scholarly journals Reciprocity and Representation Theorems for Flux- and Field-Normalised Decomposed Wave Fields

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Kees Wapenaar
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Multiple Scattering ◽  
Representation Theorems ◽  
Wave Fields ◽  
Coupled Equations ◽  
Preferred Direction ◽  
Symmetry Properties ◽  
Backward Propagation ◽  
Decomposition Operators

We consider wave propagation problems in which there is a preferred direction of propagation. To account for propagation in preferred directions, the wave equation is decomposed into a set of coupled equations for waves that propagate in opposite directions along the preferred axis. This decomposition is not unique. We discuss flux-normalised and field-normalised decomposition in a systematic way, analyse the symmetry properties of the decomposition operators, and use these symmetry properties to derive reciprocity theorems for the decomposed wave fields, for both types of normalisation. Based on the field-normalised reciprocity theorems, we derive representation theorems for decomposed wave fields. In particular, we derive double- and single-sided Kirchhoff-Helmholtz integrals for forward and backward propagation of decomposed wave fields. The single-sided Kirchhoff-Helmholtz integrals for backward propagation of field-normalised decomposed wave fields find applications in reflection imaging, accounting for multiple scattering.

Wave propagation in heterogeneous media: Effects of fine-scale heterogeneity

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. T37-T49 ◽  
Author(s):  
Florence Delprat-Jannaud ◽  
Patrick Lailly
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Multiple Scattering ◽  
Stability Condition ◽  
Heterogeneous Media ◽  
Fine Scale ◽  
Fine Grid ◽  
Scale Heterogeneity ◽  
Scattering Effects ◽  
Multiple Scattering Effects

We study the multiple scattering effects caused by fine-scale heterogeneity. For this purpose, it is unreasonable to rely on a linearization of the dependency of the wavefield in the parameters that describe the medium. Therefore, the only tools that correctly model wave propagation are based on the numerical solution of the (two-way) wave equation by finite differences or finite elements. A fine grid provides a straightforward approach to account for fine-scale heterogeneity. In this situation, there is no need for high-order schemes. Variational methods allow us to exhibit a numerical scheme that accounts for heterogeneity and that is, by construction, stable, provided a stability condition is fulfilled. This condition is a sufficient-stability condition contrary to the classical necessary-stability conditions. In addition, general mathematical results prove the finite-difference solution is close to the solution of the wave equation when the grid is fine enough. The multiple scattering effects caused by fine-scale heterogeneity are very important. In particular, we observe that imaging the so-computed synthetic data by standard migration techniques (that assume a linearization of the above-mentioned dependency) shows a strongly noise-corrupted image. This illustrates the importance of preprocessing data to remove the effect of multiple scattering. We try to improve the signal-to-noise ratio by removing multiples related to the free surface. Although significant noise reduction is achieved, even more sophisticated preprocessing is required to obtain a clear image of the subsurface.

scholarly journals Symmetry aspects in the macroscopic dynamics of magnetorheological gels and general liquid crystalline magnetic elastomers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Harald Pleiner ◽  
Helmut R. Brand
Keyword(s):  
Electric Fields ◽  
Time Reversal ◽  
Liquid Crystalline ◽  
Second Law Of Thermodynamics ◽  
Time Reversal Symmetry ◽  
Local Conservation ◽  
Continuous Symmetry ◽  
Polar Order ◽  
Preferred Direction ◽  
Symmetry Properties

Abstract We investigate theoretically the macroscopic dynamics of various types of ordered magnetic fluid, gel, and elastomeric phases. We take a symmetry point of view and emphasize its importance for a macroscopic description. The interactions and couplings among the relevant variables are based on their individual symmetry behavior, irrespective of the detailed nature of the microscopic interactions involved. Concerning the variables we discriminate between conserved variables related to a local conservation law, symmetry variables describing a (spontaneously) broken continuous symmetry (e.g., due to a preferred direction) and slowly relaxing ones that arise from special conditions of the system are considered. Among the relevant symmetries, we consider the behavior under spatial rotations (e.g., discriminating scalars, vectors or tensors), under spatial inversion (discriminating e.g., polar and axial vectors), and under time reversal symmetry (discriminating e.g., velocities from polarizations, or electric fields from magnetic ones). Those symmetries are crucial not only to find the possible cross-couplings correctly but also to get a description of the macroscopic dynamics that is compatible with thermodynamics. In particular, time reversal symmetry is decisive to get the second law of thermodynamics right. We discuss (conventional quadrupolar) nematic order, polar order, active polar order, as well as ferromagnetic order and tetrahedral (octupolar) order. In a second step, we show some of the consequences of the symmetry properties for the various systems that we have worked on within the SPP1681, including magnetic nematic (and cholesteric) elastomers, ferromagnetic nematics (also with tetrahedral order), ferromagnetic elastomers with tetrahedral order, gels and elastomers with polar or active polar order, and finally magnetorheological fluids and gels in a one- and two-fluid description.

Angle-Domain Gathers Computation Using Poynting Vector of Two-Way Acoustic Wave Equation

2014 ◽  
Vol 962-965 ◽  
pp. 2984-2987
Author(s):  
Jia Jia Yang ◽  
Bing Shou He ◽  
Ting Chen
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Real Data ◽  
Flow Density ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Wave Fields ◽  
Time Migration ◽  
Source Wave

Based on two-way acoustic wave equation, we present a method for computing angle-domain common-image gathers for reverse time migration. The method calculates the propagation direction of source wave-fields and receiver wave-fields according to expression of energy flow density vectors (Poynting vectors) of acoustic wave equation in space-time domain to obtain the reflection angle, then apply the normalized cross-correlation imaging condition to achieve the angle-domain common-image gathers. The angle gathers obtained can be used for migration velocity analysis, AVA analysis and so on. Numerical examples and real data examples demonstrate the effectiveness of this method.

A Smoothing Scheme for Seismic Wave Propagation Simulation with a Mixed FEM of Diffusionized Wave Equation

2021 ◽  
pp. 2150061
Author(s):  
Ryuta Imai ◽  
Naoki Kasui ◽  
Masayuki Yamada ◽  
Koji Hada ◽  
Hiroyuki Fujiwara
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Seismic Wave ◽  
Time Integration ◽  
Coupled System ◽  
Seismic Wave Propagation ◽  
Implicit Time ◽  
Integration Scheme ◽  
Weak Formulation ◽  
Mixed Fem

In this paper, we propose a smoothing scheme for seismic wave propagation simulation. The proposed scheme is based on a diffusionized wave equation with the fourth-order spatial derivative term. So, the solution requires higher regularity in the usual weak formulation. Reducing the diffusionized wave equation to a coupled system of diffusion equations yields a mixed FEM to ease the regularity. We mathematically explain how our scheme works for smoothing. We construct a semi-implicit time integration scheme and apply it to the wave equation. This numerical experiment reveals that our scheme is effective for filtering short wavelength components in seismic wave propagation simulation.

Abundant exact solutions to the strain wave equation in micro-structured solids

2021 ◽  
pp. 2150439
Author(s):  
Karmina K. Ali ◽  
R. Yilmazer ◽  
H. Bulut ◽  
Tolga Aktürk ◽  
M. S. Osman
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Exact Solutions ◽  
Rational Function ◽  
Physical Interpretation ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Singular Solutions ◽  
Strain Wave ◽  
Important Place

In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired.

On Steady-State Solutions of a Wave Equation by Solving a Delay Differential Equation With an Incremental Harmonic Balance Method

2018 ◽  
Author(s):  
Xuefeng Wang ◽  
Mao Liu ◽  
Weidong Zhu
Keyword(s):  
Differential Equation ◽  
Wave Propagation ◽  
Wave Equation ◽  
Steady State ◽  
Delay Differential Equation ◽  
Harmonic Balance ◽  
Incremental Harmonic Balance ◽  
Delay Differential ◽  
Ihb Method ◽  
Steady State Solutions

For wave propagation in periodic media with strong nonlinearity, steady-state solutions can be obtained by solving a corresponding nonlinear delay differential equation (DDE). Based on the periodicity, the steady-state response of a repeated particle or segment in the media contains the complete information of solutions for the wave equation. Considering the motion of the selected particle or segment as a variable, motions of its adjacent particles or segments can be described by the same variable function with different phases, which are delayed variables. Thus, the governing equation for wave propagation can be converted to a nonlinear DDE with multiple delays. A modified incremental harmonic balance (IHB) method is presented here to solve the nonlinear DDE by introducing a delay matrix operator, where a direct approach is used to efficiently and automatically construct the Jacobian matrix for the nonlinear residual in the IHB method. This method is presented by solving an example of a one-dimensional monatomic chain under a nonlinear Hertzian contact law. Results are well matched with those in previous work, while calculation time and derivation effort are significantly reduced. Also there is no additional derivation required to solve new wave systems with different governing equations.

On the construction and efficiency of staggered numerical differentiators for the wave equation

Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 107-110 ◽  
Author(s):  
M. Kindelan ◽  
A. Kamel ◽  
P. Sguazzero
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Computational Efficiency ◽  
Differential Operators ◽  
High Order

Finite‐difference (FD) techniques have established themselves as viable tools for the numerical modeling of wave propagation. The accuracy and the computational efficiency of numerical modeling can be enhanced by using high‐order spatial differential operators (Dablain,1986).

scholarly journals Solution of Acoustic Tomographic Problems on the Basis of Numerical Simulation of Particle Dynamics

2018 ◽  
Vol 155 ◽  
pp. 01019
Author(s):  
Angela Kuzovova ◽  
Ivan Kuzmenko
Keyword(s):  
Numerical Simulation ◽  
Numerical Modeling ◽  
Wave Propagation ◽  
Continuous Media ◽  
Particle Dynamics ◽  
Direct Wave ◽  
Backward Wave ◽  
Propagation Of Waves ◽  
Backward Propagation

A method is proposed for imaging of scattering heterogeneities in continuous media on the basis of numerical modeling of forward and backward wave propagation. It is shown that the combination of a solution for backward propagation of waves and direct wave propagation allows us to visualize scattering heterogeneities. The results of numerical simulation are presented.

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