Dispersion analysis for wave equations with fractional Laplacian loss operators
-
Sverre Holm
Abstract
Several wave equations for power-law attenuation have a spatial fractional derivative in the loss term. Both one-sided and two-sided spatial fractional derivatives can give causal solutions and a phase velocity dispersion which satisfies the Kramers–Kronig relation. The Chen–Holm and the Treeby–Cox equations both have the two-sided fractional Laplacian derivative, but only the latter satisfies this relation. There also exists several seemingly different expressions for the phase velocity for these equations and it is shown here that they are approximately equivalent. Causality of the Chen–Holm equation has also been a topic of some discussion and it is found that despite the lack of agreement with the Kramers–Kronig relation, it is still causal.
This paper is dedicated to the memory of the late Professor Wen Chen
References
[1] W. Cai, W. Chen, J. Fang, and S. Holm, A survey on fractional derivative modeling of power-law frequency-dependent viscous dissipative and scattering attenuation in acoustic wave propagation. Appl. Mech. Rev. 70, No 3 (2018), ID 030802; 10.1115/1.4040402.Suche in Google Scholar
[2] W. Chen and S. Holm, Modified Szabo’s wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Amer. 114, No 5 (2003), 2570–2574.10.1121/1.1621392Suche in Google Scholar PubMed
[3] W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Amer. 115, No 4 (2004), 1424–1430.10.1121/1.1646399Suche in Google Scholar PubMed
[4] T. Dethe, H. Gill, D. Green, A. Greensweight, L. Gutierrez, M. He, T. Tajima, and K. Yang, Causality and dispersion relations. Amer. J. Phys. 87, No 4 (2019), 279–290.10.1119/1.5092679Suche in Google Scholar
[5] R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, 2014.10.1007/978-3-662-43930-2Suche in Google Scholar
[6] S. Holm, Waves with Power-Law Attenuation. Springer and ASA Press, Switzerland, 2019.10.1007/978-3-030-14927-7Suche in Google Scholar
[7] S. Holm and S. P. Näsholm, A causal and fractional all-frequency wave equation for lossy media. J. Acoust. Soc. Amer. 130, No 4 (2011), 2195–2202.10.1121/1.3631626Suche in Google Scholar PubMed
[8] S. Holm and R. Sinkus, A unifying fractional wave equation for compressional and shear waves. J. Acoust. Soc. Amer. 127 (2010), 542–548.10.1121/1.3268508Suche in Google Scholar PubMed
[9] V. E. Holmes and R. J. McGough, Exact and approximate causal time-domain Green’s functions for linear with frequency attenuation. In: IEEE Int. Ultrason. Symp., IEEE (2019), 1–4.10.1109/ULTSYM.2019.8926134Suche in Google Scholar
[10] J. F. Kelly and R. J. McGough, The distinction between noncausal and nonlocal behavior in a time-fractional wave equation. In: IEEE Int. Ultrason. Symp., IEEE (2018), 1–4.10.1109/ULTSYM.2018.8579827Suche in Google Scholar
[11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.10.1142/p614Suche in Google Scholar
[12] J. Mobley, Simplified expressions of the subtracted Kramers–Kronig relations using the expanded forms applied to ultrasonic power-law systems. J. Acoust. Soc. Amer. 127, No 1 (2010), 166–173.10.1121/1.3268512Suche in Google Scholar PubMed
[13] S. P. Näsholm and S. Holm, On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 26–50; 10.2478/s13540-013-0003-1; https://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.Suche in Google Scholar
[14] V. Pandey, S. P. Näsholm, and S. Holm, Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity. Fract. Calc. Appl. Anal. 19, No 2 (2016), 498–515; 10.1515/fca-2016-0026; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.Suche in Google Scholar
[15] G. G. Stokes, On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Philos. Soc. 8, part III (1845), 287–319.10.1017/CBO9780511702242.005Suche in Google Scholar
[16] B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. J. Acoust. Soc. Amer. 127 (2010), 2741–2748.10.1121/1.3377056Suche in Google Scholar PubMed
[17] B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian. J. Acoust. Soc. Amer. 136, No 4 (2014), 1499–1510.10.1121/1.4894790Suche in Google Scholar PubMed
[18] M. Urquidi-Macdonald, S. Real, and D. D. Macdonald, Applications of Kramers–Kronig transforms in the analysis of electrochemical impedance data–III. Stability and linearity. Electrochim. Acta35, No 10 (1990), 1559–1566.Suche in Google Scholar
[19] L. M. Wiseman, J. F. Kelly, and R. J. McGough, Exact and approximate analytical time-domain Green’s functions for space-fractional wave equations. J. Acoust. Soc. Amer. 146, No 2 (2019), 1150–113.10.1121/1.5119128Suche in Google Scholar PubMed PubMed Central
[20] X. Zhao and R. J. McGough, Time-domain analysis of power law attenuation in space-fractional wave equations. J. Acoust. Soc. Amer. 144, No 1 (2018), 467–477.10.1121/1.5047670Suche in Google Scholar PubMed PubMed Central
© 2019 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA special issue – In memory of late professor Wen Chen (FCAA–Volume 22–6–2019)
- Survey Paper
- State-of-art survey of fractional order modeling and estimation methods for lithium-ion batteries
- An investigation on continuous time random walk model for bedload transport
- Tutorial Survey
- Porous functions
- Research Paper
- A time-space Hausdorff derivative model for anomalous transport in porous media
- High-order algorithms for riesz derivative and their applications (IV)
- Mass-conserving tempered fractional diffusion in a bounded interval
- Dispersion analysis for wave equations with fractional Laplacian loss operators
- Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application
- Some further results of the laplace transform for variable–order fractional difference equations
- Robust stability analysis of LTI systems with fractional degree generalized frequency variables
- Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA special issue – In memory of late professor Wen Chen (FCAA–Volume 22–6–2019)
- Survey Paper
- State-of-art survey of fractional order modeling and estimation methods for lithium-ion batteries
- An investigation on continuous time random walk model for bedload transport
- Tutorial Survey
- Porous functions
- Research Paper
- A time-space Hausdorff derivative model for anomalous transport in porous media
- High-order algorithms for riesz derivative and their applications (IV)
- Mass-conserving tempered fractional diffusion in a bounded interval
- Dispersion analysis for wave equations with fractional Laplacian loss operators
- Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application
- Some further results of the laplace transform for variable–order fractional difference equations
- Robust stability analysis of LTI systems with fractional degree generalized frequency variables
- Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
