Coupled Plane Waves Research Articles

On obtaining the coupled-mode theory using a model of coupled plane waves for symmetric resonant diffraction gratings

We consider two analytical models describing resonant scattering of light by resonant diffraction gratings possessing a horizontal symmetry plane. For the case of oblique incidence, a multiple interference model is formulated, on the basis of which, a new approach for deriving the coupled-mode theory for the considered structures is proposed. Both considered models describe resonances in the reflectance and transmittance spectra of the studied diffraction gratings arising due to the excitation of quasiguided modes and Fabry–Pérot modes. Interaction of these modes leads to the appearance of bound states in the continuum, which are also described by the proposed models.

Modeling from the first principles of the electronic properties of compositions of REE as precursors of high-temperature superconductors

Modeling of the first principles of the electronic properties of complex oxides of rare earth elements (BaY2O4, BaGd2O4, BaLu2O4) as precursors of high-temperature superconductors has been performed. The VASP software package was used as a modeling environment, in particular the method of coupled plane waves (PAW method), which allows us to obtain fairly accurate results for calculating the electron density and band structure. From the analysis of the obtained band energy structure, it follows that the studied REE oxides have a band gap width Eg = 3.29–3.84 eV, which is characteristic for dielectric materials. The studied compounds based on these rare earth elements selected from the yttrium (Y, La, Gd–Lu) and cerium (Ce–Eu) groups are characterized by an increase in Fermi energy and a decrease in the band gap as the atomic number (39, 64, 71) of the element in the periodic table increases. A method for modeling the quantum layers of the studied materials by simulating the restriction of the crystal structure along one of the coordinate axes is proposed. This representation approximates the model of the crystal lattice of REE oxides to the situation of analyzing a quantum layer whose thickness is equal to the size of the crystal cell along the specified axis. The rupture of atomic bonds in a crystal is simulated by increasing the distance between atomic layers along this axis to values at which the value of free energy is stabilized. In the quantum layer of rare earth element oxide (with its thickness close to 1 nm), a wider range of energy values is formed in which electrons are distributed than is observed in the continuous version, and the expansion of the electron distribution area extends to the energy levels of the band gap. This is explained by the fact that the geometric discretization of nanoscale structures determines the discreteness of the quantum-dimensional energy spectrum.

Reflection of plane waves in a rotating micropolar double porous thermoelastic medium with temperature dependent properties

AbstractThe objective of this empirical investigation is to explore the behavior of different waves reflected in a homogenous isotropic double porous structured micropolar thermoelastic medium with rotation and temperature‐dependent characteristics. Three‐phase‐lag theory of thermoelasticity is adopted to acquire the governing equations of the model. Six coupled plane waves have been found to propagate across the medium at distinct velocities. Derivation for energy and amplitude ratios of reflected waves were carried out by employing appropriate boundary conditions. For different angles of incidence, the outcomes for energy partition were determined numerically and displayed graphically. It is found that amplitude ratios associated with various reflected waves are affected by angular velocity, temperature‐dependent properties, micropolar parameter, and double porosity. Also, it has been discovered that throughout the reflection phenomena there is no dissipation of energy. Furthermore, this study concludes with a summary of the findings and subsequent observations. The current investigation has yielded inferences regarding specific cases that are of particular interest.

Reflection of coupled elastic waves in nonlocal rotating viscoelastic media

In this paper, propagation of homogeneous harmonic plane waves has been studied in an infinite homogeneous isotropic nonlocal viscoelastic rotating medium. Due to the rotational effects, two coupled plane waves, namely a quasi longitudinal wave and a quasi transverse wave are found in the medium under study. The dispersion equations is derived analytically and represented numerically. The problem of reflection of the quasi longitudinal wave at a stress-free solid half-space of the considered medium is also investigated. Various analytical results for the phase speed, attenuation coefficients, reflection coefficients and their corresponding energy ratios are depicted graphically for a particular model of interest to highlight the effect of nonlocal as well as rotational parameter.

Plane wave reflection in micropolar hygro-thermoelastic half-space

The research problem aims to investigate the effect of heat and moisture on the propagation of plane waves in a micropolar hygro-thermoelastic medium. The governing equations of a hygro-thermoelastic micropolar medium are developed and solved to determine the velocity equation. The plane-wave solution reveals that the medium is propagated by two coupled transverse displacement waves, namely Coupled Transverse Micropolar wave (CTM-wave) and Coupled Transverse Displacement wave (CTD-wave). Additionally, three coupled longitudinal waves are observed: thermal diffusion TD-wave, longitudinal displacement P-wave, and moisture diffusion mD-wave. To assess the characteristics of these waves in the micropolar hygrothermal medium, the speed and distance of propagation are calculated for each wave type. This analysis provides information on how fast and how far the P-wave, TD-wave, mD-wave, CTM-wave, and CTD-wave can travel in the medium. Furthermore, the research problem determines the equations for the coefficient of reflection and energy ratio when a coupled plane wave is incident on the medium. These coefficients quantify the amount of reflected energy compared to the incident energy and provide insights into the wave behavior at the interface of the medium. Finally, the variations in energy ratio and reflection coefficient are depicted graphically, allowing for a visual representation of how these quantities change with different parameters or conditions. These graphs provide a comprehensive understanding of the wave behavior and the effects of heat and moisture on the propagation characteristics in the micropolar hygrothermal medium.

From coupled plane waves to the coupled-mode theory of guided-mode resonant gratings

We establish the connection between two theoretical approaches widely used for describing resonant optical properties of guided-mode resonant gratings (GMRGs): the coupled plane waves model and the coupled-mode theory (CMT). In the coupled plane waves model (also known as the multiple interference model), the field inside and outside the structure is represented as a small number of propagating plane waves; these waves are coupled at one or several interfaces, which leads to a system of linear equations. In the CMT, the optical properties are described by several coupled differential equations, each corresponding to one of the modes supported by the structure. The approach proposed in the present work provides explicit expressions for the coupling coefficients of the CMT through “local” reflection, transmission, and diffraction coefficients of the coupled plane waves model, which is important for the design and analysis of optical resonators, filters, and sensors based on GMRGs.

Propagation of plane waves in a nonlocal transversely isotropic thermoelastic medium with voids and rotation

AbstractThe objective of present research work is to investigate the propagation of time harmonic plane waves in a rotating, nonlocal, transversely isotropic thermoelastic half‐space having void pores in the context of Lord–Shulman theory. It has been observed that four coupled plane waves travel in such type of medium with distinct speeds. Using appropriate boundary conditions, the amplitude and energy ratios for the reflected waves are presented. Numerical computations for a magnesium crystal‐like material are performed with the help of MATLAB software to study the effects of material anisotropy, nonlocal, void and rotation parameters on the amplitude ratios. The computational results are displayed and explained through graphs. The phase velocities of plane waves in transversely isotropic medium are found to depend upon the incident angle, whereas these attain constant values in an isotropic medium. The energy conservation law is also justified during the reflection phenomena.

Reflection of plane waves in a rotating nonlocal fiber-reinforced transversely isotropic thermoelastic medium

The present investigation is concerned with the reflection of plane waves from the free surface of a rotating nonlocal, fiber-reinforced, transversely isotropic thermoelastic medium in the context of Lord-Shulman theory. It has been observed that three coupled plane waves travel in the medium with distinct speeds. Using appropriate boundary conditions, the amplitude ratios for the reflected waves are derived and the numerical computations have been carried out using MATLAB software. The modulus of amplitude ratios is presented graphically to exhibit the effects due to presence of fiber-reinforcement, nonlocal and rotation parameters. The expressions of energy ratios have also been derived in explicit form and are portrayed graphically as functions of angle of incidence. The phase velocities of plane waves in transversely isotropic medium are found to depend upon the incident angle, whereas these attain constant values in isotropic medium. The energy conservation law is also justified during the reflection phenomena.

Reflection of thermoelastic coupled plane waves under the MGL model

The present study is concerned with the reflection phenomenon of thermoelastic plane waves at the thermally insulated stress-free surface of a homogeneous isotropic thermally conducting elastic half-space in the frame of modified Green-Lindasy (MGL) model of generalized thermoelasticity with strain rate. The thermoelastic coupling effect creates two types of coupled longitudinal waves which are dispersive in nature as well as exhibit attenuation. Different from the thermoelastic coupling effect, there also exist one independent vertically shear-type (SV-type) wave. In contrast to the classical Green-Lindsay (GL) and the Lord-Shulman (LS) theories of generalized thermoelasticity, the SV-type wave is not only dispersive in nature but also experiences attenuation. Analytical expressions for the amplitude ratios and their respective energy ratios for the reflected thermoelastic waves are determined when a coupled longitudinal wave is made incident on a surface. The paper concludes with the numerical results on the phase speeds, amplitude ratios and their respective energy ratios for specific parameter choices. Various graphs are plotted to analyzed the analytical solution. The characteristics of employing the MGL model are discussed by comparing the numerical results obtained for the MGL model with those obtained in case of the GL, LS and the coupled thermoelasticity (CTE) models.

Reflection at non-free boundary of a micropolar piezoelectric half-space

This paper presents a study on plane waves propagating in a micropolar piezoelectric material with transverse isotropy. The plane wave solutions of the governing field equations suggest that there exists three coupled plane waves propagating in a micropolar piezoelectric medium. Reflection of a plane wave incident at a non-free surface of a micropolar piezoelectric half-space is considered. The analytical expressions of the reflection coefficients and energy ratios are obtained. A quantitative example is set up to compute the speeds, reflection coefficients and energy ratios of reflected waves. The speeds, energy ratios and reflection coefficients are illustrated graphically against the propagation angle. The numerical results show that the boundary parameters connected to the mechanical nature of the non-free surface affect significantly the energy ratios and reflection coefficients.

Effect of Impedance Boundary on the Reflection of Plane Waves in Fraction-Order Thermoelasticity in an Initially Stressed Rotating Half-Space with a Magnetic Field

In this research article, the propagation of the plane waves in an initially stressed rotating magneto-thermoelastic solid half-space in the context of fractional-order derivative thermoelasticity is studied. The governing equations in the x–z plane are formulated and solved to obtain a cubic velocity equation that indicates the existence of three coupled plane waves. A reflection phenomenon for the incidence of a coupled plane wave for thermally insulated/isothermal surface is studied. The plane surface of the half-space is subjected to impedance boundary conditions, where normal and tangential tractions are proportional to the product of normal and tangential displacement components and frequency, respectively. The reflection coefficients and energy ratios of various reflected waves are computed numerically for a particular material and the effects of rotation, initial stress, magnetic field, fractional-order, and impedance parameters on the reflection coefficients and energy ratios are shown graphically.

Effect of impedance on the reflection of plane waves in a rotating magneto-thermoelastic solid half-space with diffusion

This research article deals with the propagation of plane waves in a rotating magneto-thermoelastic half-space with diffusion in the context of Lord–Shulman theory of thermoelasticity. The governing equations are formulated for a specialized plane and solved to obtain the velocity equation, which indicates the existence of four coupled plane waves. The wave velocities are computed for a particular medium, and the effect of the diffusion parameter is shown graphically on these waves. Impedance boundary conditions are applied to study the reflection of plane waves from the stress-free and thermally insulated/isothermal surface. Reflection coefficients and energy ratios of the reflected waves are also obtained, and the variations of reflection coefficients and energy ratios against the angle of incidence are shown graphically.

Reflection of plane waves in a nonlocal micropolar thermoelastic medium under the effect of rotation

The present investigation is concerned with the reflection of plane waves at the free surface of a homogeneous, isotropic, nonlocal, micropolar rotating thermoelastic medium. The entire thermoelastic medium is rotating with a uniform angular velocity. It is observed that there exist four coupled plane waves, which travel through the medium with distinct speeds. Using appropriate boundary conditions, the reflection coefficients and energy ratios of various reflected waves are computed numerically with the help of the software MATLAB. The numerical values of modulus of reflection coefficients are presented graphically to show the effects of nonlocal, rotation and micropolar parameters. It has been verified that during reflection phenomena, the sum of modulus of energy ratios is approximately equal to unity at each angle of incidence. The effect of micropolarity on the phase velocities is also observed and shown graphically.

Effects of rotation, voids and diffusion on characteristics of plane waves in a thermoelastic material

PurposeThe purpose of this paper is to study the effects of rotation, voids and diffusion on characteristics of plane waves in a thermoelastic material.Design/methodology/approachLord and Shulman generalization of linear thermoelasticity is used to study the plane waves in a rotating thermoelastic material with voids and diffusion. The thermoelastic solid is rotating with a uniform angular velocity. The problem is specialized in two dimensions to study wave propagation. The plane harmonic solutions of governing field equations in a plane are obtained.FindingsA velocity equation is obtained which indicates the propagation of five coupled plane waves in the medium. Reflection of an incident plane wave from stress-free surface of a half-space is also considered to obtain the amplitude ratios of various reflected waves. A numerical example is considered to illustrate graphically the effects of rotation, frequency, void and diffusion parameters on speeds and amplitude ratios of plane waves.Originality/valueThe present problem covers the combined effects of rotation, voids and diffusion on characteristics of plane waves in linear thermoelastic material in the context of Lord and Shulman (1967) and Aouadi (2010) theories, which are not studied in literature yet.

Reflection and transmission of elastic waves at an interface between two micropolar piezoelectric half-spaces

In the present paper, we consider a problem of reflection and transmission of plane waves at an interface between two different transversely isotropic micropolar piezoelectric half-spaces. The plane wave solution of governing equations for micropolar piezoelectric medium indicates the propagation of three coupled plane waves. An incident plane wave at an interface between two dissimilar half-spaces generates three reflected and three transmitted waves. The relations between amplitude ratios of reflected and transmitted waves are derived which, in general, depend on material parameters and angle of incidence. The explicit expressions for energy ratios of reflected and transmitted waves are also obtained. For relevant material parameters, the results are illustrated graphically to show the micropolar piezoelectric effects on variations of amplitude ratios and the square root of energy ratios against the angle of incidence.

Reflection and propagation of plane waves at free surfaces of a rotating micropolar fibre-reinforced medium with voids

The present paper seeks to investigate propagation and reflection of waves at free surfaces of homogeneous, anisotropic and rotating micropolar fibre-reinforced medium with voids. It has been observed that, in particular when P-wave is incident on the free surface, there exist four coupled reflected plane waves traveling in the medium; quasi-longitudinal displacement (qLD) wave, quasi-transverse displacement (qTD) wave, quasi-transverse microrotational wave and a wave due to voids. Normal mode Analysis usually called harmonic solution method is adopted in concomitant with Snell\'s laws and appropriate boundary conditions in determination of solution to the micropolar fibre reinforced modelled problem. Amplitude ratios which correspond to reflected waves in vertical and horizontal components are presented analytically. Also, the Reflection Coefficients are presented using numerical simulated results in graphical form for a particular chosen material by the help of Mathematica software. We observed that the micropolar fibre-reinforced, voids and rotational parameters have various degrees of effects to the modulation, propagation and reflection of waves in the medium. The study would have impact to micropolar fibre-reinforecd rotational-acoustic machination fields and future works about behavior of seismic waves.

Reflection at the free surface of fiber-reinforced thermoelastic rotating medium with two-temperature and phase-lag

• Wave propagation in a fiber-reinforced thermoelastic rotating medium is considered. • Two-temperature dual-phase-lag theory has been employed for addressing the mathematical analysis. • Effects of different parameters on reflection coefficients have been depicted graphically. • Reflection coefficients and energy ratios for reflected quasi waves are presented in closed form. • It is proved that at each angle of incidence the energy balance law is satisfied at the free surface. The present investigation is concerned with the reflection of plane waves from the free surface of a homogeneous, anisotropic , fiber-reinforced thermoelastic rotating medium under two-temperature and dual-phase-lag model. It has been observed that three coupled plane waves travel in the medium with distinct speeds. Using appropriate boundary conditions, the amplitude and energy ratios for the reflected waves are derived and the numerical computations have been carried out with the help of MATLAB programming. The numerical values of the modulus of reflection coefficients are presented graphically to exhibit the two-temperature, phase lag and rotation parameter effects. The expressions of energy ratios have also been obtained in explicit form and are shown graphically as functions of angle of incidence. It has been verified that during reflection phenomena, the sum of energy ratios is equal to unity at each angle of incidence.

The band gap properties of the three-component semi-infinite plate-like LRPC by using PWE/FE method

This paper applies coupled plane wave expansion and finite element (PWE/FE) method to calculate the band structure of the proposed three-component semi-infinite plate-like locally resonant phononic crystal (LRPC). In order to verify the accuracy of the result, the band structure calculated by PWE/FE method is compared to that calculated by the traditional finite element (FE) method, and the frequency range of the band gap in the band structure is compared to that of the attenuation in the transmission power spectrum. Numerical results and further analysis demonstrate that a band gap is opened by the coupling between the dominant vibrations of the rubber layer and the matrix modes. In addition, the influences of the geometry parameters on the band gap are studied and understood with the help of the simple “base-spring-mass” model, the influence of the viscidity of rubber layer on the band gap is also investigated.

Using PWE/FE Method to Calculate the Band Structures of the Semi-Infinite PCs: Periodic in x–y Plane and Finite in z -direction

Abstract This paper introduces the concept of semi-infinite phononic crystal (PC) on account of the infinite periodicity in x-y plane and finiteness in z-direction. The plane wave expansion and finite element methods are coupled and formulized to calculate the band structures of the proposed periodic elastic composite structures based on the typical geometric properties. First, the coupled plane wave expansion and finite element (PWE/FE) method is applied to calculate the band structures of the Pb/rubber, steel/epoxy and steel/aluminum semi-infinite PCs with cylindrical scatters. Then, it is used to calculate the band structure of the Pb/rubber semi-infinite PC with cubic scatter. Last, the band structure of the rubbercoated Pb/epoxy three-component semi-infinite PC is calculated by the proposed method. Besides, all the results are compared with those calculated by the finite element (FE) method implemented by adopting COMSOL Multiphysics. Numerical results and further analysis demonstrate that the proposed PWE/FE method has strong applicability and high accuracy.

Wave propagation in a transversely isotropic microstretch elastic solid

BackgroundThe theory of microstretch elastic bodies was first developed by Eringen (1971, 1990, 1999, 2004). This theory was developed by extending the theory of micropolar elastcity. Each material point in this theory has three deformable directors.MethodsThe governing equations of a transversely isotropic microstretch material are specialized in x-z plane. Plane wave solutions of these governing equations results into a bi-quadratic velocity equation. The four roots of the velocity equation correspond to four coupled plane waves which are named as Coupled Longitudinal Displacement (CLD) wave, Coupled Longitudinal Microstretch (CLM) wave, Coupled Transverse Displacement (CTD) wave and Coupled Transverse Microrotational (CTM) wave. The reflection of Coupled Longitudinal Displacement (CLD) wave is considered at a stress-free surface of half-space of material. The appropriate displacement components, microrotation component and microstretch potential for incident and four reflected waves in half-space are formulated. These solutions for incident and reflected waves satisfy the boundary conditions at a stress free surface of half-space and we obtain a non-homogeneous system of four equations in four reflection coefficients (or amplitude ratios) along with Snell’s law for the present model.ResultsThe speeds of plane waves are computed by Fortran program of bi-quadratic velocity equation for relevant physical constants of the material. The reflection coefficients of various reflected waves are also computed by Fortran program of Gauss elimination method. The speeds of plane waves are plotted against angle of propagation direction with vertical axis. The reflection coefficients of various reflected waves are plotted against the angle of incidence. These variations of speeds and reflection coefficients are also compared with those in absence of microstretch parameters.ConclusionsFor a specific material, numerical simulation in presence as well as in absence of microstretch shows that the coupled longitudinal displacement (CLD) wave is fastest wave and the coupled transverse microrotational (CTM) is observed slowest wave. The coupled longitudinal microstretch (CLM) wave is an additional wave due to the presence of microstretch in the medium. The presence of microstretch in transversely isotropic micropolar elastic solid affects the speeds of plane waves and the amplitude ratios of various reflected waves.Mathematics Subject Classification74J