Class Of Wave Equations Research Articles

Study of Love Waves in a Clamped Viscoelastic Medium with Irregular Boundaries

A mathematical model is presented to investigate the effects of sandiness, irregular boundary interfaces, heterogeneity and viscoelasticity on the phase velocity of Love waves. Geometry of the problem is consisting of an initially stressed viscoelastic layer with corrugated irregular boundaries, which is sandwiched between heterogeneous orthotropic semi-infinite half-space with initial stress and pre-stressed dry sandy half-space. Heterogeneity arises in the upper half-space is due to trigonometric variation in elastic parameters of the orthotropic medium. Inclusion of the concept of corrugated irregular viscoelastic layer clamped between two dissimilar half-spaces under different physical circumstances such as initial stress and heterogeneity brings a novelty to the existing literature related to the study of Love wave. Dispersion equation for Love wave is obtained in closed form. The obtained dispersion relation is found to be in well agreement with classical Love wave equation. Numerical example and graphical illustrations are made to demonstrate notable effect of initial stress, internal friction, wave number and amplitude of corrugations on the phase velocity of Love waves.

Krein-Schrödinger formalism of bosonic Bogoliubov–de Gennes and certain classical systems and their topological classification

To understand recent works on classical and quantum spin equations and their topological classification, we develop a unified mathematical framework for bosonic BdG systems and associated classical wave equations; it applies not just to equations that describe quantized spin excitations in magnonic crystals but more broadly to other systems that are described by a BdG hamiltonian. Because here the generator of dynamics, the analog of the hamiltonian, is para-, aka Krein-, hermitian but not hermitian, the theory of Krein spaces plays a crucial role. For systems which are thermodynamically stable, the classical equations can be expressed as a Schr\"odinger equation with a hermitian hamiltonian. We then proceed to apply the Cartan-Altland-Zirnbauer classification scheme: to properly understand what topological class these equations belong to, we need to conceptually distinguish between symmetries and constraints. Complex conjugation enters as a particle-hole constraint (as opposed to a symmetry), since classical waves are necessarily real-valued. Because of this distinction only commuting symmetries enter in the topological classification. Our arguments show that the equations for spin waves in magnonic crystals are a system of class A, the same topological class as quantum hamiltonians describing the Integer Quantum Hall Effect. Consequently, the magnonic edge modes first predicted by Shindou et al. are indeed analogs of the Quantum Hall Effect, and their net number is topologically protected.

Quantum Features of Macroscopic Fields: Entropy and Dynamics.

Macroscopic fields such as electromagnetic, magnetohydrodynamic, acoustic or gravitational waves are usually described by classical wave equations with possible additional damping terms and coherent sources. The aim of this paper is to develop a complete macroscopic formalism including random/thermal sources, dissipation and random scattering of waves by environment. The proposed reduced state of the field combines averaged field with the two-point correlation function called single-particle density matrix. The evolution equation for the reduced state of the field is obtained by reduction of the generalized quasi-free dynamical semigroups describing irreversible evolution of bosonic quantum field and the definition of entropy for the reduced state of the field follows from the von Neumann entropy of quantum field states. The presented formalism can be applied, for example, to superradiance phenomena and allows unifying the Mueller and Jones calculi in polarization optics.

Manipulating Elastic Waves with Conventional Isotropic Materials

Controlling the trajectory of general elastic waves with metamaterials is very difficult, because the classical elastic wave equations are not form-invariant. However, this study points out that the elastodynamic potential energy can nearly retain its form after conformal mapping, if the longitudinal wave velocity is much greater than the transverse. Based on this finding, an elastic-wave bender is designed and fabricated using two types of conventional rubber, and is verified as an efficient vibration isolator with a broad operating bandwidth. It even breaks the limit on damping ratios required by classical passive-vibration theory.

Torsional Waves in a Fiber Composite Medium at a Loosely Bonded Interface Constrained Between Dry Sandy Layer and Gravitating Poroelastic Substrate

The objective of this paper is to study the effect of loosely bonded interface on torsional surface wave propagation in a fiber reinforced composite medium constrained between dry sandy layer and an anisotropic gravitating poroelastic substrate. All the media are assumed to be under initial stress. The dispersion relation on this proposed multilayer ground structure has been derived in closed form under certain boundary conditions, which contain Whittaker function and its derivative, which is further expanded asymptotically, retaining up to only the linear terms. The numerical solution for the limiting case of torsional surface waves is also discussed. As a special case of the problem, when the entire medium is isotropic and one of the upper layer vanishes and removing the initial stress and gravity, the dispersion relation obtained is in agreement with the classical Love type wave equation. The influence of various technical constants, such as sandy parameter, reinforcement parameter, porosity parameter, Biot’s gravity parameter, loosely bonded parameters, initial stress of both the layers and half spaces on the phase velocity of torsional surface wave has been pointed out by means of graphs.

Green’s function technique to model Love-type wave propagation due to an impulsive point source in a piezomagnetic layered structure

The crux of the present study is to unravel the concealed characteristics on the propagation behavior of Love-type wave through a piezomagnetic material (PMM) stratum overlying a functionally graded piezomagnetic material (FGPMM) substrate under the influence of an impulsive point source at its interfacial surface. Functional gradients in the substrate is due to the spatial variation in the elastic constants and density. The elasto-magneto-dynamical equations are laid down for the piezomagnetic medium. Green’s functions are derived for the layered structure taking a point force/point charge density at the interfacial surface into account. Dispersion equation of Love-type wave is obtained with aid of admissible boundary conditions and the well known properties of Green’s function. For sake of validation, the dispersion equation is reduced to fairly match with the classical Love wave equation. To demonstrate the analytical findings numerically and set forth the parametric effects, numerical data of two PMMs are undertaken for PMM stratum and FGPMM substrate. The graphical representations reveal that phase velocity of Love-type wave is significantly affected by the material constants and the functional gradients parameters allied with the piezomagnetic layered structure. The profound effects of piezomagnetic constants, magnetic permeabilities, magnifying gradient parameters and functional gradient parameter are portrayed by graphical means. A comparative analysis to trace out the piezomagnetic effect of the PMM stratum and FGPMM substrate is accomplished owing to their transverse isotropic orientational symmetry. This study may find noteworthy dynamic and practical applications towards manufacturing and optimization of piezomagnetic sensors and transducers. In addition, the obtained result may be useful for acquiring a better performance in surface acoustic wave devices and functionally graded material waveguides.

Stabilization of Starobinsky–Vilenkin stochastic inflation by an environmental noise

We discuss the inflaton [Formula: see text] in an interaction with an infinite number of fields treated as a random environment (noise) with a friction [Formula: see text]. In a Markovian approximation we obtain a stochastic wave equation (appearing also in the warm inflation models). After the replacement of the environment by the white noise, the stochastic wave equation violates the energy conservation if [Formula: see text]. We introduce a dark energy as a compensation of the inflaton energy–momentum. We add to the classical wave equation the Starobinsky–Vilenkin noise which in the slow-roll approximation describes the quantum fluctuations in an expanding metric. We investigate the resulting consistent stochastic Einstein–Klein–Gordon system in the slow-roll regime. We obtain Fokker–Planck equation for the probability distribution of the inflaton assuming that the dependence of the scale factor [Formula: see text] and the Hubble variable [Formula: see text] on the field [Formula: see text] is known. We obtain explicit stationary solutions of the Fokker–Planck equation assuming that [Formula: see text] and [Formula: see text] can approximately be determined in a slow-roll regime with a neglect of noise. We extend the results to the multifield [Formula: see text]-dimensional configuration space. We show that in the regime [Formula: see text] the quantum noise determines the asymptotic behavior of the stationary distribution. If [Formula: see text] stays finite then the environmental noise ensures the integrability of the stationary probability. In such a case there is no need to introduce boundary conditions with the purpose to eliminate infinite inflation. The variation of [Formula: see text] could be interpreted as a sign of a transition from cold inflation to warm inflation.

Dispersion and absorption study of SH waves in sinusoidally corrugated heterogeneous viscoelastic layer sandwiched between heterogeneous isotropic half-space and magnetoelastic monoclinic half-space

The present paper employs variable separable technique to investigate the effect of corrugated boundary surface along with other affecting parameters on the phase and damped velocity of SH-wave in a three medium mathematical model, i.e. a heterogeneous viscoelastic layer with corrugated boundary surface sandwiched between a heterogeneous isotropic half-space and an anisotropic magnetoelastic monoclinic half-space. The dispersion and absorption equations have been obtained in the closed form. Some special cases of the dispersion equation have been obtained, which are in good agreement with the Classical Love Wave Equation. The absence and presence of corrugated frontier surface are determined by the term initial flatness parameter. Considerable effects of some parameters such as initial flatness parameter, heterogeneity, magneto elasticity as well as viscoelasticity associated with the layers have been investigated for both the curves, i.e. dispersion curve and attenuation curve which corresponds to the dispersion equation and the absorption equation, respectively. Vital strides have been made by the integration of both the numerical computation and graphical plots to demonstrate those effects to accomplish the objective of the problem.

Rayleigh surface wave propagation in an orthotropic rotating magneto-thermoelastic medium subjected to gravity and initial stress

The prime objective of the present article is to analyze the effects of rotation and initial stress on the propagation of Rayleigh surface waves in a homogeneous, orthotropic magneto-thermoelastic half space subjected to gravity field. The frequency equations in closed form are derived and the amplitude ratios of surface displacements, temperature change during the Rayleigh wave propagation on the surface of half space have been computed analytically. The highlights of this study are the effects of different parameters (rotation, magnetic field, initial stress, and gravity) on the velocity of Rayleigh waves. Variation in phase velocity of Rayleigh waves against a wave number is shown graphically. Some particular cases have been deduced. Also, the classical Rayleigh wave equation is obtained as a special case of the present study. Numerical example has been carried out and represented by the means of graphs. Impacts of various involved parameters appearing in the solutions are carefully analyzed. In fact, in the absence of various parameters, these equations are in agreement with the results for isotropic medium.

EFFECTIVE MASS KLEIN-GORDON EQUATION WITH POSITION DEPENDENT MAGNETIC FIELD

Recently, exact solutions of the classical and relativistic wave equations for the existance of an external potential have been widely studied in view of the position-dependent mass formalism. Such eorts have important applications in technology, especially in material science such as electronic properties of the semi-conductors and quantum dots. In the present study, we aim to obtain exact solutions of the Klein-Gordon equation in the presence of an exponential magnetic eld via effective mass formalism. Energy eigenvalues are derived by using wave functions. The studied magnetic eld and effective mass have the form ~B = B0e􀀀x^k and m(x) = (m0 + m1e􀀀x).

Detecting a hidden obstacle via the time domain enclosure method. A scalar wave case

The characterization problem of the existence of an unknown obstacle behind a known obstacle is considered by using a singe observed wave at a place where the wave is generated. The unknown obstacle is invisible from the place by using a visible ray. A mathematical formulation of the problem using the classical wave equation is given. The main result consists of two parts: (a) one can make a decision whether the unknown obstacle exists or not behind a known impenetrable obstacle by using a single wave over a finite time interval under some a‐priori information on the position of the unknown obstacle; (b) one can obtain a lower bound on the Euclidean distance of the unknown obstacle to the center point of the support of the initial data of the wave. The proof is based on the idea of the time domain enclosure method and employs some previous results on the Gaussian lower/upper estimates for the heat kernels and domination of semigroups.

On weak regularity requirements of the relaxation modulus in viscoelasticity

Abstract The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.

Analysis on propagation characteristics of the shear wave in a triple layered concentric infinite long cylindrical structure: An analytical approach

The present work delves into the propagation of a horizontally polarised shear (SH) wave in an infinitely long cylindrical structure comprised of three concentric isotropic layered media. The model has been formulated in cylindrical co-ordinates and an analytical approach is employed to achieve the closed form of the dispersion relation. The application of the Debye Asymptotic Analysis to tackle the complexity arisen due to the involvement of Hankel’s function in the solution treatment is one of the key features of the present article. With the aid of the Debye Asymptotic Analysis, the dispersion equation is shown to be in well agreement with the classical Love wave equation in the absence of the outermost layer medium. The present analysis highlights the influence of the wave number and various radii ratios of the concentric cylindrical layered elastic medium on the phase velocity of the shear wave propagating in the embraced structure. Numerical computations have been carried out to accomplish the graphical demonstration unravelling some important peculiarities associated with the propagation characteristics of the shear wave in the considered cylindrical structure.

Classical Quantum Field Theory Based on the Hypothesis of the Absolute Reference System

The quantum field theory based on the hypothesis of the absolute reference system is a classical non-relativistic theory, which is compatible with current quantum theory. This conclusion arises when one compares the theoretical results of quantum electrodynamics using the basic principles of this hypothesis. Wave equation, which replaces this of Schrodinger, is the classical wave equation of a peculiar electromagnetic wave, derived from the study of particle structure.

Noether’s Theorem Applied to the Classical and Schrödinger Wave Equations

Noether’s Theorem, which relates continuous transformations to conservation laws, is applied to the classical wave equation and the Schrödinger equation. Transformations are derived that lead to invariances and conservation laws.

Symmetry reductions and exact solutions to the systems of nonlinear partial differential equations**This work was supported by the National Natural Science Foundation of China under Grant No. 11171041 and the high-level personnel foundation of Liaocheng University under grant No. 31805.

In this paper, the Lie symmetry analysis is performed on the four systems of nonlinear partial differential equations (PDEs). One is the system of nonlinear coupled PDEs in a real physical application. The other system describes the unsteady liquid and gas flows in long pipelines. The third is the approximate equations for long waves in shallow water. The fourth is the general classical dispersiveless long wave equations. All of the point symmetries of the systems are presented for the first time, then the symmetry reductions and exact solutions to the systems are investigated.

Wave-equation traveltime inversion with multifrequency bands: Synthetic and land data examples

We have developed a wave-equation traveltime inversion method with multifrequency bands to invert for the shallow or intermediate subsurface velocity distribution. Similar to the classical wave-equation traveltime inversion, this method searches for the velocity model that minimizes the squared sum of the traveltime residuals using source wavelets with progressively higher peak frequencies. Wave-equation traveltime inversion can partially avoid the cycle-skipping problem by recovering the low-wavenumber parts of the velocity model. However, we also use the frequency information hidden in the traveltimes to obtain a more highly resolved tomogram. Therefore, we use different frequency bands when calculating the Fréchet derivatives so that tomograms with better resolution can be reconstructed. Results are validated by the zero-offset gathers from the raw data associated with moderate geometric irregularities. The improved wave-equation traveltime method is robust and merely needs a rough estimate of the starting model. Numerical tests on the synthetic and field data sets validate the above claims.

Complex Fractional Zener Model of Wave Propagation in ℝ

Abstract The classical wave equation is generalized within fractional framework, by using fractional derivatives of real and complex order in the constitutive equation, so that it describes wave propagation in one dimensional infinite viscoelastic rod. We analyze existence, uniqueness and properties of solutions to the corresponding initial-boundary value problem for generalized wave equation. Also, we provide a comparative analysis with the case of the same equation but considered on a bounded or half-bounded spatial domain. We conclude our investigation with a numerical example that illustrates obtained results.

Influence of loose bonding, initial stress and reinforcement on Love-type wave propagating in a functionally graded piezoelectric composite structure

This present study investigates Love-type wave propagation in composite structure consists of a loosely bonded functionally graded piezoelectric material (FGPM) stratum lying over a functionally graded initially-stressed fibre-reinforced material (FGIFM) substrate. The closed-form expressions of the dispersion relation have been obtained analytically for both the cases of electrically open and electrically short conditions. Some special cases of the problem have also been studied and the obtained results are found in well-agreement with the classical Love wave equation. The emphatic influence of wave number, bonding parameter associated with bonding of stratum with substrate of the composite structure, piezoelectric coefficient as well as dielectric constant of the piezoelectric stratum, horizontal initial stresses, and functional gradedness of the composite structure on the phase velocity of Love-type wave has been reported and illustrated through numerical computation along with graphical demonstration in both the cases of electrically open and electrically short condition for the reinforced and reinforced-free composite structure. Comparative study has been carried out to analyze the distinct cases associated with functional gradedness of the composite structure and also various cases which reveals the influence of piezoelectricity, reinforcement and horizontal initial stress acting in the composite structure, and bonding of the stratum and substrate of the composite structure in context of the present problem which serves as one of the major highlights of the study.

Nonlocal operators with local boundary conditions in higher dimensions

We present novel nonlocal governing operators in 2D/3D for wave propagation and diffusion. The operators are inspired by peridynamics. They agree with the original peridynamics operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of separable kernel functions together with even and odd parts of bivariate functions on rectangular/box domains. The operators are bounded and self-adjoint. We present all possible 36 different types of BC in 2D which include pure and mixed combinations of Neumann, Dirichlet, periodic, and antiperiodic BC. Our construction is systematic and easy to follow. We provide numerical experiments that verify our theoretical findings. We also compare the solutions of the classical wave and heat equations to their nonlocal counterparts.