First-order Wave Equation Research Articles

Composition Methods, Maxwell's Equations, and Source Terms

This paper is devoted to high-order numerical time integration of first-order wave equation systems originating from spatial discretization of Maxwell's equations. The focus lies on the accuracy of high-order composition in the presence of source functions. Source functions are known to generate order reduction, and this is most severe for high-order methods. For two methods based on two well-known fourth-order symmetric compositions, convergence results are given assuming simultaneous space-time grid refinement. Herewith physical sources and source functions emanating from Dirichlet boundary conditions are distinguished. Among other things it is shown that the reduction can cost two orders. On the other hand, when a certain perturbation of a source function is used, the reduction is generally diminished by one order. In that case, reduction is absent for physical sources and for Dirichlet sources the order is equal to at least three under stable simultaneous space-time grid refinement.

On the Linear Forms of the Schrödinger Equation

Generalizing the linearisation procedure used by Dirac and later by L\'evy-Leblond, we derive the first-order non-relativistic wave equations for particles of spin 1 and spin 3/2 starting from the Schrodinger equation.

The influence of plasma defocusing in high harmonic generation

We numerically investigate the influence of plasma defocusing in high harmonic generation (HHG) by solving the first-order wave equation in an ionized medium and defining an enhancement factor to quantitatively analyze the influence of plasma defocusing. While degrading the driver pulse intensity, plasma also has a strong impact on HHG phase-matching. Combined with the HHG wavelength scaling law, our results give an estimate of HHG efficiencies with different driver wavelengths and show a limited HHG efficiency in high density media.

Solutions of Podolsky's electrodynamics equation in the first-order formalism

The Podolsky generalized electrodynamics with higher derivatives is formulated in the first-order formalism. The first-order relativistic wave equation in the 20-dimensional matrix form is derived. We prove that the matrices of the equation obey the Petiau–Duffin–Kemmer algebra. The Hermitianizing matrix and Lagrangian in the first-order formalism are given. The projection operators extracting solutions of field equations for states with definite energy–momentum and spin projections are obtained, and we find the density matrix for the massive state. The 13 × 13-matrix Schrödinger form of the equation is derived, and the Hamiltonian is obtained. Projection operators extracting the physical eigenvalues of the Hamiltonian are found.

Optical pulse propagation with minimal approximations

Propagation equations for optical pulses are needed to assist in describing applications in ever more extreme situations---including those in metamaterials with linear and nonlinear magnetic responses. Here I show how to derive a single first-order propagation equation using a minimum of approximations and a straightforward ``factorization'' mathematical scheme. The approach generates exact coupled bidirectional equations, after which it is clear that the description can be reduced to a single unidirectional first-order wave equation by means of a simple ``slow evolution'' approximation, where the optical pulse changes little over the distance of one wavelength. It also allows a direct term-to-term comparison of an exact bidirectional theory with the approximate unidirectional theory.

A note on Lie superalgebras

We treat the possible Lie superalgebras where in addition to Poincar´ e generators there are n supergenerators. These superalgebras are determined with the help of relativistic wave equations. It is shown that structure constants are connected with the matrices of first-order relativistic wave equations. Some of these Lie superalgebras may be interesting from mathematical point of view.

Entrainment waves in decelerating transient turbulent jets

A simplified one-dimensional partial differential equation for the integral axial momentum flux during the deceleration phase of single-pulsed transient incompressible jets is derived and solved analytically. The wave speed of the derived first-order nonlinear wave equation shows that the momentum flux transient from the deceleration phase propagates downstream at twice the initial jet penetration rate. Transient-jet velocity data from the existing literature is shown to be consistent with this derivation, and an algebraic analytical solution matches the measured timing and decay of axial velocity after the deceleration transient. The solution also shows that a wave of increased entrainment accompanies the deceleration transient as it travels downstream through the jet. In the long-time limit, the peak entrainment rate at the leading edge of this ‘entrainment wave’ approaches an asymptotic value of three times that of the initial steady jet. The rate of approach to the asymptotic behaviour is controlled by the deceleration rate, which suggests that rate-shaping may be tailored to achieve a desired mixing state at a given time after the end of a single-pulsed jet. In the wake of the entrainment wave, the absolute entrainment rate eventually decays to zero. The local injected fluid concentration also decays, however, so that entrainment rate relative to the local concentration of injected fluid remains higher than in the initial steady jet. An analysis of diesel engine fuel-jets is provided as one example of a transient-jet application in which the considerable increase in the mixing rate after the deceleration phase has important implications.

Sea Level Variations in the Tropical Pacific Ocean and the Madden–Julian Oscillation

Abstract Satellite observations of sea level and surface wind from the tropical Pacific Ocean, and their relationship to the Madden–Julian oscillation (MJO), are analyzed using a combination of statistical techniques and a simple, physically based model. Wavenumber–frequency analysis reveals that sea level variations at the equator contain prominent eastward-propagating signals as the intraseasonal Kelvin waves. The component of sea level variation that is coherent with the MJO (ηMJO) is concentrated in a narrow strip along the equator between 150°E and 110°W. To explain the physical forcing of ηMJO, the component of zonal wind stress that is coherent with the MJO is also calculated. It is shown that is strongest in the western Pacific, but the MJO accounts for a higher percentage of the wind variance in the central equatorial Pacific. A simple linear model of the Kelvin waves, based on a first-order wave equation forced by and with a linear damping term included, successfully reproduces ηMJO. It is also shown that zonal wind variations to the east of the date line act to increase the apparent propagation speeds of the Kelvin waves.

Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme

To deal with the numerical dispersion problem, by combining the staggered-grid technology with the compact finite difference scheme, we derive a compact staggered-grid finite difference scheme from the first-order velocity-stress wave equations for the transversely isotropic media. Comparing the principal truncation error terms of the compact staggered-grid finite difference scheme, the staggered-grid finite difference scheme, and the compact finite difference scheme, we analyze the approximation accuracy of these three schemes using Fourier analysis. Finally, seismic wave numerical simulation in transversely isotropic (VTI) media is performed using the three schemes. The results indicate that the compact staggered-grid finite difference scheme has the smallest truncation error, the highest accuracy, and the weakest numerical dispersion among the three schemes. In summary, the numerical modeling shows the validity of the compact staggered-grid finite difference scheme.

Simple five-dimensional wave equation for a Dirac particle

A first-order relativistic wave equation is constructed in five dimensions. Its solutions are eight-component spinors, interpreted as single-particle fermion wave functions in four-dimensional space-time. Use of a “cylinder condition” (the removal of explicit dependence on the fifth coordinate) reduces each eight-component solution to a pair of degenerate four-component spinors. It is shown that, when the cylinder condition is applied, the results obtained from the new equation are the same as those obtained from the Dirac equation. Without the cylinder condition, on the other hand, the equation implies the existence of a scalar potential, and for zero-mass particles it leads to a four-dimensional fermionic equation analogous to Maxwell’s equation with sources.

A Simple Illustration of a Weak Spectral Cascade

The textbook first encounter with nonlinearity in a partial differential equation (PDE) is the first-order wave equation: $u_t + u u_x = 0$. Often referred to as the inviscid Burgers equation, this equation is familiar to many in the theoretical contexts of characteristics, wavebreaking, or shock propagation. Another canonical behavior contained within this simplest of PDEs is the spectral cascade. Surprisingly, buried in a little-known 1964 article by G.W. Platzman is an elegant example of an exact Fourier series solution associated with a purely sinusoidal initial condition. This Fourier representation, valid prior to wavebreaking, is generalized to arbitrary continuous initial conditions on both the periodic and infinite domains. For the specific example of Platzman's original problem, the Fourier coefficients decay exponentially with increasing wavenumber, and the decay rate flattens to zero precisely at the time of wavebreaking. It is demonstrated that two simplified descriptions, a downscale truncation and a linearization from initial conditions, also produce an exponential spectral cascade uniformly to large wavenumbers. This weak cascade is responsible for the initial generation of Fourier harmonics in the viscous Burgers equation.

A Note on Wave Equation and Convolutions

We study the first‐order nonhomogenous wave equation. We extend the convolution theorem into a general case with a double convolution as the nonhomogenous term. The uniqueness and continuity of the solution are proved and we provide some examples in order to validate our results.

Multisymplectic Structures and Discretizations for One-way Wave Equations

Multisymplectic structures for one-way wave equations are presented in this letter. Based on the multisymplectic formulation, we obtain the corresponding multisymplectic discretizations. The structure-preserving property of a finite difference scheme for the first-order one-way wave equation is proved. Implications and applications of this result are explored.

A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tilt anisotropic media

The paper presents a staggered-grid any even-order accurate finite-difference scheme for two-dimensional (2D), three-component (3C), first-order stress-velocity elastic wave equation and its stability condition in the arbitrary tilt anisotropic media; and derives a perfectly matched absorbing layer (PML) boundary condition and its staggered-grid any even-order accurate difference scheme in the 2D arbitrary tilt anisotropic media. The results of numerical modeling indicate that the modeling precision is high, the calculation efficiency is satisfactory and the absorbing boundary condition is better. The wave-front shapes of elastic waves are complex in the anisotropic media, and the velocity of qP wave is not always faster than that of qS wave. The wave-front triplication of qS wave and its events in both reflected domain and propagated domain, which are not commonly hyperbola, is a common phenomenon. When the symmetry axis is tilted in the TI media, the phenomenon of S-wave splitting is clearly observed in the snaps of three components and synthetic seismograms, and the events of all kinds of waves are asymmetric.

Oscillations in a Simple Microvascular Network

We have identified the simplest topology that will permit spontaneous oscillations in a model of microvascular blood flow that includes the plasma skimming effect and the Fahraeus-Lindqvist effect and assumes that the flow can be described by a first-order wave equation in blood hematocrit. Our analysis is based on transforming the governing partial differential equations into delay differential equations and analyzing the associated linear stability problem. In doing so we have discovered three dimensionless parameters, which can be used to predict the occurrence of nonlinear oscillations. Two of these parameters are related to the response of the hydraulic resistances in the branches to perturbations. The other parameter is related to the amount of time necessary for the blood to pass through each of the branches. The simple topology used in this study is much simpler than networks found in vivo. However, we believe our analysis will form the basis for understanding more complex networks.

Simulation seismic wave propagation in topographic structures using asymmetric staggered grids

A new 3 D finite-difference (FD) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures. The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise. Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme. The method had no interpolation between the fine and coarse grids. All grids were computed at the same spatial iteration. Complicated geometrical structures like rough submarine interface, fault and nonplanar interfaces were treated with fine irregular grids. Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time, at the same time, has satisfactory stability and accuracy.

3‐D Irregular Grids Finite Difference Method for Elastic Wave Propagation in Anisotropic Media

AbstractThis paper presents a new 3‐D finite‐difference (FD) method of spatially irregular grids to simulate elastic wave propagation in heterogeneous anisotropic media with topographic structures. The method approximates the first‐order elastic wave equations with irregular grids finite difference operator with second‐order time precision and fourth‐order spatial precision. Unlike the multi‐grid scheme, this method has no interpolation between the fine and coarse grids. All grids are computed at the same spatial iteration. Complicated geometrical structures like rough submarine interface, fault and nonplanar interfaces are treated with fine irregular grids. Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time, at the same time, has satisfactory stability and accuracy. The proposed scheme is more efficient than conventional methods in simulating seismic wave propagation in topographic structures.

Bounds for solutions of a six-point partial-difference scheme

This paper is concerned with a class of partial-difference equations which arise from discretising the one-dimensional heat and the first-order wave equations. Explicit bounds are found for their solutions and subexponential solutions are considered. Using the Green's function, we derive a stability criteria for arbitrary solutions of this equation.

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Leakage-flow-induced vibrations were examined using linear stability analysis in the case of a one-dimensional narrow parallel passage in which a plate could vibrate in beam mode. It was found that unlike the case of the single-degree-of-freedom system, instability could occur for the parallel passage in a continuous system. Destabilizing mechanisms were considered by deriving a wave equation of plate displacement, and the characteristics of the waves were examined. The wave equation was expressed using first- and second-order wave equations, and abeam equation. The energy supplied to the wave components of the vibration mode and the dispersion relation was examined. It was found that the wave expressed by the first-order wave equation, which was derived from the dissipation term of the leakage flow, could act as a negative damping force to the wave on the plate propagating forward when the wave speed of the first-order wave equation was greater than that of the plate vibration. The mechanism was greatly different from that of the single-degree-of-freedom system where the delay due to fluid inertia could induce unstable vibration. It was suggested that the leakage-flow-induced vibration of continuous system be similar to the Oil whip of the journal bearing.

On Alfvén waves in the solar breeze

The application to the solar wind motivates the consideration of Alfvén waves in a radial background magnetic field and radial mean flow, in two cases, viz., with velocity and magnetic field perturbations along parallels, or also with perturbations along meridians, combined in the radial components of vorticity and electric current. In both cases the same second-order Alfvén wave equation is obtained; it has, in general, two singularities. If the mean flow velocity is taken to be a power of radial distance, with exponent other than zero or unity, there is a transition layer. In general there is a second singularity, viz., a critical layer, where the Alfvén speed equals the mean flow velocity. There is one exceptional case in which the critical layer does not exist, namely a homogeneous medium, for which the mean flow velocity decays on the inverse square of the radial distance, and then Alfvén speed also decays in the same way, so that their ratio is a constant, leading to two possibilities: (i) the ratio is not unity, and the wave equation remains of the second-order; (ii) the wave equation becomes of first-order in the case the mean flow velocity and Alfvén speed are equal everywhere, because then the waves can propagate only in one direction. Case (i) corresponds to Alfvén waves in the solar breeze. Exact solutions of the wave equations are obtained for all values of the radius, as a single expression for the first-order wave equation, whereas for the second-order wave equation it is possible to obtain solutions for small and large radius; the transition level limits the radius of convergence of one of these solutions, but the two solutions together cover the full range of radial distances. The choices of boundary conditions are discussed and the wavefields plotted vs dimensionless distance for several values of the two dimensionless parameters of the problem, viz., the Alfvén number and dimensionless frequency, which appear in one combination only. The analytical results of the present paper are compared with the mostly numerical results in the literature. It is shown that the Alfvén waves exhibit some properties observed in the solar wind, like the nonequipartition of kinetic and magnetic energies.