Three-dimensional Wave Equation Research Articles

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

Non-polynomial Cubic Spline Method for Three-Dimensional Wave Equation

Non-polynomial Cubic Spline Method for Three-Dimensional Wave Equation

Identification and extraction of lateral target signals in tunnel geological prediction with the Karhunen-Loéve beamforming method

Before the construction of a tunnel, effective and accurate advanced prospecting techniques are required to detect unexpected geological heterogeneity around the tunnel. However, because the data collected during tunnel advance prediction include waves reflected from different directions and a large amount of environmental noise, the seismic records collected are extremely complex, and it is difficult to extract the target signals from the lateral area of the tunnel effectively. The beamforming method can create a focused wave front through the stack of the wavefield, improving the data quality of the target signal. This paper incorporates the Karhunen-Loéve method in the beamforming procedure to extract the lateral signal of the tunnel. First, the test principle of Karhunen-Loéve beamforming (KLBF) based on the tunnel seismic prediction (TSP) observation system is discussed, and the corresponding delay parameters are estimated according to different application conditions. In addition, ray tracing and three-dimensional wave equation methods are used to carry out numerical simulation research on the models of faults, karst caves and underground rivers. Finally, the corresponding KLBF method can be used to improve the quality of effective signals by different delay time calculation methods, which can better identify and extract the lateral reflected signals.

The Importance of Weber–Maxwell Electrodynamics in Electrical Engineering

Weber-Maxwell electrodynamics combines classical Weber electrodynamics and Maxwell’s equations, including all four field equations and the Lorentz force, into a single de facto equivalent three-dimensional wave equation. From classical Weber electrodynamics, Weber-Maxwell electrodynamics inherits properties in which the concept of the magnetic field is unnecessary, and Newton’s third law is satisfied under all circumstances. From Maxwell’s electrodynamics, Weber-Maxwell electrodynamics inherits the ability to be compatible with electromagnetic waves. This article shows that in Weber-Maxwell electrodynamics, all conservation laws are satisfied, and that electromagnetic waves in an isolated system do not possess energy and momentum, but only mediate them between particles of matter. Furthermore, the article shows that the modern formulation of Weber electrodynamics is clearly superior to standard electrodynamics in electrical engineering, because it not only eliminates internal contradictions, but also represents considerable simplification and compression.

Unified and extended trigonometric B-spline DQM for the numerical treatment of three-dimensional wave equations

In this work, we propose a new differential quadrature method (DQM) based on the Unified and Extended Trigonometric B-spline (UETB-spline) functions for the numerical approximation of 3D wave equations. The UETB -spline functions are modified and then utilized in DQM to compute the weighting coefficients of spatial derivatives. After inverting the obtained coefficients, we acquire systems of ODEs that are solved by the SSP-RK5,4 scheme. Some numerical examples are considered to examine the accuracy and proficiency of the proposed approach. The obtained solutions give excellent agreement with analytical solutions. Moreover, the analysis for the rate of convergence (ROC) is performed. Computational complexity exhibits that the approach is not complex in the view of the computational cost. All calculations have been done using Dev-C++ 6.3 version and graphs have been plotted using MATLAB 2015b software.

Improvement of a soundproofing vent hole

In tropical climate countries, the house structure with a vent hole is widely used. The vent hole has a structure that penetrates the house's outer wall, allowing fresh air from outside to enter the room. However, external traffic noise also enters the living spaces through the vents. A novel type of soundproof vent hole unit with both soundproofing and ventilation functions was proposed in previous work. An elliptical cavity was used for this unit, but it is difficult to manufacture an elliptical cavity with a high eccentricity; also, the installation and adjustment work are expected to be difficult. A rectangular cavity with an inlet and outlet in orthogonal planes fitted to the house walls is used in this study to provide soundproofing functionality to the vent hole. First, the generation mechanism of sound pressure components propagating inside the unit was theoretically elucidated using the three-dimensional wave equation. Second, to maximize soundproofing ability, methods for reducing the number of resonances and reducing the effect of higher-order sound pressure components are described.

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences.

Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.

Size effects on the mixed modes and defect modes for a nano-scale phononic crystal slab

The size-dependent band structure of an Si phononic crystal (PnC) slab with an air hole is studied by utilizing the non-classic wave equations of the nonlocal strain gradient theory (NSGT). The three-dimensional (3D) non-classic wave equations for the anisotropic material are derived according to the differential form of the NSGT. Based on the the general form of partial differential equation modules in COMSOL, a method is proposed to solve the non-classic wave equations. The bands of the in-plane modes and mixed modes are identified. The in-plane size effect and thickness effect on the band structure of the PnC slab are compared. It is found that the thickness effect only acts on the mixed modes. The relative width of the band gap is widened by the thickness effect. The effects of the geometric parameters on the thickness effect of the mixed modes are further studied, and a defect is introduced to the PnC supercell to reveal the influence of the size effects with stiffness-softening and stiffness-hardening on the defect modes. This study paves the way for studying and designing PnC slabs at nano-scale.

The Solvability of the 3-D Elastic Wave Equations in Inhomogeneous Media

In this research, the three-dimensional elastic wave equations with variable coefficients (i.e. propagate through inhomogeneous media) are solved with the application of the Fourier transform in the spatial coordinates. The wave equation is coupled variable coefficients PDEs whose solutions may have significant in engineering applications. The method utilizes the second order ODE as the baseline for obtaining the complete solution. The solution of second order ODEs is expressed in one integration because the variable coefficients are broken down into several functions and resulted in first order reduction. Moreover, the coupled equations are performed by the order reduction of the higher order ODEs into the second order. The extended procedure for integral equation is implemented for the solutions from the transformed wave equations to generate the explicit expression. It is shown that the proposed method of integral evaluation is resulted in finding the roots of polynomials. Hence, it is concluded that the solvability of the elastic wave equations is ensured by the proposed method.

Inverse moving point source problem for the wave equation

In this paper, we consider the problem of identifying a single moving point source for a three-dimensional wave equation from boundary measurements. Precisely, we show that the knowledge of the field generated by the source at six different points of the boundary over a finite time interval is sufficient to determine uniquely its trajectory. We also derive a Lipschitz stability estimate for the inversion.

Exploiting temporal data reuse and asynchrony in the reverse time migration

Reverse Time Migration (RTM) is a state-of-the-art algorithm used in seismic depth imaging in complex geological environments for the oil and gas exploration industry. It calculates high-resolution images by solving the three-dimensional acoustic wave equation using seismic datasets recorded at various receiver locations. Reverse Time Migration’s computational phases are predominantly composed of stencil computational kernels for the finite-difference time-domain scheme, applying the absorbing boundary conditions, and I/O operations needed for the imaging condition. In this paper, we integrate the asynchronous Multicore Wavefront Diamond (MWD) tiling approach into the full RTM workflow. Multicore Wavefront Diamond permits to further increase data reuse by leveraging spatial with Temporal Blocking (TB) during the stencil computations. This integration engenders new challenges with a snowball effect on the legacy synchronous RTM workflow as it requires rethinking of how the absorbing boundary conditions, the I/O operations, and the imaging condition operate. These disruptive changes are necessary to maintain the performance superiority of asynchronous stencil execution throughout the time integration, while ensuring the quality of the subsurface image does not deteriorate. We assess the overall performance of the new MWD-based RTM and compare against traditional Spatial Blocking (SB)-based RTM on various shared-memory systems using the SEG Salt3D model. The MWD-based RTM achieves up to 70% performance speedup compared to SB-based RTM. To our knowledge, this paper highlights for the first time the applicability of asynchronous executions with temporal blocking throughout the whole RTM. This may eventually create new research opportunities in improving hydrocarbon extraction for the petroleum industry.

Learning coordinate systems for fast and accurate acoustic modeling

Many popular numerical methods, such as finite difference and spectral methods, rely on simple domain geometry and environmental smoothness. Unfortunately, these features are rarely found in real-world simulations. We propose learning coordinate transformations with deep neural networks to facilitate acoustic modeling in complex media. Using automatic differentiation, we obtain new coordinate systems by solving various optimization problems that map the computational domain to a rectangular grid. Different choices of the objective function are utilized to attain different goals, including (i) mapping complicated boundaries such as the seafloor to straight lines and (ii) reducing the rank of two-dimensional slices of the refractive index. The first choice allows for domain decomposition to accurately model sharp discontinuities in density and sound speed. The second drastically accelerates a non-intrusive reduced-order modeling technique referred to as the dynamical low-rank approximation. Using realistic ocean test cases, we compare the performance of our learned coordinate systems with domain decomposition to the classic approach of smoothing sharp discontinuities, and we compare full-rank, low-rank, and coordinate-system-accelerated low-rank solutions of the three-dimensional parabolic wave equation.

An accurate and efficient local one-dimensional method for the 3D acoustic wave equation

Abstract We establish an accurate and efficient scheme with four-order accuracy for solving three-dimensional (3D) acoustic wave equation. First, the local one-dimensional method is used to transfer the 3D wave equation into three one-dimensional wave equations. Then, a new scheme is obtained by the Padé formulas for computation of spatial second derivatives and the correction of the truncation error remainder for discretization of temporal second derivative. It is compact and can be solved directly by the Thomas algorithm. Subsequently, the Fourier analysis method and the Lax equivalence theorem are employed to prove the stability and convergence of the present scheme, which shows that it is conditionally stable and convergent, and the stability condition is superior to that of most existing numerical methods of equivalent order of accuracy in the literature. It allows us to reduce computational cost with relatively large time step lengths. Finally, numerical examples have demonstrated high accuracy, stability, and efficiency of our method.

An explicit high-order compact finite difference scheme for the three-dimensional acoustic wave equation with variable speed of sound

In this paper, the correction for the remainder of the truncation error of the second-order central difference scheme is employed to discretize temporal derivative, while the fourth-order Padé schemes are directly used to compute spatial derivatives, an explicit high-order compact finite difference scheme to solve the three-dimensional acoustic wave equation with variable speed of sound is proposed. This new scheme has the fourth-order accuracy in both temporal and spatial directions. It has high computational efficiency since the Thomas algorithm is employed to solve three tridiagonal linear systems formed by the Padé schemes on the (n)th time step and then an explicit time advancement process is conducted for the th time step. The stability and convergence condition of the proposed scheme are proved. We extend the proposed method to solve problems with nonlinear source terms and systems. Numerical experiments are conducted to demonstrate theoretical analysis results of the proposed scheme.

On Non-Detectability of Non-Computability and the Degree of Non-Computability of Solutions of Circuit and Wave Equations on Digital Computers

It is known that there exist mathematical problems of practical relevance which cannot be computed on a Turing machine. An important example is the calculation of the first derivative of continuously differentiable functions. This paper precisely classifies the non-computability of the first derivative, and of the maximum-norm of the first derivative in the Zheng-Weihrauch hierarchy. Based on this classification, the paper investigates whether it is possible that a Turing machine detects this non-computability of the first derivative by observing the data of the problem, and whether it is possible to detect upper bounds for the peak value of the first derivative of continuously differentiable functions. So from a practical point of view, the question is whether it is possible to implement an exit-flag functionality for observing non-computability of the first derivative. This paper even studies two different types of exit-flag functionality. A strong one, where the Turing machine always has to stop, and a weak one, where the Turing machine stops if and only if the input lies within the corresponding set of interest. It will be shown that non-computability of the first derivative is not detectable by a Turing machine for two concrete examples, namely for the problem of computing the input–output behavior of simple analog circuits and for solutions of the three-dimensional wave equation. In addition, it is shown that it is even impossible to detect an upper bound for the maximum norm of the first derivative. In particular, it is shown that all three problems are not even semidecidable. Finally, we briefly discuss implications of these results for analog and quantum computing.

An accurate and efficient time-domain numerical model for simulating water-axisymmetric structure subjected to horizontal earthquakes

The seismic response of offshore structures is generally influenced by water-structure interaction. This study focuses on the earthquake-induced hydrodynamic forces on axisymmetric offshore structures. Based on the finite element method, an accurate and efficient time-domain numerical model for simulating water-axisymmetric structures subjected to horizontal earthquakes is developed. First, according to interface boundary condition and using the method of separation of variables, the three-dimensional (3D) wave equation governing the compressible water is transformed into a two-dimensional (2D) governing equation in the vertical and radial directions, where the solution in the circumferential direction is analytical. Secondly, an exact artificial boundary condition (ABC) in frequency domain is developed by using the separation of variables method. Third, a high-accuracy ABC that is local in time but global in space is obtained by applying the temporal localization method. Furthermore, the high-accuracy artificial boundary condition is discretized and coupled with the finite element equation of the near field, leading to a symmetric second-order ordinary differential equations. Finally, the coupled finite element equation for the water-axisymmetric cylinder is presented. The proposed method is used to study the effect of water compressibility on the seismic response of and evaluate the seismic responses of a composite bucket foundation.

A numerical method for the solution of the three-dimensional acoustic wave equation in a marine environment considering complex sources

The need to understand how complex acoustic sources propagate noise in a realistic environment is of growing interest. In this work, we propose a numerical model for the simulation of noise generated by sources with directivity and propagating in ocean waveguides. The numerical model solves the wave equation for the acoustic pressure in the physical space using a second-order accurate finite difference method (FDTD). The source is implemented using an improved form of the hard-source method, which implicitly takes into consideration the reflection of acoustic waves associated to the presence of the ocean free surface, through the use of the method of images. This novel method is shown to improve the results with respect to the standard hard source implementation. We first validate the numerical method considering both analytical solutions and benchmark cases for the case of a monopole and then explore the acoustic energy patterns developed in the case of dipole and quadrupole sources. Specifically, the algorithm and the implementation of complex sources are evaluated first in a semi-infinite fluid layer and then considering two classical waveguides: the Ideal one and the Pekeris one. The comparison with analytical results shows the numerical method’s accuracy and that incorporating free surface effects in the hard source implementation improves results. In addition, the study shows that the acoustic response in the near field, of the order of few kilometers from the source, is strongly influenced by the source’s directivity and orientation relative to the free surface. The results of this paper have implications for future research aimed at characterizing and quantifying ship propeller noise in realistic waveguides. Indeed, this preliminary work is necessary to proceed to more complex numerical experiments, such as considering a real propeller signal as a source or considering stratification of the medium or propagation in confined domains such as experimental tanks.

Analytical study of nonlinear water wave equations for their fractional solution structures

This paper examines the three-dimensional nonlinear time-fractional water wave equations for their analytical wave solutions. These are the equations of the names [Formula: see text]-Zakharov–Kuznetsov–Burgers equation and [Formula: see text]-Yu–Toda–Sasa–Fukuyama equation. The obtained wave solutions are in the form of kink, periodic and singular waves by utilizing the [Formula: see text]-expansion approach. The aforesaid solutions are verified and demonstrated graphically via symbolic soft computations.

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus {mathbb {T}}^3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on {mathbb {T}}^3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Phi ^4_3-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.