scholarly journals Crime pays; homogenized wave equations for long times

2021 ◽  
pp. 1-42
Author(s):  
Grégoire Allaire ◽  
Agnes Lamacz-Keymling ◽  
Jeffrey Rauch
Keyword(s):  
Exact Solution ◽  
Asymptotic Expansion ◽  
Wave Equation ◽  
Wave Equations ◽  
High Order ◽  
Periodic Homogenization ◽  
Accurate Approximation ◽  
Systematic Analysis ◽  
Wave Numbers ◽  
Small Period

This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, ϵ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times t of order ϵ − 2 + δ for any δ > 0. Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of ϵ, yields an approximation of the exact solution with error O ( ϵ N ) for times ϵ − N with N as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms.

scholarly journals ASYMPTOTIC EXPANSION FOR DAMPED WAVE EQUATIONS WITH PERIODIC COEFFICIENTS

2001 ◽  
Vol 11 (07) ◽  
pp. 1285-1310 ◽  
Author(s):  
R. ORIVE ◽  
E. ZUAZUA ◽  
A. F. PAZOTO
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Heat Kernel ◽  
Wave Equations ◽  
Bloch Wave ◽  
Periodic Coefficients ◽  
First Approximation ◽  
Wave Decomposition ◽  
Dissipative Wave Equation

We consider a linear dissipative wave equation in ℝN with periodic coefficients. By means of Bloch wave decomposition, we obtain an expansion of solutions as t→∞ and conclude that, in a first approximation, the solutions behave as the homogenized heat kernel.

scholarly journals Interaction of mesons with a potential field

1948 ◽  
Vol 193 (1035) ◽  
pp. 559-579 ◽  
Keyword(s):  
Exact Solution ◽  
Wave Equation ◽  
Elastic Scattering ◽  
Electrostatic Field ◽  
Potential Field ◽  
Born Approximation ◽  
Wave Equations ◽  
Scattering Problem ◽  
Coulomb Field ◽  
Radial Wave

The Takata-Saketani wave equation for mesons is developed to allow the solution of a variety of problems concerning the behaviour of charged vector mesons in an electrostatic field. In particular the form of the radial wave equations is discussed. The general elastic scattering problem is also formulated, and the Born approximation for the phases is discussed for the Coulomb field, for which, as is known, no exact solution is possible. The exact treatment of scattering by a spherical well is considered. This is the only case for which exact treatment in terms of known functions seems practicable, and might be of service in indicating the limits of the validity of approximate processes in other cases.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

On the existence of compressional MHD oscillations in an inhomogeneous magnetoplasma

1990 ◽  
Vol 44 (2) ◽  
pp. 361-375 ◽  
Author(s):  
Andrew N. Wright
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Density Distribution ◽  
Cold Plasma ◽  
Wave Equations ◽  
Perturbation Field ◽  
Plasma Density Distribution ◽  
Background Magnetic Field ◽  
Transverse Fields ◽  
The Magnetic Field

In a cold plasma the wave equation for solely compressional magnetic field perturbations appears to decouple in any surface orthogonal to the background magnetic field. However, the compressional fields in any two of these surfaces are related to each other by the condition that the perturbation field b be divergence-free. Hence the wave equations in these surfaces are not truly decoupled from one another. If the two solutions happen to be ‘matched’ (i.e. V.b = 0) then the medium may execute a solely compressional oscillation. If the two solutions are unmatched then transverse fields must evolve. We consider two classes of compressional solutions and derive a set of criteria for when the medium will be able to support pure compressional field oscillations. These criteria relate to the geometry of the magnetic field and the plasma density distribution. We present the conditions in such a manner that it is easy to see if a given magnetoplasma is able to executive either of the compressional solutions we investigate.

Pseudoacoustic tilted transversely isotropic modeling with optimal k-space operator-based implicit finite-difference schemes

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. T139-T157 ◽  
Author(s):  
Shigang Xu ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Spatial Modeling ◽  
Wave Equations ◽  
Transversely Isotropic ◽  
Dispersion Relations ◽  
High Order ◽  
Space Operator ◽  
Modeling Accuracy ◽  
Tti Media ◽  
Temporal And Spatial

Current temporal high-order finite-difference (FD) stencils are mainly designed for isotropic wave equations, which cannot be directly extended to pseudoacoustic wave equations (PWEs) in tilted transversely isotropic (TTI) media. Moreover, it is difficult to obtain the time-space domain FD coefficients for anisotropic PWEs based on nonlinear dispersion relations in which anisotropy parameters are coupled with FD coefficients. Therefore, a second-order FD for temporal derivatives and a high-order FD for spatial derivatives are commonly used to discretize PWEs in TTI media. To improve the temporal and spatial modeling accuracy further, we have developed several effective FD schemes for modeling PWEs in TTI media. Through combining the [Formula: see text] (wavenumber)-space operators with the conventional implicit FD stencils (i.e., regular-grid [RG], staggered-grid [SG], and rotated SG [RSG]), we establish novel dispersion relations and determine FD coefficients using least-squares (LS). Based on [Formula: see text]-space operator compensation, we adopt the modified LS-based implicit RG-FD, implicit SG-FD, and implicit RSG-FD methods to respectively solve the second- and first-order PWEs in TTI media. Dispersion analyses indicate that the modified implicit FD schemes based on [Formula: see text]-space operator compensation can greatly increase the numerical accuracy at large wavenumbers. Modeling examples in TTI media demonstrate that the proposed FD schemes can adopt a short FD operator to simultaneously achieve high temporal and spatial modeling accuracy, thus significantly improve the computational efficiency compared with the conventional methods.

scholarly journals Some arguments for the wave equation in Quantum theory

2021 ◽  
Vol 5 (1) ◽  
pp. 314-336
Author(s):  
Tristram de Piro ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Continuity Equation ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Atomic System ◽  
Balmer Series ◽  
Inertial Frames ◽  
The Stability

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

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