Phenomenological Power-Law Wave Equations

  • Sverre HolmEmail author


As mentioned in the introduction of Chap.  5, this chapter is concerned with wave equations that are made to fit frequency domain power laws with little consideration for time domain properties. First wave equations which are found by manipulating the common wave equations will be discussed. This is done by replacing ordinary derivatives in time or space with fractional ones. In some cases, this will give the same solutions as those found from fractional constitutive laws, and in other cases new equations will result.


  1. T.M. Atanacković, S. Konjik, S. Pilipović, S. Simić, Variational problems with fractional derivatives: Invariance conditions and Noether’s theorem. Nonlinear Anal.: Theory Methods Appl. 71(5), 1504–1517 (2009)CrossRefGoogle Scholar
  2. W. Cai, W. Chen, J. Fang, 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. (2018)Google Scholar
  3. W. Chen, S. Holm, Modified Szabo’s wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am. 114(5), 2570–2574 (2003)ADSCrossRefGoogle Scholar
  4. W. Chen, S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115(4), 1424–1430 (2004)ADSCrossRefGoogle Scholar
  5. W. Chen, J. Fang, G. Pang, S. Holm, Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation. J. Acoust. Soc. Am. 141(1), 244–253 (2017)ADSCrossRefGoogle Scholar
  6. S. Holm, S.P. Näsholm, Comparison of fractional wave equations for power law attenuation in ultrasound and elastography. Ultrasound Med. Biol. 40(4), 695–703 (2014)CrossRefGoogle Scholar
  7. J.F. Kelly, R.J. McGough, M.M. Meerschaert, Analytical time-domain Green’s functions for power-law media. J. Acoust. Soc. Am. 124(5), 2861–2872 (2008)ADSCrossRefGoogle Scholar
  8. T.L. Szabo, Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96, 491–500 (1994)ADSCrossRefGoogle Scholar
  9. V.E. Tarasov, G. Zaslavsky, Conservation laws and Hamilton’s equations for systems with long-range interaction and memory. Commun. Nonlinear Sci. Numer. Simul. 13(9), 1860–1878 (2008)ADSMathSciNetCrossRefGoogle Scholar
  10. B.E. Treeby, B.T. Cox, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. J. Acoust. Soc. Am. 127, 2741–2748 (2010)ADSCrossRefGoogle Scholar
  11. B.E. Treeby, B.T. Cox, Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian. J. Acoust. Soc. Am. 136(4), 1499–1510 (2014)ADSCrossRefGoogle Scholar
  12. B.E. Treeby, J. Jaros, A.P. Rendell, B.T. Cox, Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method. J. Acoust. Soc. Am. 131(6), 4324–4336 (2012)ADSCrossRefGoogle Scholar
  13. S.W. Wheatcraft, M.M. Meerschaert, Fractional conservation of mass. Adv. Water Resour. 31(10), 1377–1381 (2008)ADSCrossRefGoogle Scholar
  14. M.G. Wismer, Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. J. Acoust. Soc. Am. 120, 3493–3502 (2006)ADSCrossRefGoogle Scholar
  15. X. Zhang, W. Chen, C. Zhang, Modified Szabo’s wave equation for arbitrarily frequency-dependent viscous dissipation in soft matter with applications to 3D ultrasonic imaging. Acta Mech. Solida Sin. 25(5), 510–519 (2012)CrossRefGoogle Scholar
  16. X. Zhao, R.J. McGough, Numerical evaluation of time-domain Green’s functions for space-fractional wave equations. J. Acoust. Soc. Am. 140(4), 3187–3187 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of OsloOsloNorway

Personalised recommendations