum, makes stable some propagation features under time-asymptotic approximations and permits a better description of the time-asymptotic behaviour of the solutions. The results are achieved firstly by making apparent the cone origin in the solution formula, secondly by applying precisely an adapted version of the stationary phase method with a new error bound, and finally by minimising the error bound with respect to the cone origin.

Peculiarities of the nonlinear surface wave propagation in the composite waveguide with different nonlinearity types of the layers

In this paper, we explore the photorefractive crystal with diffusion nonlinearity mechanism covered by self-focusing Kerr-type crystal contacting with dielectric medium. Four types of nonlinear surface TM electromagnetic waves can propagate along the interfaces of such three-layered composite waveguide. The dispersion equations and conditions of existence of the waves depending on waveguide characteristics are analyzed. This allows to find exact and approximate solutions of dispersion equations in different cases and to obtain the dependence of propagation constant of the surface wave in explicit form. Conditions to control the field distribution across the interface planes by varying the waveguide temperature or waveguide optical and structure characteristics are presented.

Localized states in symmetric three-layered structure consisting of linear layer between focusing media separated by interfaces with nonlinear response

We analyze the localization in three-layered symmetric structure consisting of linear layer between focusing nonlinear media separated by nonlinear interfaces. The mathematical formulation of the model is a one-dimensional boundary value problem for the nonlinear Schrödinger equation. We find nonlinear localized states of two types of symmetry. We derive the energies of obtained stationary states in explicit form. We obtain the localization energies as exact solutions of dispersion equations choosing the amplitude of the interface oscillations as a free parameter. We analyze the conditions of their existence depending on the combination of signs of interface parameters.

An efficient plasmonic waveguide mode solution using rational function fitting of Green’s function

A novel approach to find plasmonic waveguide modes is presented in this paper. The method is based on rational function fitting of spectral Green’s function (RFF-SGF) of slab waveguide with the use of modified VECTFIT algorithm. Dispersion equation solutions including complex modes of structures with material/metal loss can be obtained with no extra complexity during the poles and residues extraction in the fitting approach. Moreover, this method can be used in multilayered plasmonic waveguides straightforwardly which are widely utilized in nanophotonic devices. The main advantage of this method lies in its simple implementation, speed as well as controllable accuracy. Surface plasmon and also surface wave modes of various plasmonic slab waveguides are investigated, and excellent agreements with rigorous methods are shown.

Computation of Propagating and Non-Propagating Lamb-Like Wave in a Functionally Graded Piezoelectric Spherical Curved Plate by an Orthogonal Function Technique.

Non-propagating waves have great potential for crack evaluation, but it is difficult to obtain the complex solutions of the transcendental dispersion equation corresponding to the non-propagating wave. This paper presents an analytical approach based on the orthogonal function technique to investigate non-propagating Lamb-like waves in a functionally graded piezoelectric spherical curved plate. The presented approach can transform the set of partial differential equations for the acoustic waves into an eigenvalue problem that can give the generally complex wave numbers and the field profiles. A comparison of the obtained results with the well-known ones in plates is provided. The obtained solutions of the dispersion equation are shown graphically in three dimensional frequency-complex wave number space, which aids in understanding the properties of non-propagating waves better. The properties of the guided wave, including real, purely imaginary, and complex branches in various functionally graded piezoelectric spherical curved plates, are studied. The effects of material piezoelectricity, graded fields, and mechanical and electrical boundary conditions on the dispersion characteristics, are illustrated. The amplitude distributions of displacement and electric potential are also discussed, to analyze the specificities of non-propagating waves.

Study of Scholte wave dispersion curves and modal energy distribution using a wavefield numerical simulation method

Surface waves travelling along the subsurface interface between water and sediment are called Scholte (or generalised Rayleigh) waves. In this paper, we examine the characteristics of Scholte wave dispersion curves and their modal energy distribution through synthetic simulation tests. Analytical solutions of the dispersion equation for a homogeneous isotropic layered medium describe the propagation characteristics of the fundamental and higher modes, but do not provide information about the relative energy distributions of those wave modes. In real situations, the dispersion curves with different modes have different energies corresponding to different frequency bands. It is instructive to study and analyse the behaviours of Scholte wave dispersion curves by using joint dispersion equation solutions and numerical simulations results. Dispersion equation solutions are used to validate the numerical results, which in turn are used to assess energy distribution for both the fundamental and the higher modes. We observe that the thickness of the overlying water layer and the elastic properties of the ocean bottom play important roles in the dispersion characteristics of Scholte waves. The dispersion of the soft ocean-bottom model (whose S-wave velocity of the first solid layer is lower than the P-wave velocity of water) is dominated by the fundamental mode and appears to be less dependent on the thickness of the overlying water layer. In contrast, the dispersion of the hard ocean-bottom model contains significant energy in the higher modes. Higher modes will shift to lower frequencies and will be spatially closer (in the frequency direction) when the water layer thickness increases. The dispersion characteristics of the real data are in a good agreement with the conclusion of a soft ocean-bottom model. Our modelling approach of combining analytical solutions of the dispersion equation with numerical modelling can facilitate Scholte wave exploration in water-covered areas.

Symmetric surface waves along a metamaterial dielectric waveguide and a perfectly conducting cylinder covered by a metamaterial layer


 
 
 Existence of symmetric complex waves in a metamaterial dielectric rod and a perfectly conducting cylinder of circular cross section covered by a concentric layer of metamaterial, a metamaterial Goubau line, is proved. Analytical investigation and numerical solution of dispersion equations reveal several important properties of running waves inherent to open metal-metamaterial waveguides which have not been reported for waveguides filled with standard media.
 
 

Effect of small anisotropy and absorption on metamaterial applications: “non-ideal” features of propagation and tunneling of electromagnetic waves

For cases of isotropic, uniaxial, and biaxial electromagnetic metamaterials (MM), a comparative analysis of the effect of small deviations of local material parameters from “ideal” values on the realization of MM applications (“zero” media, the Veselago-Pendry superlens) has been carried out. On the basis of the detailed investigation of the solutions of dispersion equations, it is established that even a very small dielectric and (or) magnetic anisotropy of a general form is the universal “non-ideal” factor determining (to a much greater extent than small losses) the operability of those MM applications where the wave misphasing in the effective medium is undesirable. The characteristics of wave attenuation in the absorbing isotropic and weakly anisotropic MM are mainly comparable for the applications. Limitations of the traditional approaches using the second-order curves (or surfaces) for analytic modeling of the absorbing MM dispersion equations are shown.

Energy spectrum of localized quasiparticles renormalized by multi-phonon processes at finite temperature

The theory of renormalized energy spectrum of localized quasi-particle interacting with polarization phonons at finite temperature is developed within the Feynman-Pines diagram technique. The created computer program effectively takes into account multi-phonon processes, exactly defining all diagrams of mass operator together with their analytical expressions in arbitrary order over the coupling constant. Now it is possible to separate the pole and non-pole mass operator terms and perform a partial summing of their main terms. The renormalized spectrum of the system is obtained within the solution of dispersion equation in the vicinity of the main state where the high- and low-energy complexes of bound states are observed. The properties of the spectrum are analyzed depending on the coupling constant and the temperature.

Wave propagation in a three-layer semi-infinite hydrodynamic system with a rigid lid

Research into propagation and interaction of waves in a three-layer hydrodynamic system is one of the relevant problems of modern theoretical and experimental hydrodynamics. The authors studied propagation and interaction of waves along contact surfaces of the three-layer hydrodynamic system "liquid half-space – layer – layer with a rigid lid". By applying a method of large-scale approximations, the first three linear approximations of the correspondent weakly nonlinear problem were obtained. The structure of wave motions on contact surfaces was explored. Dependence of amplitudes of waves-responses on contact surfaces at various geometrical and physical parameters was analyzed. In particular, for large values of thickness of the upper layer, it was found that a change in value of the wave number leads to rapid convergence of amplitudes of waves-responses to the common limited value. The authors showed the need for a detailed study of the limited case in the absence of density jump, in which one of the solutions of dispersion equation tends to zero. Results of the present research can be used in the design of algorithms for detection of wave motions in various liquid media.

Start current of dielectric-loaded grating in Smith-Purcell radiation

In this paper, a three-dimensional dielectric loaded grating (DLG) is proposed for the Smith-Purcell (SP) device. Taking into the considerations of thickness and width of electron beam, the dispersion equation is derived by using field matches method. The complex frequency is obtained by the numerical solution of dispersion equation, in which the imaginary part represents linear growth rate. The impacts of the electron beam filling factor (EBFF) on growth rate are discussed under the condition that the beam current and beam current density are kept as constants, respectively. In addition, the start current for SP oscillator is obtained by using the dispersion relation combined with boundary conditions. The relationship between the start current and other parameters is discussed and compared with the conventional metal grating. The results show that with the increasing of EBFF, the peak growth rate increases rapidly firstly and then decreases slowly, in which the current and current density are kept as constants, respectively. For the SP oscillator, the start current is increased with the shifting up beam voltage, but it is decreased with the improved EBFF, and only it has a slightly increasing trend when EBFF is close to 1. In addition, the start current is decreased with the increasing of relative dielectric constant, which indicates that by introducing DLG, the start current can be effectively reduced. Theoretical results are in good agreement with that of the simulations.

Nonexistence of small doubly periodic solutions for dispersive equations

We study the question of existence of time-periodic, spatially periodic solutions for dispersive evolution equations, and in particular, we introduce a framework for demonstrating the nonexistence of such solutions. We formulate the problem so that doubly periodic solutions correspond to fixed points of a certain operator. We prove that this operator is locally contracting, for almost every temporal period, if the Duhamel integral associated to the evolution exhibits a weak smoothing property. This implies the nonexistence of nontrivial, small-amplitude time-periodic solutions for almost every period if the smoothing property holds. This can be viewed as a partial analogue of scattering for dispersive equations on periodic intervals, since scattering in free space implies the nonexistence of small coherent structures. We use a normal form to demonstrate the smoothing property on specific examples, so that it can be seen that there are indeed equations for which the hypotheses of the general theorem hold. The nonexistence result is thus established through the novel combination of small divisor estimates and dispersive smoothing estimates. The examples treated include the Korteweg-de Vries equation and the Kawahara equation.

First integrals for nonlinear dispersive equations

Given a solution of a semilinear dispersive partial differential equation with a real analytic nonlinearity, we relate its Cauchy data at two different times by nonlinear representation formulas in terms of convergent series. These series are constructed by means of generating functions. All this theory is based on a new suitable formulation of the dynamics of solutions of dispersive equations.