Wave Equation Research Articles

RHDLPP: A multigroup radiation hydrodynamics code for laser-produced plasmas

In this paper, we introduce the RHDLPP, a flux-limited multigroup radiation hydrodynamics numerical code designed for simulating laser-produced plasmas in diverse environments. The code bifurcates into two packages: RHDLPP-LTP for low-temperature plasmas generated by moderate-intensity nanosecond lasers, and RHDLPP-HTP for high-temperature, high-density plasmas formed by high-intensity laser pulses. The core radiation hydrodynamic equations are resolved in the Eulerian frame, employing an operator-split method. This method decomposes the solution into two substeps: first, the explicit resolution of the hyperbolic subsystems integrating radiation and fluid dynamics; second, the implicit treatment of the parabolic part comprising stiff radiation diffusion, heat conduction, and energy exchange. Laser propagation and energy deposition are modeled through a hybrid approach, combining geometrical-optics ray-tracing in sub-critical plasma regions with a one-dimensional solution of the Helmholtz wave equation in super-critical areas. The thermodynamic states are ascertained using an equation of state, based on either the real gas approximation or the quotidian equation of state (QEOS). For ionization calculations, the code employs a steady-state collisional-radiation (CR) model using the screened-hydrogenic approximation. Additionally, RHDLPP includes RHDLPP-SpeIma3D, a three-dimensional spectral simulation post-processing module, for generating both temporally-spatially resolved and time-integrated spectra and imaging, facilitating direct comparisons with experimental data. The paper showcases a series of verification tests to establish the code's accuracy and efficiency, followed by application cases, including simulations of laser-produced aluminium (Al) plasmas, pre-pulse-induced target deformation of tin (Sn) microdroplets relevant to extreme ultraviolet lithography light sources, and varied imaging and spectroscopic simulations. These simulations highlight RHDLPP's effectiveness and applicability in fields such as laser-induced breakdown spectroscopy, extreme ultraviolet lithography sources, and high-energy-density physics.

Global existence and stability of the wave equation with boundary variable damping

AbstractIn this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities.

P-wave propagation in attenuative elliptical VTI media

Elliptical anisotropy is convenient to use as the reference medium in perturbation methods designed to study P-wave propagation for transverse isotropy (TI). We make the elliptically anisotropic TI model attenuative and discuss the corresponding P-wave dispersion relation and the wave equation. Our analysis leads to two conditions in terms of the Thomsen-type parameters, which guarantee that the P-wave slowness surface and the dispersion relation satisfy elliptical equations. We also obtain the viscoacoustic wave equation for such elliptically anisotropic media and solve it for point-source radiation using the correspondence principle. For the constant- Q TI model, we use the weighting function method to derive the viscoacoustic wave equation in differential form. Numerical examples validate the proposed elliptical conditions and illustrate the behavior of the P wavefield in attenuative elliptical TI models.

Quantum approach to bound states in field theory

It is well known that (possibly nonunique) suitable field dynamics can be prescribed in spacetimes with timelike boundaries by means of appropriate boundary conditions. In [R. M. Wald, ], Wald derived a conserved energy functional for each prescribed dynamics. This conserved energy is related to the positive self-adjoint extensions of the spatial part A of the wave equation ∂2Φ/∂t2=−AΦ (A may not be, in principle, essentially self-adjoint). This is quite surprising since the canonical energy is not conserved in these cases. In this paper, we rederive this energy functional from an action principle (with appropriate boundary terms) following [A. A. Saharian, ] and consider field dynamics arising from nonpositive self-adjoint extensions of A. The spectrum of the resulting theory fails to be positive and unstable mode solutions for classical fields come to light. By studying fields in half-Minkowski spacetime, we illustrate that these unstable classical solutions come as a consequence of an inverted parabolic potential governing their dynamics. From the quantum mechanical point of view, this leads to an effective inverted harmonic oscillator at the boundary. We then explore these unstable modes behavior, as well as their instabilities, at the quantum level. Published by the American Physical Society 2024

Uniqueness of an inverse cavity scattering problem for the time-harmonic biharmonic wave equation

This paper addresses an inverse cavity scattering problem associated with the time-harmonic biharmonic wave equation in two dimensions. The objective is to determine the domain or shape of the cavity. The Green’s representations are demonstrated for the solution to the boundary value problem, and the one-to-one correspondence is confirmed between the Helmholtz component of biharmonic waves and the resulting far-field patterns. Two mixed reciprocity relations are deduced, linking the scattered field generated by plane waves to the far-field pattern produced by various types of point sources. Furthermore, the symmetry relations are explored for the scattered fields generated by point sources. Finally, we present two uniqueness results for the inverse problem by utilizing both far-field patterns and phaseless near-field data.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

Limiting behavior of quasilinear wave equations with fractional-type dissipation

Abstract In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.

Reconstruction of the initial data from the trace of the solutions on an infinite time cylinder of damped wave equations *

In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation. We recover the initial velocity from the trace of the solution on a space-time cylinder. This inverse problem is related to photoacoustic tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

Greybody factors, quasi-normal modes and thermal fluctuations of quantum-corrected Schwarzschild black hole surrounded by quintessence

In this study, we examine the effects of a quintessence field-enclosed Schwarzschild black hole that has undergone quantum corrections on a number of different physical phenomena. The effective potential and its graphical features across different values of the quantum correction and quintessence field as well as relevant physical parameters are analyzed by deriving the radial equation from the Klein–Gordon equation and converting it into a Schrodinger wave equation using the tortoise coordinate. Both the event and cosmic horizons are found in the solutions to the radial equation. We provide a detailed analysis of the greybody component and its behavior by merging these solutions in the intermediate regime. By analyzing the black hole metric function, effective potential, and absorption probabilities, the study uncovers the complex relationship between black hole characteristics, quantum corrections, and the surrounding quintessence field. Investigating the BH’s quasinormal modes and thermal fluctuations provides insights into its stability as well as thermodynamic properties. The findings offer a comprehensive understanding of how the black hole interacts with its environment and responds to environmental changes, shedding light on the rich dynamics of black hole systems in the presence of quantum corrections and quintessence fields.

Shear wave anisotropy response characteristics of laminated shale and its correction methods

Laminated shale exhibits strong anisotropic characteristics in horizontal wells, significantly affecting the application of shear wave travel time data. Using equivalent medium theory and the elastic wave equation, we constructed a physical model for laminated shale, simulating the impact of lamination angle and lamination density on shear wave velocity and shear wave anisotropy coefficient. We established the relationship between the shear wave anisotropy coefficient and the cosine of the lamination angle. The simulation results indicate that shear wave velocity decreases with an increase in lamination angle and density, while the shear wave anisotropy coefficient gradually decreases with an increase in lamination angle and initially increases and then decreases with an increase in lamination density. Moreover, it exhibits a power exponential relationship with the cosine of the lamination angle. After correction, the minimum horizontal principal stress and fracture pressure calculated from shear wave data show an average relative error reduction of 6.72% and 7.77%, respectively, compared to measured data. This demonstrates that the correction method effectively eliminates the impact of laminated shale anisotropy on shear waves.

Well‐posedness and error analysis of wave equations with Markovian switching

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems.

A Set of Accurate Dispersive Nonlinear Wave Equations

In this study, a set of accurate dispersive nonlinear wave equations is established, using the wave velocity and free surface elevation as variables. These equations improve upon previous equations in which the velocity potential is used as a variable by considering the rotational wave motion and by adding a second-order bottom slope term that applies to general situations, allowing the equations to consider the influence of rapidly changing, horizontal, two-dimensional bottom topographies. The problem of the inaccuracy of the integral calculations used in previous equations in nearshore areas is solved by approximating the integral calculations into differential calculations, and a set of coupled wave equations is established by keeping the free surface elevation and the horizontal velocity constant, thus allowing the calculation of nearshore wave-generated currents. The benefits of the current model are confirmed through comparisons with corresponding laboratory experimental findings and are illustrated through a comparison with the numerical outcomes of other pertinent models.

Hilbert transform view of water-wave theory

A general rethinking of the mathematical foundations of water surface waves from the perspective of the Hilbert transform uncovers shortcomings of the standard multiple-scale approach as well as elucidates the interplay of non-local and dispersive effects. Application of the Hilbert transforms to planar and cylindrical settings allows us to deduce new weakly nonlinear models, including an alternative to Zakharov's equation and an envelope equation for cylindrical waves on deep water, as well as to highlight the crucial differences between these geometries.

New solitary waves in a convecting fluid

It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations.

Recovery of a general nonlinearity in the semilinear wave equation

We study the inverse problem of recovery a nonlinearity f ( t , x , u ), which is compactly supported in x, in the semilinear wave equation u tt − Δ u + f ( t , x , u ) = 0. We probe the medium with either complex or real-valued harmonic waves of wavelength ∼ h and amplitude ∼ 1. They propagate in a regime where the nonlinearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits supp x f. We show that one can recover f ( t , x , u ) when it is an odd function of u, and we can recover α ( x ) when f ( t , x , u ) = α ( x ) u 2 m . This is done in an explicit way as h → 0.

Analysis of the co-rotating vortex pair as a test case for computational aeroacoustics

The co-rotating vortex pair arrangement is a frequently applied test case in computational aeroacoustics. Its flow quantities and far-field radiated sound pressure can be computed analytically, assuming delta-distribution potential vortices. When this configuration is used for verification in a numerical computation, the singular vortex cores cannot be modeled directly; they have to be desingularized, which affects the associated flow and acoustical fields. Closed-form expressions for the aeroacoustic source terms are derived for the Scully vortex model and used for detailed numerical computations of the radiated sound pressure. The effects of desingularization and finite source region size are analyzed in the frequency domain for Lighthill’s equation and the perturbed convective wave equation by means of an efficient numerical approach that computes the convolution of the aeroacoustic source terms and the 2D free-field Green’s function. It is shown that the deviations of the far-field radiated sound pressure from the idealized analytical solution depend on the Mach number, vortex core radius, and truncation limit of the source region. A significant increase in the effective size of the source region due to the desingularization is demonstrated. Minimal error bounds compared to the analytic far-field pressure solution are established. The presented converged numerical results serve as reference solutions for validating implementations of hybrid computational aeroacoustic workflows.

A new highly accurate and efficient pure visco‐acoustic wave equation for tilted transversely isotropic attenuating media

AbstractThe propagation of seismic waves in attenuating anisotropic media exhibits amplitude dissipation and phase dispersion. To describe its effects, the fractional Laplacian pure visco‐acoustic wave equations capable of producing stable and noise‐free wavefields have been derived. However, except for acoustic approximation, previous wave equations utilize the approximations with lower accuracy in simplifying the denominator of the approximate complex‐valued dispersion relation, resulting in reduced accuracy. To address this concern, we use a combination of complex stiffness coefficients to replace the denominator term of the approximate complex‐valued dispersion relation. This approximation effectively reduces the loss of accuracy caused by ignoring the influence of the velocity anisotropy parameter ε and the attenuation anisotropy parameter εQ in the denominator term, leading to a wave equation with high accuracy in media with large anisotropic parameters ε and δ. In addition, the new wave equation only contains two high‐order spatial partial derivatives and has high computational efficiency. Theoretical analysis and numerical examples demonstrate that the proposed pure visco‐acoustic tilted transversely isotropic wave equation outperforms the previous pure visco‐acoustic wave equation in terms of simulation accuracy. The newly developed wave equation is well suited for the application of Q‐compensated reverse time migration and full waveform inversion in attenuating anisotropic media.

Q−Relativistic wave equation of the form i∂^q·ψq + mψ_0 = Eψ

In this paper we introduce a q relativistic wave equation of the form i∂^q·ψq + mψ_0 = Eψ. We present the q-spinorial solutions using the method of separated variables in the q-relativistic wave equation. Some comments are mentioned at the end of the paper.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.