Time-domain Modeling of Constant- Q Seismic Waves Using Fractional Derivatives

Carcione, José M.; Cavallini, F.; Mainardi, Francesco; Hanyga, Andrzej

doi:10.1007/s00024-002-8705-z

Cited by 183 publications

(66 citation statements)

References 8 publications

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“…We described them in the framework of a fractional Fokker-Planck equation (FFPE). A general form of the real-time-space FFPE (3) describes both waves and relaxation processes in a variety of applications of diffusion-wave phenomena in inhomogeneous media [24][25][26][27][28][29]35]. The suggested scenario of the analysis is as follows.…”

Section: Discussionmentioning

confidence: 99%

“…The FFPE (3) is a general form of space-time fractional equations. It describes both waves and relaxation processes in a variety of applications like diffusion-wave phenomena in inhomogeneous media [24][25][26][27][28][29]. It should be stressed that fractional space derivatives describe Lévy flights [6].…”

Section: Ffpe In Slab Geometry: Parabolic Equation Approximationmentioning

confidence: 99%

“…For example, fractional calculus description has been introduced to describe the wave propagation at anomalous diffusive oscillations in the Lorentz (Cauchy) pair plasma with dust impurities [25], random walk of optic rays in Lévy lens [26], fractional wave-diffusion in heterogeneous media [27,28] as well as seismic waves [29,30]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Lévy Transport in Slab Geometry of Inhomogeneous Media

Iomin

Sandev

2016

Math. Model. Nat. Phenom.

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We present a physical example, where a fractional (both in space and time) Schrödinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, Lévy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.

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Section: Discussionmentioning

confidence: 99%

Section: Ffpe In Slab Geometry: Parabolic Equation Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lévy Transport in Slab Geometry of Inhomogeneous Media

Iomin

Sandev

2016

Math. Model. Nat. Phenom.

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“…The study of lossy waves in the time domain often leads to wave equations with fractional time derivatives [34,35]. Therefore the appearance of the operator ∂ 1 2 t in (69) is expected, due to the inclusion of viscous and thermal dissipation.…”

Section: Suggestion Of a Time-domain Formulationmentioning

confidence: 99%

Acoustics in a Two-Deck Viscothermal Boundary Layer over an Impedance Surface

Khamis

2017

AIAA Journal

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The acoustics of a mean flow boundary layer over an impedance surface or acoustic lining are considered. By considering a thick mean flow boundary layer (possibly due to turbulence), the boundary layer structure is separated asymptotically into two decks, with a thin weakly viscous mean flow boundary layer and an even thinner strongly viscous acoustic sublayer, without requiring a high-frequency. Using this, analytic solutions are found for the acoustic modes in a cylindrical lined duct. The mode shapes in each region compare well with numerical solutions of the linearised compressible Navier-Stokes equations, as does a uniform composite asymptotic solution. A closed-form effective impedance boundary condition is derived which can be applied to acoustics in inviscid slipping flow to account for both shear and viscosity in the boundary layer. The importance of the boundary layer is demonstrated in the frequency domain, and the new boundary condition is found to correctly predict the attenuation of upstream-propagating cuton modes, which are poorly predicted by existing inviscid boundary conditions. Stability is also investigated, and the new boundary condition is found to yield good results away from the critical layer. A time-domain formulation of a simplified version of the new impedance boundary condition is proposed.

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“…Physical models are generally very efficient to describe the attenuation process (frequency dependence, causality, etc.) but they are nevertheless not very easy to implement or cost effective due to the use of memory variables for instance [10,26,27,28,29]. We propose to use the Caughey damping formulation [3,4] to describe the attenuation of the waves in an absorbing layer of finite thickness.…”

Section: 2mentioning

confidence: 99%

A simple numerical absorbing layer method in elastodynamics

Semblat

Gandomzadeh

Lenti

2009

Comptes Rendus. Mécanique

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Abstract:The numerical analysis of elastic wave propagation in unbounded media may be difficult to handle due to spurious waves reflected at the model artificial boundaries. Several sophisticated techniques such as nonreflecting boundary conditions, infinite elements or absorbing layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this Note, a simple and efficient absorbing layer method is proposed in the framework of the Finite Element Method. This method considers Rayleigh/Caughey damping in the absorbing layer and its principle is presented first. The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types (surface waves, body waves) and incidences (normal to grazing). The method is shown to be efficient for different types of elastic waves and may thus be used for various elastodynamic problems in unbounded domains.Abstract: La simulation numérique de la propagation d'ondes élastiques en milieux infinis peut s'avérer délicate du fait des réflexions d'ondes parasites sur les frontières du modèle discrétisé. Plusieurs méthodes sophistiquées, telles que les frontières absorbantes, les éléments infinis ou les couches absorbantes (e.g. « Perfectly Matched Layers ») permettent une réduction importante des réflexions parasites. Dans cette note, une méthode de couche absorbante simple et efficace est proposée dans le cadre de la méthode des éléments finis. Cette méthode s'appuie sur une formulation de l'amortissement dans la couche de type Rayleigh/Caughey et ses principes sont d'abord détaillés. L'efficacité de la méthode est alors démontrée grâce à des simulations unidimensionnelles en considérant un amortissement homogène ou variable dans la couche absorbante. Des modèles bidimensionnels permettent ensuite d'apprécier l'efficacité de la méthode de couche absorbante proposée pour différents types d'ondes (ondes de surface, ondes de volume) et des incidences variées (normale à rasante). La méthode s'avère ainsi efficace pour différents types d'ondes élastiques et pourrait être utilisée pour traiter divers problèmes élastodynamiques en milieux non bornés. 1.Numerical methods for elastic waves in unbounded domains Various numerical methods are available to simulate elastic wave propagation in solids: finite differences [1,2], finite elements [3,4], boundary elements [5,6,7], spectral elements [8,9], etc. Such methods as finite or spectral elements have strong advantages (for complex geometries [9], nonlinear media [10], etc) but may have such drawbacks as numerical dispersion [4,11,12] for low order finite elements, or spurious reflections at the mesh boundaries [4,13]. The problem of spurious reflections may be dealt with using the Boundary Element Method [5,6,7] or coupling it with other numerical methods [14]. Domain Reduction Methods are also available in th...

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Time-domain Modeling of Constant- Q Seismic Waves Using Fractional Derivatives

Cited by 183 publications

References 8 publications

Lévy Transport in Slab Geometry of Inhomogeneous Media

Lévy Transport in Slab Geometry of Inhomogeneous Media

Acoustics in a Two-Deck Viscothermal Boundary Layer over an Impedance Surface

A simple numerical absorbing layer method in elastodynamics

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