Representation of Matrix Potentials in the Rayleigh Wave Equation by a Symmetric Matrix

Approximate symmetry analysis of nonlinear Rayleigh-wave equation

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988781850055x ◽

2018 ◽

Vol 15 (04) ◽

pp. 1850055

Author(s):

Saeede Rashidi ◽

S. Reza Hejazi ◽

Elham Dastranj

Keyword(s):

Wave Equation ◽

Power Series ◽

Small Parameter ◽

Conservation Laws ◽

Rayleigh Wave ◽

Symmetry Analysis ◽

Approximate Symmetry ◽

Series Method ◽

Power Series Method ◽

New Exact Solutions

In this paper, the Lie approximate symmetry analysis is applied to investigate the new exact solutions of the Rayleigh-wave equation. The power series method is employed to solve some of the obtained reduced ordinary differential equations with a small parameter. We yield the new analytical solutions with small parameter which is effectively obtained by the proposed method. The concept of nonlinear self-adjointness is used to construct the conservation laws for Rayleigh-wave equation. It is shown that this equation is approximately nonlinearly self-adjoint and therefore desired conservation laws can be found using appropriate formal Lagrangians.

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An improved approximation for the Rayleigh wave equation

Ultrasonics ◽

10.1016/j.ultras.2006.09.006 ◽

2007 ◽

Vol 46 (1) ◽

pp. 23-24 ◽

Cited By ~ 9

Author(s):

Daniel Royer ◽

Dominique Clorennec

Keyword(s):

Wave Equation ◽

Rayleigh Wave

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Researching on Characteristics of Rayleigh Waves

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.721.472 ◽

2014 ◽

Vol 721 ◽

pp. 472-475

Author(s):

Xu Fang Zhu ◽

Bing Yan

Keyword(s):

Wave Equation ◽

Rayleigh Wave ◽

Dispersion Equation ◽

Rayleigh Waves ◽

Low Frequency ◽

Secondary Wave ◽

Reflected Wave ◽

Strong Energy ◽

Geological Information ◽

Interface Distance

Rayleigh wave is a secondary wave characterized by low frequency and strong energy, propagating mainly along the interface of medium and rapid attenuation of energy with increase in interface distance. The same as reflected wave and refracted wave, Rayleigh wave also contain subsurface geological information. In this paper, the concept of the Rayleigh wave, wave equation, dispersion equation, the frequency bulk characteristics and the application of the actual data are used to indicate the characteristics of Rayleigh wave and its application.

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The Rayleigh wave equation—an analysis

Nonlinear Analysis ◽

10.1016/0362-546x(78)90061-5 ◽

1978 ◽

Vol 2 (2) ◽

pp. 129-156 ◽

Cited By ~ 8

Author(s):

William S. Hall

Keyword(s):

Wave Equation ◽

Rayleigh Wave

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Wave-equation Rayleigh-wave dispersion inversion using fundamental and higher modes

Geophysics ◽

10.1190/geo2018-0506.1 ◽

2019 ◽

Vol 84 (4) ◽

pp. EN57-EN65 ◽

Cited By ~ 7

Author(s):

Zhen-Dong Zhang ◽

Tariq Alkhalifah

Keyword(s):

Wave Equation ◽

Rayleigh Wave ◽

Surface Waves ◽

Wave Velocity ◽

Minimum Problem ◽

Velocity Spectrum ◽

Local Similarity ◽

Inversion Algorithm ◽

S Wave ◽

Near Surface

Recorded surface waves often provide reasonable estimates of the S-wave velocity in the near surface. However, existing algorithms are mainly based on the 1D layered-model assumption and require picking the dispersion curves either automatically or manually. We have developed a wave-equation-based inversion algorithm that inverts for S-wave velocities using fundamental and higher mode Rayleigh waves without picking an explicit dispersion curve. Our method aims to maximize the similarity of the phase velocity spectrum ([Formula: see text]) of the observed and predicted surface waves with all Rayleigh-wave modes (if they exist) included in the inversion. The [Formula: see text] spectrum is calculated using the linear Radon transform applied to a local similarity-based objective function; thus, we do not need to pick velocities in spectrum plots. As a result, the best match between the predicted and observed [Formula: see text] spectrum provides the optimal estimation of the S-wave velocity. We derive S-wave velocity updates using the adjoint-state method and solve the optimization problem using a limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Our method excels in cases in which the S-wave velocity has vertical reversals and lateral variations because we used all-modes dispersion, and it can suppress the local minimum problem often associated with full-waveform inversion applications. Synthetic and field examples are used to verify the effectiveness of our method.

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Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

Symmetry ◽

10.3390/sym12020202 ◽

2020 ◽

Vol 12 (2) ◽

pp. 202

Author(s):

Cheng Sun ◽

Zailin Yang ◽

Guanxixi Jiang

Keyword(s):

Differential Equation ◽

Wave Equation ◽

Elastic Wave ◽

Symmetric Matrix ◽

Perfectly Matched Layer ◽

High Order ◽

Matrix Form ◽

Discrete Approximation ◽

Complex Frequency ◽

Elastic Wave Equation

In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations.

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