Wave Equations For Fields Research Articles

Scalar's quasibound states in cosmological black hole background

In this letter, we present in detail a novel exact solution of scalar quasibound states of a static spherically symmetric Schwarzschild de Sitter black hole. We investigate and work out the governing covariant scalar field wave equation, i.e., the Klein-Gordon equation and isolate the radial equation. In this letter, we show in detail, the derivation of the successfully obtained exact radial wave solutions which are expressed in terms of General Heun functions. Having the exact solutions in hand, the energy levels expression is obtained from special function's polynomial condition. In the last part, the Hawking radiation is investigated via the Damour-Ruffini method and the Hawking temperature is obtained from radiation distribution function.

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Dynamic response of semi-cylindrical depression, cylindrical cavity and type-III crack to SH wave in half-space anisotropic media

In this study, the anti-plane dynamic response of an elastic half-space anisotropic medium containing surface semi-cylindrical depressions and internal cylindrical cavity and type-III crack is solved analytically. The wave function expansion method, the complex function method and the Green's function method can be used to effectively construct the free wave field equation and the scattered wave field equation in the case of SH wave incidence, in which the scattered wave field generated by the crack is constructed by the Green's function method and the "crack cut" method. For the scattered wave field generated by a circular cavity is constructed using the mirror method, where the positional offset coefficient in the mirror method of a half-space anisotropic medium is introduced, which well solves the problem of the asymmetry of the scattered wave field. Finally, the effects of anisotropic strength of the medium, SH wave incidence angle and dimensionless incident wave number on the surface displacement amplitude (|W|), dynamic stress concentration factor (DSCF) at the boundary of the circular cavity and dynamic stress intensity factor (DSIF) at the crack tip are investigated in both frequency and time domains under different working conditions, and conclusions different from those of isotropic medium are obtained, which provide some theoretical references for the earthquake-resistant design of the underground structure with canyon topography and the surface building.

Research on the Time-Domain Explicit and Implicit Solution Methods of the Shallow Water Seismic Wavefield Equations

The current time-domain solution methods for the wavefield equations of a single medium do not apply to the wavefield equations of shallow water seismic with a fluid–elastomer coupling. To solve this problem, based on the explicit central difference method and implicit Newmark method, the explicit–explicit method, implicit–implicit method, and explicit–implicit method time-domain expressions for the local solution are derived, and the time-domain expressions for the explicit and implicit methods in the global solution are derived.The stability and computational efficiency of different time-domain solving methods for the shallow water seismic wavefield equations are theoretically analyzed. The numerical results are compared with the simulation data from the multiphysics field simulation software COMSOL 6.0, and the numerical stability, computational efficiency and accuracy of the different solving methods are also analyzed theoretically. The results show that the implicit method in the global solution is relatively optimal among the methods proposed in this paper, which ensures numerical stability at the larger step size for improving the computational efficiency and considers the higher computational efficiency and accuracy.

Exact scalar quasibound states solutions of f(R) theory's static spherically symmetric black hole

Following the recent developments in black holes' quasibound states exact solutions by Senjaya (2023b,c), in this letter, we present a novel exact solution of quasibound states of a scalar around a static spherically symmetric black hole solution of f(R) theory of gravity. We start with the relativistic scalar field wave equation, i.e. the Klein-Gordon equation, isolating the radial equation and finally successfully obtain the exact radial wave solutions in terms of General Heun functions. Having the exact solutions in hand, the energy levels expression is obtained from the radial wave polynomial conditions. In the last part of this letter, the Hawking radiation is investigated via the Damour-Ruffini method and the Hawking temperature is obtained from radiation distribution function.

Seismic Velocity Inversion via Physical Embedding Recurrent Neural Networks (RNN)

Seismic velocity inversion is one of the most critical issues in the field of seismic exploration and has long been the focus of numerous experts and scholars. In recent years, the advancement of machine learning technologies has infused new vitality into the research of seismic velocity inversion and yielded a wealth of research outcomes. Typically, seismic velocity inversion based on machine learning lacks control over physical processes and interpretability. Starting from wave theory and the physical processes of seismic data acquisition, this paper proposes a method for seismic velocity model inversion based on Physical Embedding Recurrent Neural Networks. Firstly, the wave equation is a mathematical representation of the physical process of acoustic waves propagating through a medium, and the finite difference method is an effective approach to solving the wave equation. With this in mind, we introduce the architecture of recurrent neural networks to describe the finite difference solution of the wave equation, realizing the embedding of physical processes into machine learning. Secondly, in seismic data acquisition, the propagation of acoustic waves from multiple sources through the medium represents a high-dimensional causal time series (wavefield snapshots), where the influential variable is the velocity model, and the received signals are the observations of the wavefield. This forms a forward modeling process as the forward simulation of the wavefield equation, and the use of error back-propagation between observations and calculations as the velocity inversion process. Through time-lapse inversion and by incorporating the causal information of wavefield propagation, the non-uniqueness issue in velocity inversion is mitigated. Through mathematical derivations and theoretical model analyses, the effectiveness and rationality of the method are demonstrated. In conjunction with simulation results for complex models, the method proposed in this paper can achieve velocity inversion in complex geological structures.

Deriving an Electric Wave Equation from Weber’s Electrodynamics

Weber’s electrodynamics presents an alternative theory to the widely accepted Maxwell–Lorentz electromagnetism. It is founded on the concept of direct action between particles, and has recently gained some momentum through theoretical and experimental advancements. However, a major criticism remains: the lack of a comprehensive electromagnetic wave equation for free space. Our motivation in this research article is to address this criticism, in some measure, by deriving an electric wave equation from Weber’s electrodynamics based on the axiom of vacuum polarization. Although this assumption has limited experimental evidence and its validity remains a topic of debate among researchers, it has been shown to be useful in the calculation of various quantum mechanical phenomena. Based on this concept, and beginning with Weber’s force, we derive an expression which resembles the familiar electric field wave equation derived from Maxwell’s equations.

Foundations for QED, Feynman operator calculus, Dyson conjectures, and Einstein’s dual theory

This paper reviews research on the foundations of quantum electrodynamics (QED). We show that there are three definitions of the proper time that follow from Einstein’s theory. The first definition is used to prove that the universe has a unique clock (Newton-Horwitz-Fanchi time) available to all observers. This clock is used to briefly discuss the mathematical foundations for Feynman’s time ordered operator calculus. We use this calculus to solve the first and second conjectures of Dyson for QED: that the renormalized perturbation series is asymptotic and, that the ultra-violet divergence is caused by a violation of the time-energy uncertainly relationship. The second definition gives Minkowski’s version of Einstein’s theory and its problems are briefly reviewed. The third definition gives the dual Newton, dual Maxwell and dual quantum theories. The theory is dual in that, for a set of n particles, every observer has two unique sets of global variables (X, t) and (X, τ) to study the system, where X is the canonical center of mass. Using (X, t) time is relative with speed c, while in (X, τ), time is unique with relative speed b. The dual Maxwell theory contains a longitudinal (dissipative) term in the E field wave equation, which appears instantaneously with acceleration and we predict that radiation from a cyclotron will not produce photoelectrons. It is shown that this term gives an effective mass for the photon. A major outcome is the dual unification of Newtonian mechanics and classical electrodynamics with Einstein’s theory and without the need for point particles or a self-energy divergency. This means that a second quantized version will not produce a self-energy or infrared divergency. These results along with the proof of Dyson’s second conjecture resolves all the problems with QED. The dual Dirac theory provides a new formula for the anomalous magnetic moment of a charged particle, which can give exact values for the electron, muon and proton g-factors.

A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> but also on the magnetic field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> residing over the interface. Thereby, another equation is essential for solving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> simultaneously. Realizing that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> is a volume variable, while <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.

Full Waveform Inversion of Viscoelastic Media Based on Gradient Preconditioning

In viscoelastic media, reverse time migration (RTM) requires accurate Q value for energy compensation and phase correction. Full waveform inversion (FWI) has the potential to provide accurate Q value. In the wide frequency band seismic data, the quality factor Q almost does not change with frequency, which is called constant Q model, but its computational complexity is large. In the narrow frequency band data, when the memory variable order L is assumed to be 1 in the standard linear solid (SLS) model, the attenuation simulation effect of the subsurface Q value is acceptable, and the calculation amount is small. In this paper, we derive the back propagation wave field equation and gradient formula of full waveform inversion of viscoelastic medium based on the SLS model with memory variable order L = 1, and analyzes the effectiveness and feasibility of this method. In viscoelastic medium, low velocity body will produce low frequency noise in seismic record, which will lead to heterogeneous gradient energy. This low frequency noise is similar to the low frequency noise produced near the source of shallow strata. This will reduce the accuracy of the inversion results. The gradient preconditioning method using the pseudo-Hessian operator can make the gradient energy more balanced. Therefore, we introduce the pseudo-Hessian operator in gradient preconditioning and derives the gradient preconditioning formula for the full waveform inversion in viscoelastic media, which solves the problem of the non-uniform gradient energy. Examples show that the method can suppress the non-uniform gradient energy in the full waveform inversion of viscoelastic media.

Double null evolution of a scalar field in Kerr spacetime

We have developed and implemented a numerical scheme and code that computes the evolution of the massless scalar field wave equation in Kerr spacetime, using double null coordinates. The results show a smooth behavior across the event horizon, making it possible to give initial data outside the event horizon, evolve the scalar field across the horizon, and keep the evolution even inside the black hole. This is the first time that a double null evolution is performed for a scalar field in Kerr spacetime.

A Finite Element-Based Characteristic Mode Analysis

A novel Green’s function-free characteristic modes formulation is introduced in this work. The desired impedance or admittance matrix is obtained utilizing and appropriately modifying the versatile finite element method. For this purpose, the generalized eigenvalue problem of the electric or magnetic field vector wave equation is formulated. In the case of the electric field wave equation, using the Schur complement, the system is reformulated and expressed only in terms of the tangential electric field over the radiating apertures, retaining the equivalent magnetic currents. Similarly, in the case of the magnetic field wave equation, the electric current density on radiating metallic surfaces is isolated using the Schur complement. In both cases, the obtained matrix is split into its real and imaginary part to yield the characteristic modes eigenvalue problem. Key advantage of the proposed formulation is that it does not require the evaluation of Green’s function, thereby the study of any arbitrarily shaped, multilayered geometry loaded with anisotropic and inhomogeneous materials is feasible. To prove the validity of the proposed methodology various classical structures, with both homogeneous, and inhomogeneous and anisotropic materials, published in the bibliography are studied. Both the eigenvalues and eigenvectors compared with the published results show good agreement.

Far-IR to deep-UV adaptive supercontinuum generation using semiconductor nano-antennas via carrier injection rate modulation

Supercontinuum generating sources, which incorporate a non-linear medium that can generate a wideband intensity spectrum under high-power excitation, are ideal for many applications of photonics such as spectroscopy and imaging. Supercontinuum generation using ultra-miniaturized devices is of great interest for on-chip imaging, on-chip measurement, and for future integrated photonic devices. In this study, semiconductor nano-antennas are proposed for ultra-broadband supercontinuum generation via analytical and numerical investigation of the electric field wave equation and the Lorentz dispersion model, incorporating semiconductor electron dynamics under optical excitation. It is shown that by a rapid modulation of the carrier injection rate for a semiconductor nano-antenna, one can generate an ultra-wideband supercontinuum that extends from the far-infrared (Far-IR) range to the deep-ultraviolet (Deep-UV) range for an infrared excitation of arbitrary intensity level. The modulation of the injection rate is achieved by high-intensity pulsed-pump irradiation of the nano-antenna, which has a fast nonradiative electron recombination mechanism that is on the order of sub-picoseconds. It is shown that when the pulse period of the pump irradiation is of the same order with the electron recombination time, rapid modulation of the free electron density occurs and electric energy accumulates in the nano-antenna, allowing for the generation of a broad supercontinuum. The numerical results are compared with the semiempirical second harmonic generation efficiency results for validation and a mean accuracy of 99.7% is observed. The aim of the study is to demonstrate that semiconductor nano-antennas can be employed to achieve superior supercontinuum generation performance at the nanoscale and the process can be programmed in an adaptive manner for continuous spectral shaping via tuning the pulse period of the pump irradiation.

A classical mechanical model of two interacting massless particles in de Sitter space and its quantization

A conformally invariant model of two interacting massless particles in Minkowski space was proposed by Casalbuoni and Gomis [1]. We generalize this model to the case of de Sitter space from the perspective of geodesic distance, in such a way that the resulting, generalized action reduces to the original action in a limit that de Sitter radius goes to infinity. We analyze the Hamiltonian formulation in accordance with Dirac's prescription for constrained Hamiltonian systems and carry out its subsequent canonical quantization in coordinate representation following DeWitt. As the result, we derive a fourth-order differential wave equation for bilocal fields that, in the infinite radius limit, reproduces one obtained in the original model for Minkowski space case.

Performance of 3D Wave Field Modeling Using the Staggered Grid Finite Difference Method with General-Purpose Processors

This paper aims to provide a quantitative understanding of the performance of numerical modeling of a wave field equation using general-purpose processors. In particular, this article presents the most important aspects related to the memory workloads and execution time of the numerical modeling of both acoustic and fully elastic waves in isotropic and anisotropic mediums. The results presented in this article were calculated for the staggered grid finite difference method. Our results show that the more realistic the seismic wave simulations that are performed, the more the demand for memory and the computational capacity of the computing environment increases. The results presented in this article allow the estimation of the memory requirements and computational time of wavefield modeling for the considered model (acoustic, elastic or anisotropic) so that their feasibility can be assessed in a given computing environment and within an acceptable time. Understanding the numerical modeling performance is especially important when graphical processing units (GPU) are utilized to satisfy the intensive calculations of three-dimensional seismic forward modeling.

Hidden symmetries for transparent de Sitter space

It is known that odd-dimensional de Sitter space acts as a transparent potential for free fields. Previous studies have explained this phenomena by relating de Sitter free field equations of motion to the time-independent Schrödinger equation with known transparent potentials. In this work we show that de Sitter’s transparency is a consequence of an infinite set of ‘hidden’ symmetries. These symmetries arise from an accidental symmetry for the zero-mode of matter fields, as well as the boost isometry of de Sitter space. For simplicity, we consider the case of massive Klein–Gordon theory. We show that the Noether charges associated with these hidden symmetries distinguish the two linearly-independent solutions of the free field wave equation in the asymptotic past and future of de Sitter. Conservation of these charges requires that the asymptotic behavior of any solution be identical, up to a constant phase, in the future and the past, which is the property of transparency. In the quantized theory, these charges act trivially on particle states belonging to the in/out vacuum Fock space. For particle states constructed from other vacua, the action of the charges generates particles.

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the tracefree matter case

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4D anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary problem for a system of quasilinear wave equations for various conformal fields by exploiting the conformal and coordinate gauges. By analysing the conformal constraints we show a systematic procedure to prescribe initial and boundary data. This analysis is complemented by the propagation of the constraints, showing that a solution to the wave equations implies a solution to the Einstein field equations. In addition, we study three explicit tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang–Mills field. For each one of these we identify the basic data required to couple them to the system of wave equations. As our main result, we establish the local existence and uniqueness of solutions for the evolution system in a neighbourhood around the corner, provided compatibility conditions for the initial and boundary data are imposed up to a certain order.

Scalar fields and parametrized spherically symmetric black holes: Can one hear the shape of space-time?

In this work we study whether parametrized spherically symmetric black hole solutions in metric theories of gravity can appear to be isospectral when studying perturbations. From a theory agnostic point of view, the test scalar field wave equation is the ideal starting point to approach the quasinormal mode spectrum in alternative black hole solutions. We use a parametrization for the metric proposed by Rezzolla and Zhidenko, as well as the higher order WKB method in the determination of the quasinormal mode spectra. We look for possible degeneracies in a tractable subset of the parameter space with respect to the Schwarzschild quasinormal modes. Considering the frequencies and damping times of the expected observationally most relevant quasinormal modes, we find such degeneracies. We explicitly demonstrate that the leading Schwarzschild quasinormal modes can be approximated by alternative black hole solutions when their mass is treated as free parameter. In practice, we conclude that the mass has to be known with extremely high precision in order to restrict the leading terms in the metric expansion to currently known limits coming from the PPN expansion. Possible limitations of using the quasinormal mode ringdown to investigate black hole space-times are discussed.

An Efficient Mode-Based Domain Decomposition Hybrid 2-D/Q-2D Finite-Element Time-Domain Method for Power/Ground Plate-Pair Analysis

Generally, with a surface magnetic current excitation, only the $\mathrm {TM}_{\mathrm {zmn}}$ modes are activated between the power/ground plate pair. As a result, the magnetic field only has azimuthal component while the longitude component is zero. To make full use of this quasi-2-D (Q-2D) property, a Q-2D finite-element time-domain (FETD) method combined with the basic 3-D triangular prism mesh elements is firstly proposed to solve the magnetic field wave equation. It is further noted that the higher order modes are confined in the proximity of the antipads (labeled as via domain) due to the exponential attenuation property along the propagating direction. Therefore, in the region sufficiently far from the antipad (marked as plate-pair domain), only the fundamental mode needs to be considered. In this way, a Q-2D FETD analysis is required in the via domain while in the plate-pair domain, only a 2-D FETD solver is needed. As a result, the proposed algorithm is actually a hybrid 2-D and Q-2D FETD method. Besides, to handle arbitrarily shaped antipads, the wave-port excitation is implemented, and the corresponding mode-based S-parameter extraction methodology is developed according to the orthogonal property of different eigenmodes. The efficiency, accuracy, and generality of the proposed approach are verified by several representative examples.

Complex Symmetric Formulation of Maxwell Equations for Fields and Potentials

Maxwell equations have two types of asymmetries between the electric and magnetic fields. The first asymmetry is the inhomogeneity induced by the absence of magnetic charge sources. The second asymmetry is due to parity. We show how both asymmetries are naturally resolved under an alternative formulation of Maxwell equations for fields or potentials that uses a compact complex vector operator representation. The developed complex symmetric operator formalism can be easily applied to performing the continuity equation, the field wave equations, the Maxwell equations for potentials, the gauge transformations, and the 4-momentum representation; in general, the developed formalism constitutes a simple way of unfolding the Maxwell theory. Finally, we provide insights for extending the presented analysis within the context of (i) bicomplex numbers and tessarine algebra; and (ii) Lp-spaces in nonlinear Maxwell equations.