Wave Equation

A wave equation on a one-dimensional interval I has a van der Pol type nonlinear boundary condition at the right end. At the left end, the boundary condition is fixed. At exactly the midpoint of the interval I, energy is injected into the system through a pair of transmission conditions in the feedback form of anti-damping. We wish to study chaotic wave propagation in the system. A cause of chaos by snapback repellers has been identified. These snapback repellers are repelling fixed points possessing homoclinic orbits of the non-invertible map in 2D corresponding to wave reflections and transmissions at, respectively, the boundary and the middleof-the-span points. Existing literature ͓F. R. Marotto, J. Math. Anal. Appl. 63, 199-223 ͑1978͔͒ on snapback repellers contains an error. We clarify the error and give a refined theorem that snapback repellers imply chaos. Numerical simulations of chaotic vibration are also illustrated.

We consider a hydrogen atom in the background spacetimes generated by an infinitely thin cosmic string and by a point-like global monopole. In both cases, we find the solutions of the corresponding Dirac equations and we determine the energy levels of the atom. We investigate how the geometric and topological features of these spacetimes leads to shifts in the energy levels as compared with the flat Minkowski spacetime.

A model based on the nite volume method combined with a rst-order approximate Riemann solver is used to solve the two-dimensional shallow w ater wave equation on an unstructured triangular grid. Veri cation of the model is achieved by comparing the model results with analytical solutions as well as documented published results with very good agreement. The model is applied to a case study involving the sudden collapse of a water supply reservoir. These reservoirs are generally located in elevated positions in residential areas and represent a potential risk to life and property. The model is used to predict the progress of the ood ensuing from the instantaneous collapse of a water supply reservoir. The model is conservative, robust, e cient, capable of simulating the wetting and drying processes, resolves shocks, simulates ow around complex geometry and obstacles and includes the in uence of steep bed slopes and friction. 1991 Mathematics Subject Classi cation. 65C20, 65M99.

In this study, the analytical solutions of the Dirac equation have been presented for the Hulthén and Eckart potentials by applying an approximation to centrifugal-like term in the case of spin symmetry, Delta(r) = C = constant, and pseudospin symmetry, Sigma(r) = C = constant, for any spin-orbit quantum number kappa. The energy eigenvalues and corresponding spinor wave functions are

We investigated the bound-state spectrum of a particle with spin Fу 1 2 in the magnetic field of a thin currentcarrying wire. We prove the existence of infinitely many bound states for any F. The bound-state energies closely follow a Coulomb-like behavior with an effective angular momentum l *, suggesting a high adiabaticity of the system. Numerical results (Fϭ 1 2 ,...,3) are well approximated by a single simple formula for the quantum defect, which is more accurate than the approximations given in the literature. We find a qualitative disagreement with the result obtained by adiabatic approximation.

This paper presents a multidimensional state-space approach for the numerical simulation of sound propagation in enclosures. The simulation algorithm is essentially based on the wave digital filter principle however we will give a more direct access to the numerical solution of the wave equation here. To simulate sound propagation in enclosures, a detailed treatment of the boundary conditions is necessary. We focus in this paper on the treatment of memoryless boundary conditions in the new simulation algorithm.

A fluid-structure-interaction problem comprising a membrane interacting with an aerodynamic flow is of interest for research focused on the behavior of flexible aerospace structures. However, there are no systematic studies available in literature considering this type of problem. The present work examines the numerical solution method and the typical solutions for a 1D vibrating membrane, i.e. a vibrating string, subject to an Euler flow. To assess the vibrating string behavior we present a non-dimensionalization of the fluid-structure-interaction equations, showing us 3 similarity parameters characterizing the interaction system. Furthermore we present a specific space-time finite-element discretization technique for the structure part based on the discontinuous Galerkin method. To relate the discontinuous elements we introduce Riemann solutions on the inter-element boundaries. Accordingly we rewrite the vibrating string equation into a system of first order wave equations. In the...

A simple non-quasi-static small-signal equivalent circuit model is derived for the ideal MOSFET wave equation under the gradual channel approximation. This equivalent circuit represents each Y-parameter by its DC small-signal value shunted by a (trans) capacitor in series with a charging (trans) resistor. A large-signal model for the intrinsic MOSFET is derived by first implementing this RC topology in the

We use an energy method to solve a three-point boundary-value problem for a hyperbolic equation with a Bessel operator and an integral condition. The proof is based on an energy inequality and on the fact that the range of the operator generated is dense. * Mathematics Subject Classifications: 35L20, 35L67, 34B15.

A method is proposed to augment the proper orthogonal decomposition basis functions with discontinuity modes to better capture moving discontinuities in reduced-order models. Moving discontinuities can be shocks in unsteady gas flows or bubbles in multiphase flow. The method is shown to work for a simple test problem using the first-order wave equation. A method for detecting discontinuities numerically is developed using mathematical morphology. This method is shown to properly identify the edges of bubbles in multiphase flow.

The main goal of this paper is to present a numerical model describing the major physical phenomena involved in electromagnetic casting industrial processes as precisely as possible. Under suitable physical assumptions, we derive the set of equations in the two-dimensional case; we describe in detail the numerical methods used to solve such equations and derive an iterative algorithm. Numerical results describing the case of an aluminium ingot are presented in order to show the efliciency of the method. lc'

Linear and nonlinear stoehastie wave equations given by a spaee-time Gaussian white noise are eonsidered in a spaee of dimension d 2: 2. In the linear ease the solution is a random Sehwartz distribution. In the nonlinear ease existence and uniqueness of solutions is proven in the framework of Colombeau random generalized funetions. In the ease where the nonlinear drift is given by the Fourier transform f of a eomplex measure (for instanee when f is a trigonometrie funetion) the solution is proven to be IJ1 associated with the solution of the free equation, for any p 2: I.

This paper presents an extension to the rectangular dielectric waveguide technique to obtain the permittivity of samples iteratively from the effective refractive index measurements by using the solution of the wave equation. The effective refractive index is presented for several samples in the 33 GHz to 50 GHz frequency range. The method is very fast and simple and provides an accuracy of better than 1% for the determination of permittivity of materials.

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock.

R. Ikehata recently proved some integral estimate for the di¤erence between the solution of an abstract heat equation and the solution of an abstract wave equation which results from the heat equation by a time singular perturbation. The estimate is obtained if the initial values are chosen appropriately. We prove a pointwise estimate which improves the above result for large times into several directions, and we also establish the optimality of this estimate for the wave equation in an exterior domain. Our proofs rely on the spectral theorem for unbounded self-adjoint operators.

The proposed paper presents the unobserved inadequacies in de Broglie's given concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc 2 is shown to be inappropriate for matter waves and is acceptable only for photons, where the symbols have their usual meanings. The superluminal phase velocity expression c 2 /υ, for matter waves, is investigated in detail and is also reported to be inadequate in the proposed paper. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality concepts, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications provided in the paper. A new relation between the Kinetic energy and Momentum of the moving body is also proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems, presented in the paper, provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product. 2 (1.6) Date: May 26, 2018. 1991 Mathematics Subject Classification. Primary 81Q05, 33D45, 35C05, 42A38; Secondary 81Q15, 20C35. Key words and phrases. The Cauchy initial value problem, the Schrödinger equation, Hamiltonian, harmonic oscillator, the hypergeometric function, hyperspherical harmonics, Bargmann function, the Laguerre polynomials, the Meixner polynomials, the Meixner-Pollaczek polynomials.

The dispersive properties of finite element semidiscretizations of the two-dimensional wave equation are examined. Both bilinear quadrilateral elements and linear triangular elements are considered with diagonal and nondiagonal mass matrices in uniform meshes. It is shown that mass diagonalization and underintegration of the stiffness matrix of the quadrilateral element markedly increases dispersive errors. The dispersive properties of triangular meshes depends on the mesh layout; certain layouts introduce optical modes which amplify numerically induced oscillations and dispersive errors. Compared to the five-point Laplacian finite difference operator, rectangular finite element semidiscretizations with consistent mass matrices provide superior fidelity regardless of the wave direction.

A magnetomechanical model for the design and control of Villari-effect magnetostrictive sensors is presented. The model quantifies the magnetization changes that a magnetostrictive material undergoes when subjected to a dc excitation field and variable stresses. The magnetic behavior is characterized by considering the Jiles-Atherton mean field theory for ferromagnetic hysteresis, which is constructed from a thermodynamic balance between the energy available for magnetic moment rotation and the energy lost as domain walls attach to and detach from pinning sites. The effect of stress on magnetization is quantified through a law of approach to the anhysteretic magnetization. Elastic properties of the sensor are incorporated by means of a wave equation that quantifies the strains and stresses arising in response to moment rotations. This yields a nonlinear PDE system for the strains, stresses and magnetization state of a magnetostrictive transducer as it drives or is driven by external loads. Because the model addresses the magnetoelastic coupling, it is applicable to both magnetostrictive sensors and actuators. Properties of the model and approximation method are illustrated by comparison with experimental data collected from a Terfenol-D sensor.

The formalism based on the equal-time Wigner function of the two-point correlation function for a quantized Klein-Gordon field is presented. The notion of the gauge-invariant Wigner transform is introduced and equations for the corresponding phase-space calculus are formulated. The equations of motion governing the Wigner function of the Klein-Gordon field are derived. It is shown that they lead to a relativistic transport equation with electric and magnetic forces and quantum corrections. The governing equations are much simpler than in the fermionic case which has been treated earlier. In addition the newly developed formalism is applied towards the description of spontaneous symmetry breakdown.

The spectral element method, which provides an accurate solution of the elastodynamic problem in heterogeneous media, is implemented in a code, called RegSEM, to compute seismic wave propagation at the regional scale. By regional scale we here mean distances ranging from about 1 km (local scale) to 90 • (continental scale). The advantage of RegSEM resides in its ability to accurately take into account 3-D discontinuities such as the sediment-rock interface and the Moho. For this purpose, one version of the code handles local unstructured meshes and another version manages continental structured meshes. The wave equation can be solved in any velocity model, including anisotropy and intrinsic attenuation in the continental version. To validate the code, results from RegSEM are compared to analytical and semi-analytical solutions available in simple cases (e.g. explosion in PREM, plane wave in a hemispherical basin). In addition, realistic simulations of an earthquake in different tomographic models of Europe are performed. All these simulations show the great flexibility of the code and point out the large influence of the shallow layers on the propagation of seismic waves at the regional scale.

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

The Salnt-Venant equations are used to describe river waves. Generally, for flood muting in rivers, the Saint-Venant system is reduced to the diffusive wave equation which can be resolved using finite-difference algorithms. The choice of a numerical method, and of the space and time steps to be retained, depends essentially on the form of flood hydrographs and the hydraulic properties of the river. This paper investigates these areas; two sets of criteria are proposed, the first to define parameter ranges representing each wave type and then, in the particular case of the diffusive wave model, to define criteria for the choice of numerical algorithm and appropriate space and time steps. The first analysis was based on the concept that river wave behaviour is determined by the balance between friction and inertia. The conclusions relate to the magnitude of temporal character° istics of flood waves, expressed as a function of the Froude number of the steady uniform flow and a dimensionless wave number of the unsteady component of the motion. The second part discussed questions related to the diffusive wave problem and to numerical instabilities. A technique is proposed to guide the user in the choice of the computational algorithm and specifies the error introduced by numerical methods. The technique was applied to flood-roaring simulation for the Loire river in France. In this case, two finite-difference algorithm~ were compared to the exact solution given by the analytical method. Comparisons between results show the efficiency of the technique to optimise the choice of the finite-dif:ference method and the adequate space and time steps.

The partial differential equation of diffusion is generalized by replacing the first order time derivative by a fractional derivative of order a, 0 < a 6 2. An approximate solution based on the decomposition method is given for the generalized fractional diffusion (diffusion-wave) equation. The fractional derivative is described in the Caputo sense. Numerical example is given to show the application of the present technique. Results show the transition from a pure diffusion process (a = 1) to a pure wave process (a = 2).

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion BH (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in R d , like, for example, (B1( , deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly we prove that some processes like C(|B1(|B2(...|Bn+1(t)|...)|)|) are governed by fractional diffusion equations.

In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero.

Autor: Eric Brandão. Revisão Técnica: William D'Andrea Fonseca. "Este livro aborda os princípios para a modelagem e caracterização da propagação do som em ambientes, bem como os fundamentos para o desenvolvimento de projetos de recintos com uma qualidade acústica adequada. A obra apresenta conceitos básicos sobre o som e processamento de sinais, propagação sonora em ambientes (em baixas, médias e altas frequências), cálculo e medição de parâmetros objetivos, como tempo de reverberação, claridade, espacialidade e inteligibilidade da fala etc. Além disso, aborda a modelagem, a caracterização e o projeto de dispositivos utilizados no tratamento acústico de ambientes (absorvedores e difusores). O capítulo final trata de diretrizes para o projeto de vários tipos de ambientes, como estúdios, salas de concerto, teatros, salas de aula, cinemas etc. A obra é destinada a professores, alunos e profissionais que atuam no projeto acústico de ambientes, como os engenheiros acústicos, engenheiros civis, arquitetos, físicos e técnicos de áudio; também serve de base para pessoas que almejam um bom entendimento sobre o tema da acústica de salas. O livro conta com sólido desenvolvimento matemático e diversos exemplos e discussões, contribuindo para solidificar os conceitos apresentados." Preview/amostra em: https://www.blucher.com.br/livro/detalhes/acustica-de-salas-1163

In this article, we implement relatively a new analytical technique, the Adomian decomposition method, for solving the boundary value problems of time-fractional wave equation. The fractional derivative is described in the Caputo sense. The decomposition method is used to construct analytical approximate solutions of time-fractional wave equation subject to specified boundary conditions. The solutions are calculated in the form of a convergent series with easily computable components. Some examples are given. The results reveal that the Adomian method is very effective and convenient.

In this study, absorption of high frequency radio waves in the ionospheric plasma have been investigated. The wave equation was obtained in terms of ionospheric parameters. The numerical values of the absorption have been calculated for 4 MHz, 4.5 MHz and 5 MHz waves. The necessary parameters for calculation have been obtained using an International Reference Ionosphere (IRI) Model. The altitudinal, diurnal, seasonal and the variations of absorption with frequency have been examined. The calculations show that the highest absorption occurs in the D-region. The absorption is higher in summer than in other seasons and is maximum at daylight. In addition, absorption decreases with the increase of frequency.

A technique is described for the solution of the wave equation with time dependent boundary conditions. The finite element solution accompanied by the numerical Laplace inversion process Seems to be an efficient procedure to treat such problems. 'The programming involved is straightforward in the sense that numerical Laplace inversion routines can be directly used as a time integration procedure after obtaining standard finite element differential equation solutions inthe transformed domain.

Prospecting for oil and gas resources poses the problem of determining the geological structure of the earth&#39;s crust from indirect measurements. Seismic migration is an acoustic image reconstruction technique based on the inversion of the scalar wave equation. Extensive computation is necessary before reliable information can be extracted from large sets of recorded data. In this paper a collection of &quot;industrial&quot; migration techniques, each giving rise to a data parallel algorithm, is outlined. Computer simulations on synthetic seismic data illustrate the problem and the approach

It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of ``shape-invariant&#39;&#39; potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.

Sound wave propagation in a relativistic perfect fluid with a non-homogeneous isentropic flow is studied in terms of acoustic geometry. The sound wave equation turns out to be equivalent to the equation of motion for a massless scalar field propagating in a curved space-time geometry. The geometry is described by the acoustic metric tensor that depends locally on the equation of state and the four-velocity of the fluid. For a relativistic supersonic flow in curved space-time the ergosphere and acoustic horizon may be defined in a way analogous to the non-relativistic case. A general-relativistic expression for the acoustic analog of surface gravity has been found.

The book was written in a textbook format based on course notes used in an advance level and graduate course at Iowa State University. In an introductory section a brief review of the physical significance and the mathematical behaviour of the most common partial differential equations encountered in fluid mechanics and heat transfer is presented, followed by the introduction of the basic fundamentals of finite difference techniques. These techniques are then applied to obtain the solution of selected model equations (wave, conduction, Laplace and Burgers). The individual characteristics of the various schemes are illustrated and discussed.

Symmetry and Separation of Variables: Encyclopedia of Mathematics and its Applications: Volume 4 Willard Miller Frontmatter More information vi Contents Chapter 3 The Three-Variable Helmholtz and Laplace Equations.. . 160 3.1 The Helmholtz Equation (A 3 + <o 2)^ = 0 160 3.2 A Hilbert Space Model: The Sphere S 2 169 3.3 Lame Polynomials and Functions on the Sphere 3.4 Expansion Formulas for Separable Solutions of the Helmoltz Equation 3.5 Non-Hilbert Space Models for Solutions of the Helmholtz Equation 3.6 The Laplace Equation A 3^ = 0 204 3.7 Identities Relating Separable Solutions of the Laplace Equation 213 Exercises 222 Chapter 4 The Wave Equation 223 4.1 The Equation * "-A 2 * = 0 223 4.2 The Laplace Operator on the Sphere 4.3 Diagonalization of P 0 , P 2 , and D 4.4 The Schrodinger and EPD Equations 4.5 The Wave Equation (3 //-A 3)^(x) = 0 Exercises Chapter 5 The Hypergeometric Function and Its Generalizations.. .

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all