Scalar Wave Equation Research Articles

Cylindrical Unidirectional Wave Transmission and Experiments Based on UWB-SWPL

Current studies have found that electromagnetic waves propagate in both positive and negative directions on the surface of magneto-optical media, and even have unidirectional propagation characteristics. This unique physical phenomenon has attracted the attention of many scholars in the industry. Through the derivation and theoretical analysis of vector potential equation, scalar wave equation and dielectric wrapped conductor waveguide, the existing TM mode and TE mode are shown. Then the millimeter wave energy is effectively radiated by the horn antenna to achieve a high efficiency coupling device. The electromagnetic energy is gathered on the surface of the conductor and transmitted along the conductor. At last, the validation of surface wave transmission is realized through experiments, which proves its authenticity.

Deductions from the Quaternion Form of Maxwell’s Electromagnetic Equations

It is often claimed that Maxwell’s electromagnetic equations were originally written in terms of quaternions. Once returned to that form and treated with left and right hand operators as in the mathematics of P. M. Jack, a new seventh scalar electromagnetic field component emerges with possible relations to clean energy extraction and gravitation. It is the purpose here to examine this approach afresh and see how it might link up with other fairly recent, but little known, work in the field. Again, as with the usual form of Maxwell’s equations, a new scalar wave equation is derived but, on this occasion, due to the presence of the scalar component of the quaternion, that equation exhibits a wave speed greater than the speed of light. Historical and present uses within military and humanitarian contexts are considered briefly.

Efficient and Accurate Algorithm for the Full Modal Green's Kernel of the Scalar Wave Equation in Helioseismology

In this work, we provide an algorithm to compute efficiently and accurately the full outgoing modal Green's kernel for the scalar wave equation in local helioseismology under spherical symmetry. Du...

Local energy decay for scalar fields on time dependent non-trapping backgrounds

Author(s): Metcalfe, J; Sterbenz, J; Tataru, D | Abstract: We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a “two point” local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.

EXACT NON-REFLECTING BOUNDARY CONDITIONS WITH AN FDTD SCHEME FOR THE SCALAR WAVE EQUATION IN WAVEGUIDE PROBLEMS

EXACT NON-REFLECTING BOUNDARY CONDITIONS WITH AN FDTD SCHEME FOR THE SCALAR WAVE EQUATION IN WAVEGUIDE PROBLEMS

Viscoacoustic anisotropic wave equations

The wave equation plays a central role in seismic modeling, processing, imaging and inversion. Incorporating attenuation anisotropy into the acoustic anisotropic wave equations provides a choice for acoustic forward and inverse modeling in attenuating anisotropic media. However, the existing viscoacoustic anisotropic wave equations are obtained for a specified viscoacoustic model. We have developed a relatively general representation of the scalar and vector viscoacoustic wave equations for orthorhombic anisotropy. We also obtain the viscoacoustic wave equations for transverse isotropy as a special case. The viscoacoustic orthorhombic wave equations are flexible for multiple viscoacoustic models. We take into account the classic visocoacoustic models such as the Kelvin-Voigt, Maxwell, standard-linear-solid and Kjartansson models, and we derive the corresponding viscoacoustic wave equations in differential form. To analyze the wave propagation in viscoacoustic models, we derive the asymptotic point-source solution of the scalar wave equation. Numerical examples indicate a comparison of the acoustic waveforms excited by a point source in the viscoacoustic orthorhombic models and the corresponding nonattenuating model, and the effect of the attenuation anisotropy on the acoustic waveforms.

The acoustic analogy in four dimensions

The classical acoustic analogy theory is incomplete in the sense that the original research on the subject focused only on the prediction of acoustic pressure. There were no provisions for predicting the three components of acoustic velocity which are needed as input for aeroacoustic scattering applications. This is because the scalar wave equations of Lighthill and Ffowcs Williams and Hawkings were derived from the fluid conservation equations by eliminating three of the four governing differential equations from which the acoustic velocity could have been obtained. Recently developed acoustic analogy methods for predicting the acoustic velocity lead to complex formulations whose numerical evaluation can be problematic when providing input for large-scale scattering problems. Their calculation can overwhelm the numerical scattering process when the incident sound has high-frequency content, such as that produced by the rotating blades of various propulsion devices. To obtain improved acoustic analogy formulations for scattering and other applications, the three discarded differential equations are returned to the analysis in this paper and the historical acoustic analogy equations are derived anew using four-dimensional tensor methods. The 4-D formulation is an application of the electromagnetic analogy (EMA), a concept based on the equivalence of Maxwell’s equations and the fluid conservation equations of mass and momentum. The scalar equations of Lighthill, Ffowcs Williams and Hawkings, Farassat, and Kirchhoff are extended to four equations – one equation for the acoustic pressure (or acoustic density) and three equations for the acoustic velocity. The 4-D tensor representation provides significant theoretical and computational simplification relative to the classical approach. For each of the original acoustic analogy results, a single, concise formulation in 4D is derived that enables the simultaneous prediction of acoustic pressure and the three components of acoustic velocity.

An interval finite element method for electromagnetic problems with spatially uncertain parameters

During the manufacturing process of dielectric materials used in electromagnetic engineering, the electromagnetic parameters are often spatially uncertain due to the processing technology, environmental temperature, personal operations, etc. Traditionally, the random field model can be used to measure the spatial uncertainties, but its construction requires a large number of samples. On the contrary, the interval field model only needs the upper and lower bounds of the spatially uncertain parameters, which requires much less samples and furthermore is easy to understand and use for engineers. Therefore, in this paper, the interval field model is introduced to describe the spatial uncertainties of dielectric materials, and then an interval finite element method (IFEM) is proposed to calculate the upper and lower bounds of electromagnetic responses. Firstly, the interval field of the dielectric material is represented by the interval K-L expansion and inserted into the scalar Helmholtz wave equations, and thus the interval equilibrium equations are constructed according to the node-based finite element method. Secondly, a perturbation interval finite element method is developed for calculating the upper and lower bounds of electromagnetic responses such as the electric strength and magnetic strength. Finally, the effectiveness of the proposed method is verified by three numerical examples.

True-amplitude linearized waveform inversion with the quasi-elastic wave equation

We present a quasi-elastic wave equation as a function of the pressure variable, which can accurately model PP reflections with elastic amplitude variation with offset effects under the first-order Born approximation. The kinematic part of the quasi-elastic wave equation accurately models the propagation of P waves, whereas the virtual-source part, which models the amplitudes of reflections, is a function of the perturbations of density and Lamé parameters [Formula: see text] and [Formula: see text]. The quasi-elastic wave equation generates a scattering radiation pattern that is exactly the same as that for the elastic wave equation, and only requires the solution of two acoustic wave equations for each shot gather. This means that the quasi-elastic wave equation can be used for true-amplitude linearized waveform inversion (also known as least-squares reverse time migration) of elastic PP reflections, in which the corresponding misfit gradients are with respect to the perturbations of density and the P- and S-wave impedances. The perturbations of elastic parameters are iteratively updated by minimizing the [Formula: see text]-norm of the difference between the recorded PP reflections and the predicted pressure data modeled from the quasi-elastic wave equation. Numerical tests on synthetic and field data indicate that true-amplitude linearized waveform inversion using the quasi-elastic wave equation can account for the elastic PP amplitudes and provide a robust estimate of the perturbations of P- and S-wave impedances and, in some cases, the density. In addition, true-amplitude linearized waveform inversion provides images with a wider bandwidth and fewer artifacts because the PP amplitudes are accurately explained. We also determine the 2D scalar quasi-elastic wave equation for P-SV reflections and the 3D vector equation for PS reflections.

Upcoming-wave equation of seismic migration

Upcoming-wave is the basis of several seismic migration methods that use the principle of 1-way-time by propagating only from the event point to the receiver. The migration method that uses this upcoming wave theory is the finite difference migration and Fourier transformation migration. In this paper, we will elaborate how the theories of these two migrations are related and described from scalar wave equations to various forms of upcoming wave approximations, both differential equation form of space-time (x-t) and frequency - wave number (f-k) form, along with their transformations.

Global existence for systems of quasilinear wave equations in (1 + 4)-dimensions

Hörmander proved global existence of solutions for sufficiently small initial data for scalar wave equations in (1+4)-dimensions of the form □u=Q(u,u′,u″) where Q vanishes to second order and (∂u2Q)(0,0,0)=0. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms u∂αu=12∂αu2 and as such do not immediately generalize to systems. The current study remedies such and extends both results to the case of multiple speed systems.

Wave optics in spacetimes with compact gravitating object

We investigate the wave optics in spherically symmetric spacetimes: Schwarzschild black hole, spherical star with a perfect absorbing surface, and massless/massive Ellis wormholes. Assuming a point wave source, wave pattern and power spectrums for scattering waves are obtained by solving the scalar wave equation numerically. We found that the power spectrum at the observer in the forward direction shows oscillations with two characteristic periods determined by the interference effect associated with the photon sphere and the diffraction effect due to the absorbing boundary condition inside of the photon sphere.

Analog black holes and energy extraction by super-radiance from Bose Einstein condensates (BEC) with constant density

This paper investigates the acoustic superradiance of the density and phase fluctuations from the single vortex state of a Bose-Einstein condensate, by employing full time-domain and asymptotic frequency domain numerical calculations. The draining bathtub model of an incompressible barotropic fluid is adopted to describe the vortex. The propagation of the axisymmetric density and phase fluctuations in the condensate are governed by the massless scalar Klein-Gordon wave equation, which establishes the rotating black-hole analogy. Hence, the amplified scattering of these fluctuations from the vortex comprise the superradiance effect. A particular coordinate transformation is applied to reveal the event horizon and the ergosphere termwise in the metric and the respective asymptotic spectral solutions. A comparative analysis of the time domain and asymptotic frequency domain results are given for a range of rotational speed of the vortex and the frequency of the impinging fluctuations. The agreement at low rotational speeds of the vortex is shown to be very good, which starts to deteriorate at higher rotational speeds due to increasing constraint violations of the time-domain calculations. We further demonstrate an asymptotic upper bound for the superradiance as a function of vortex rotational speed, provided that the vortex remains stable.

Forward Modeling of GPR Data by Unstaggered Finite Difference Frequency Domain (FDFD) Method: An Approach towards an Appropriate Numerical Scheme

Forward modeling of ground penetrating radar (GPR) is an important part to the inversion/modeling of the observed data. The aim of this study is to establish specific numerical schemes for forward modeling of GPR data by finite difference frequency domain (FDFD) method which were originally developed for seismic or finite difference time domain (FDTD) method. A total number of six modified and improved FDFD techniques have been used to discretize the two-dimensional (2D) transverse electric (TE)-mode scalar wave equation in order to find the suitable method for this. These techniques include five-point classical to nine-point mixed unstaggered-grid configurations. The numerical schemes for three unsplit perfectly matched layer (PML) for nine-point mixed unstaggered-grid configurations are also presented. The applicability of these techniques is tested by using the underground models of relative permittivity and conductivity for the two cases of homogeneous and 2-cross models. GPR shot gather data for these two models are also produced for this study. The relative reflection errors of the numerical schemes are also estimated for the homogeneous model to comprehend the appropriate method for the modeling. The algorithm for complex-frequency shifted PML (CFSPML) gives the least error in case of the forward modeling of the GPR data.

Peeling on Kerr Spacetime: Linear and Semi-linear Scalar Fields

We study the peeling on Kerr spacetime for fields satisfying conformally invariant linear and nonlinear scalar wave equations. We follow an approach initiated by L.J. Mason and the first author for the Schwarzschild metric, based on a Penrose compactification and energy estimates. This approach provides a definition of the peeling at all orders in terms of Sobolev regularity near ${\mathscr I}$ instead of ${\cal C}^k$ regularity at ${\mathscr I}$, allowing to characterise completely and without loss the classes of initial data ensuring a certain order of peeling at ${\mathscr I}$. This paper extends the construction to the Kerr metric, confirms the validity and optimality of the flat spacetime model (in the sense that the same regularity and fall-off assumptions on the data guarantee the peeling behaviour in flat spacetime and on the Kerr metric) and does so for the first time for a nonlinear equation. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics.

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.

A new perspective on the Ermakov-Pinney and scalar wave equations

The first part of the paper proves that a subset of the general set of Ermakov-Pinney equationscan be obtained by differentiation of a first-order non-linear differential equation. The second partof the paper proves that, similarly, the equation for the amplitude function for the parametrix ofthe scalar wave equation can be obtained by covariant differentiation of a first-order non-linearequation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi) = 0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2 = 0. The auxiliary 1-forms (psi,rho) are evaluated explicitlyin Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained.Thus, the novel method developed in this paper can be used with profit in physical applications.

A stable high-order absorbing boundary based on continued fraction for scalar wave propagation in 2D and 3D unbounded layers

PurposeThe purpose of this paper is to propose a stable high-order absorbing boundary condition (ABC) based on new continued fraction for scalar wave propagation in 2D and 3D unbounded layers.Design/methodology/approachThe ABC is obtained based on continued fraction (CF) expansion of the frequency-domain dynamic stiffness coefficient (DtN kernel) on the artificial boundary of a truncated infinite domain. The CF which has been used to the thin layer method in [69] will be applied to the DtN method to develop a time-domain high-order ABC for the transient scalar wave propagation in 2D. Furthermore, a new stable composite-CF is proposed in this study for 3D unbounded layers by nesting the above CF for 2D layer and another CF.FindingsThe ABS has been transformed from frequency to time domain by using the auxiliary variable technique. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the ABC-FEM coupled system is discussed and cured.Originality/valueThis manuscript establishes a stable high-order time-domain ABC for the scalar wave equation in 2D and 3D unbounded layers, which is based on the new continued fraction. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the coupled system is discussed and cured.

An Electromagnetic Stochastic Finite Element Method for Helmholtz-Type Wave Propagation Analysis

In this paper, an electromagnetic stochastic finite element method is presented to calculate the statistical moments of electromagnetic problems with spatially uncertain dielectric parameters. First, the random field model of the dielectric material is represented by the Karhunen–Loeve expansion and inserted into the scalar Helmholtz wave equations. The stochastic equilibrium equations are thus constructed according to the node-based finite element method. Second, the first-order and second-order perturbation stochastic finite element methods are developed for calculating the statistical moments of electromagnetic responses. Finally, the feasibility and validity of the proposed method are verified by three numerical examples.

Higher Order Cut Finite Elements for the Wave Equation

The scalar wave equation is solved using higher order immersed finite elements. We demonstrate that higher order convergence can be obtained. Small cuts with the background mesh are stabilized by adding penalty terms to the weak formulation. This ensures that the condition numbers of the mass and stiffness matrix are independent of how the boundary cuts the mesh. The penalties consist of jumps in higher order derivatives integrated over the interior faces of the elements cut by the boundary. The dependence on the polynomial degree of the condition number of the stabilized mass matrix is estimated. We conclude that the condition number grows extremely fast when increasing the polynomial degree of the finite element space. The time step restriction of the resulting system is investigated numerically and is concluded not to be worse than for a standard (non-immersed) finite element method.