Fractional Calculus and Applied Analysis

, Volume 16, Issue 1, pp 26–50 | Cite as

On a fractional Zener elastic wave equation

Survey Paper


This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.

Key Words and Phrases

fractional calculus acoustical wave equations elastic wave equations fractional wave equations fractional viscoelasticity fractional ordinary and partial differential equations 

MSC 2010

Primary 26A33 Secondary 33E12, 34A08, 34K37, 35L05, 92C50, 92C55, 35R11, 74J10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. H. Abel, Auflösung einer mechanischen Aufgabe (Resolution of a mechanical problem). J. Reine. Angew. Math. 1 (1826), 153–157.MATHCrossRefGoogle Scholar
  2. [2]
    K. Adolfsson, M. Enelund, and P. Olsson, On the fractional order model of viscoelasticity. Mech. Time-Dep. Mater. 9, No 1 (2005), 15–34.CrossRefGoogle Scholar
  3. [3]
    M. Ainslie and J. G. McColm, A simplified formula for viscous and chemical absorption in sea water. J. Acoust. Soc. Am. 103, No 3 (1998), 1671–1672.CrossRefGoogle Scholar
  4. [4]
    T. M. Atanackovic, A modified Zener model of a viscoelastic body. Continuum Mech. Therm. 14, No 2 (2002), 137–148.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    T. M. Atanackovic, S. Konjik, L. Oparnica, and D. Zorica, Thermodynamical restrictions and wave propagation for a class of fractional order viscoelastic rods. Abstr. Appl. Anal. 2011 (2011), Article ID 975694.Google Scholar
  6. [6]
    R. L. Bagley. The thermorheologically complex material. Int. J. Eng. Sci. 29, No 7 (1991), 797–806.MATHCrossRefGoogle Scholar
  7. [7]
    R. L. Bagley and P. J. Torvik, Fractional calculus — A different approach to the analysis of viscoelastically damped structures. AIAA J. 21, No 5 (1983), 741–748.MATHCrossRefGoogle Scholar
  8. [8]
    R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, No 1 (1986), 133–155.MATHCrossRefGoogle Scholar
  9. [9]
    J. C. Bamber, Attenuation and Absorption, Ch. 4, pp. 93–166. John Wiley & Sons, Chichester, 2005.Google Scholar
  10. [10]
    C. T. Barry, B. Mills, Z. Hah, R. A. Mooney, C. K. Ryan, D. J. Rubens, and K. J. Parker, Shear wave dispersion measures liver steatosis. Ultrasound Med. Biol. 38, No 2 (2012), 175–182.CrossRefGoogle Scholar
  11. [11]
    H. Bass, L. Sutherland, A. Zuckerwar, D. Blackstock, and D. Hester, Atmospheric absorption of sound: Further developments. J. Acoust. Soc. Am. 97 (1995), 680–683.CrossRefGoogle Scholar
  12. [12]
    P. Beard, Biomedical photoacoustic imaging. Interface Focus 1, No 4 (2011), 602–631.CrossRefGoogle Scholar
  13. [13]
    M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, Luminescence decays with underlying distributions of rate constants: General properties and selected cases. In: M.N. Berberan-Santos, M. Hof, Eds., Fluorescence of Supermolecules, Polymers, and Nanosystems, Vol. 4. Springer Ser. on Fluorescence, pp. 67–103, Springer, Berlin-Heidelberg, 2008.CrossRefGoogle Scholar
  14. [14]
    J. Bercoff, M. Tanter, and M. Fink, Supersonic shear imaging: a new technique for soft tissue elasticity mapping. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 51, No 4 (2004), 396–409.CrossRefGoogle Scholar
  15. [15]
    S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, Generalized fractional order bloch equation with extended delay. Int. J. Bifurcat. Chaos 22, No 04 (2012), 1250071-1–1250071-15.Google Scholar
  16. [16]
    M. Caputo, Linear models of dissipation whose Q is almost frequency independent — II. Geophys. J. Roy. Astr. S. 13, No 5 (1967), 529–539; Reprinted in: Fract. Calc. Appl. Anal. 11, No 1 (2008), 3–14; Scholar
  17. [17]
    M. Caputo, J. M. Carcione, and F. Cavallini, Wave simulation in biologic media based on the Kelvin-Voigt fractional-derivative stress-strain relation. Ultrasound Med. Biol. 37, No 6 (2011), 996–1004.CrossRefGoogle Scholar
  18. [18]
    M. Caputo and F. Mainardi. A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, No 1 (1971), 134–147; Reprinted in: Fract. Calc. Appl. Anal. 10, No 3 (2007), 309–324; at Scholar
  19. [19]
    J.M. Carcione, A generalization of the Fourier pseudospectral method. Geophysics 75, No 6 (2010), A53–A56.CrossRefGoogle Scholar
  20. [20]
    S. Chatelin, S. A. Lambert, L. Jugé, X. Cai, S. P. Näsholm, V. Vilgrain, B. E. Van Beers, L. E. Maitre, X. Bilston, B. Guzina, S. Holm, and R. Sinkus. Measured elasticity and its frequency dependence are sensitive to tissue microarchitecture in mr elastography. In: Proc. 20th Annual Meeting of ISMRM, May 2012.Google Scholar
  21. [21]
    A. Chatterjee, Statistical origins of fractional derivatives in viscoelasticity. J. Sound. Vib. 284, No 3–5 (2005), 1239–1245.CrossRefGoogle Scholar
  22. [22]
    S. Chen, M. Fatemi, and J. F. Greenleaf, Quantifying elasticity and viscosity from measurement of shear wave speed dispersion. J. Acoust. Soc. Am. 115, No 6 (2004), 2781–2785.CrossRefGoogle Scholar
  23. [23]
    W. Chen and S. Holm, Modified Szabo’s wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am. 114, No 5 (2003), 2570–2574.CrossRefGoogle Scholar
  24. [24]
    K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics I. Alternating current characteristics. J. Chem. Phys. 9, No 4 (1941), 341–351.CrossRefGoogle Scholar
  25. [25]
    C. Coussot, S. Kalyanam, R. Yapp, and M. Insana, Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 56, No 4 (2009), 715–725.CrossRefGoogle Scholar
  26. [26]
    D. O. Craiem, F. J. Rojo, J. M. Atienza, G. V. Guinea, and R. L. Armentano, Fractional calculus applied to model arterial viscoelasticity. Latin. Am. Appl. Res. 38, No 2 (2008), 141–145.Google Scholar
  27. [27]
    G. B. Davis, M. Kohandel, S. Sivaloganathan, and G. Tenti, The constitutive properties of the brain paraenchyma. Part 2. Fractional derivative approach. Med. Eng. Phys. 28, No 5 (2006), 455–459.CrossRefGoogle Scholar
  28. [28]
    E. C. de Oliveira, F. Mainardi, and J. Vaz, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. J. Phys. 193 (2011), 161–171.Google Scholar
  29. [29]
    F. Dinzart and P. Lipinski, Improved five-parameter fractional derivative model for elastomers. Arch. Mech. 61, No 6 (2009), 459–474.MathSciNetGoogle Scholar
  30. [30]
    V. D. Djordjević, J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31 (2003), 692–699.CrossRefGoogle Scholar
  31. [31]
    M. M. Djrbashian, Integral Transforms and Representations of Functions in the Complex Domain, Chs. 3–4. Nauka, Moscow, 1966 (In Russian).Google Scholar
  32. [32]
    M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in the Complex Domain, Ch. 1. Birkhäuser, Basel, 1993.MATHCrossRefGoogle Scholar
  33. [33]
    F. A. Duck, Physical Properties of Tissue. Academic Press, 1990.Google Scholar
  34. [34]
    R. L. Ehman, K. J. Glaser, and A. Manduca, Review of MR elastography applications and recent developments. J. Magn. Reson. 36, No 4 (2012), 757–774.CrossRefGoogle Scholar
  35. [35]
    Y. Feldman, Y. A. Gusev, and M. A. Vasilyeva, Dielectric Relaxation Phenomena in Complex Systems. Tutorial, Kazan Federal University, Institute of Physics, 2012.Google Scholar
  36. [36]
    C. Friedrich and H. Braun, Generalized cole-cole behavior and its rheological relevance. Rheol. Acta 31 (1992), 309–322.CrossRefGoogle Scholar
  37. [37]
    J. Garnier and K. Sølna, Effective fractional acoustic wave equations in one-dimensional random multiscale media. J. Acoust. Soc. Am. 127, No 1 (2010), 62–72.CrossRefGoogle Scholar
  38. [38]
    W. G. Glöckle and T. F. Nonnenmacher, Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24, No 24 (1991), 6426–6434CrossRefGoogle Scholar
  39. [39]
    N. M. Grahovac and M. Zigic, Modelling of the hamstring muscle group by use of fractional derivatives. Comput. Math. Appl. 59, No 5 (2010), 1695–1700.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications. J. of Appl. Math. 2011 (2011), 1–51.MathSciNetCrossRefGoogle Scholar
  41. [41]
    S. Holm and S. P. Näsholm, A causal and fractional all-frequency wave equation for lossy media. J. Acoust. Soc. Am. 130, No 4 (2011), 2195–2202.CrossRefGoogle Scholar
  42. [42]
    S. Holm and R. Sinkus, A unifying fractional wave equation for compressional and shear waves. J. Acoust. Soc. Am. 127 (2010), 542–548.CrossRefGoogle Scholar
  43. [43]
    L. Jugé, S. A. Lambert, S. Chatelin, L. ter Beek, V. Vilgrain, B. E. Van Beers, L. E. Bilston, B. Guzina, S. Holm, and R. Sinkus, Sub-voxel micro-architecture assessment by diffusion of mechanical shear waves. In: Proc. 20th Annual Meeting of ISMRM, May 2012. Google Scholar
  44. [44]
    J. F. Kelly and R. J. McGough, Fractal ladder models and power law wave equations. J. Acoust. Soc. Am. 126, No 4 (2009), 2072–2081.CrossRefGoogle Scholar
  45. [45]
    D. Klatt, U. Hamhaber, P. Asbach, J. Braun, and I. Sack, Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography: A study of brain and liver viscoelasticity. Phys. Med. Biol. 52, No 24 (2007), 7281–7294.CrossRefGoogle Scholar
  46. [46]
    M. Kohandel, S. Sivaloganathan, G. Tenti, and K. Darvish, Frequency dependence of complex moduli of brain tissue using a fractional Zener model. Phys. Med. Biol. 50, No 12 (2005), 2799–2805.CrossRefGoogle Scholar
  47. [47]
    S. Konjik, L. Oparnica, and D. Zorica, Waves in fractional Zener type viscoelastic media. J. Math. Anal. Appl. 365, No 1 (2010), 259–268.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    R. Kowar and O. Scherzer, Attenuation models in photoacoustics. In: H. Ammari, Ed., Mathematical Modeling in Biomedical Imaging II, Vol. 2035 of L.N.M., pp. 85–130, Springer, Berlin-Heidelberg, 2012.CrossRefGoogle Scholar
  49. [49]
    M. Liebler, S. Ginter, T. Dreyer, and R. E. Riedlinger, Full wave modeling of therapeutic ultrasound: Efficient time-domain implementation of the frequency power-law attenuation. J. Acoust. Soc. Am. 116 (2004), 2742–2750.CrossRefGoogle Scholar
  50. [50]
    J. G. Liu and M. Y. Xu, Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions. Mech. Time-Depend. Mat. 10 (2006), 263–279.CrossRefGoogle Scholar
  51. [51]
    Y. Luchko, Fractional wave equation and damped waves. ArXiv e-prints, May 2012.Google Scholar
  52. [52]
    J. A. T. Machado and A. Galhano, Fractional dynamics: A statistical perspective. J. Comput. Nonlin. Dynam. 3, No 2 (2008), 021201-1–021201-4.Google Scholar
  53. [53]
    F. Mainardi, Fractional relaxation in anelastic solids. Journal of Alloys and Compounds 211 (1994), 534–538.CrossRefGoogle Scholar
  54. [54]
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models, pp. 1–347. Imperial College Press, London, 2010.MATHCrossRefGoogle Scholar
  55. [55]
    F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 15, No 4 (2012), 712–717; DOI:10.2478/s13540-012-0048-6; at Scholar
  56. [56]
    F. Mainardi and G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur. J. Phys. 193 (2011), 133–160.Google Scholar
  57. [57]
    T. Meidav, Viscoelastic properties of the standard linear solid. Geophys. Prospect. 12, No 1 (1964), 1365–2478.CrossRefGoogle Scholar
  58. [58]
    S. I. Meshkov, G. N. Pachevskaya, V. S. Postnikov, and U. A. Rossikhin, Integral representations of ∋γ-functions and their application to problems in linear viscoelasticity. Int. J. Eng. Sci. 9, No 4 (1971), 387–398.MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    R. Metzler and T. F. Nonnenmacher, Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Int. J. Plasticity 19, No 7 (2003), 941–959.MATHCrossRefGoogle Scholar
  60. [60]
    M. G. Mittag-Leffer, Sur la nouvelle fonction E α(x) (On the new function E α(x)). C. R. Acad. Sci. Paris 137 (1903), 554–558.Google Scholar
  61. [61]
    R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. Science 269, No 5232 (1995), 1854–1857.CrossRefGoogle Scholar
  62. [62]
    A. I. Nachman, J. F. Smith III, and R. C. Waag, An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88 (1990), 1584–1595.CrossRefGoogle Scholar
  63. [63]
    S. P. Näsholm and S. Holm, Linking multiple relaxation, power-law attenuation, and fractional wave equations. J. Acoust. Soc. Am. 130, No 5 (2011), 3038–3045.CrossRefGoogle Scholar
  64. [64]
    S. P. Näsholm and S. Holm, A fractional acoustic wave equation from multiple relaxation loss and conservation laws. Proc. 5th Int. Workshop on Fractional Differentiation and its Applications’ 2012, China.Google Scholar
  65. [65]
    R. R. Nigmatullin, Theory of dielectric relaxation in non-crystalline solids: From a set of micromotions to the averaged collective motion in the mesoscale region. Physica B 358, No 1–4 (2005), 201–215.CrossRefGoogle Scholar
  66. [66]
    R. F. O’Doherty and N. A. Anstey, Reflections on amplitudes. Geophys. Prosp. 19 (1971), 430–458.CrossRefGoogle Scholar
  67. [67]
    M. L. Palmeri and K. R. Nightingale, Acoustic radiation force-based elasticity imaging methods. Interface Focus 1, No 4 (2011), 553–564.CrossRefGoogle Scholar
  68. [68]
    S. Papazoglou, S. Hirsch, J. Braun, and I. Sack, Multifrequency inversion in magnetic resonance elastography. Phys. Med. Biol. 57, No 8 (2012), 2329–2346.CrossRefGoogle Scholar
  69. [69]
    K. Papoulia, V. Panoskaltsis, N. Kurup, and I. Korovajchuk, Rheological representation of fractional order viscoelastic material models. Rheol. Acta 49 (2010), 381–400.CrossRefGoogle Scholar
  70. [70]
    H. Pauly and H. P. Schwan, Mechanism of absorption of ultrasound in liver tissue. J. Acoust. Soc. Am. 50, No 2B (1971), 692–699.CrossRefGoogle Scholar
  71. [71]
    L. M. Petrovic, D. T. Spasic, and T. M. Atanackovic, On a mathematical model of a human root dentin. Dent. Mater. 21, No 2 (2005), 125–128.CrossRefGoogle Scholar
  72. [72]
    I. Podlubny, Fractional Differential Equations, Ch. 10.2. Academic Press, New York, 1999.MATHGoogle Scholar
  73. [73]
    I. Podlubny, Fractional Differential Equations, Chs. 1–2. Academic Press, New York, 1999.MATHGoogle Scholar
  74. [74]
    F. Prieur and S. Holm, Nonlinear acoustic wave equations with fractional loss operators. J. Acoust. Soc. Am. 130, No 3 (2011), 1125–1132.CrossRefGoogle Scholar
  75. [75]
    F. Prieur, G. Vilenskiy, and S. Holm, A more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators. J. Acoust. Soc. Am. 132 (2012), 2169–2172.CrossRefGoogle Scholar
  76. [76]
    T. Pritz, Analysis of four-parameter fractional derivative model of real solid materials. J. Sound. Vib. 195, No 1 (1996), 103–115.MATHCrossRefGoogle Scholar
  77. [77]
    T. Pritz, Loss factor peak of viscoelastic materials: Magnitude to width relations. J. Sound. Vib. 246, No 2 (2001), 265–280.CrossRefGoogle Scholar
  78. [78]
    T. Pritz, Five-parameter fractional derivative model for polymeric damping materials. J. Sound. Vib. 265, No 5 (2003), 935–952.CrossRefGoogle Scholar
  79. [79]
    H. Roitner, J. Bauer-Marschallinger, T. Berer, and P. Burgholzer, Experimental evaluation of time domain models for ultrasound attenuation losses in photoacoustic imaging. J. Acoust. Soc. Am. 131 (2012), 3763–3774.CrossRefGoogle Scholar
  80. [80]
    Y. A. Rossikhin, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Applied Mech. Rev. 63, No 1 (2010), 010701-1–010701-12.Google Scholar
  81. [81]
    Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50 (1997), 15–67.CrossRefGoogle Scholar
  82. [82]
    Y. A. Rossikhin and M. V. Shitikova, Analysis of rheological equations involving more than one fractional parameters by the use of the simplest mechanical systems based on these equations. Mech. Time-Depend. Mat. 5, No 2 (2001), 131–175.CrossRefGoogle Scholar
  83. [83]
    Y. A. Rossikhin and M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Appl. Mech. Rev. 63 (2010), 010801–1-25.CrossRefGoogle Scholar
  84. [84]
    D. Royer and E. Dieulesaint, Elastic Waves in Solids, Vol. I. Springer, Berlin, 2000.CrossRefGoogle Scholar
  85. [85]
    I. Sack, B. Beierbach, J. Wuerfel, D. Klatt, U. Hamhaber, S. Papazoglou, P. Martus, and J. Braun, The impact of aging and gender on brain viscoelasticity. NeuroImage 46, No 3 (2009), 652–657.CrossRefGoogle Scholar
  86. [86]
    M. Sasso, G. Palmieri, and D. Amodio, Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend. Mat. 15 (2011), 367–387.CrossRefGoogle Scholar
  87. [87]
    H. Schiessel and A. Blumen, Hierarchical analogues to fractional relaxation equations. J. Phys. A 26, No 19 (1993), 5057–5069.CrossRefGoogle Scholar
  88. [88]
    H. Schiessel and A. Blumen, Mesoscopic pictures of Sol-Gel transition: Ladder models and fractal networks. Macromolecules 28 (1995), 4013–4019.CrossRefGoogle Scholar
  89. [89]
    M. Seredyńska and A. Hanyga, Relaxation, dispersion, attenuation, and finite propagation speed in viscoelastic media. J. Math. Phys. 51, No 9 (2010), 092901.MathSciNetCrossRefGoogle Scholar
  90. [90]
    R. Sinkus, J.-L. Daire, V. Vilgrain, and B. E. Van Beers, Elasticity imaging via MRI: Basics, overcoming the waveguide limit, and clinical liver results. Curr. Med. Imaging Rev. 8, No 1 (2012), 56–63.CrossRefGoogle Scholar
  91. [91]
    R. Sinkus, J. Lorenzen, D. Schrader, M. Lorenzen, M. Dargatz, and D. Holz, High-resolution tensor MR elastography for breast tumour detection. Phys. Med. Biol. 45, No 6 (2000), 1649–1664.CrossRefGoogle Scholar
  92. [92]
    R. Sinkus, K. Siegmann, T. Xydeas, M. Tanter, C. Claussen, and M. Fink, MR elastography of breast lesions: Understanding the solid/liquid duality can improve the specificity of contrast-enhanced MR mammography. Magn. Res. in Med. 58, No 6 (2007), 1135–1144.CrossRefGoogle Scholar
  93. [93]
    A. A. Stanislavsky, The stochastic nature of complexity evolution in the fractional systems. Chaos Soliton Fract. 34, No 1 (2007), 51–61.MathSciNetMATHCrossRefGoogle Scholar
  94. [94]
    T. L. Szabo and J. Wu, A model for longitudinal and shear wave propagation in viscoelastic media. J. Acoust. Soc. Am. 107 (2000), 2437–2446.CrossRefGoogle Scholar
  95. [95]
    M. Tabei, T. D. Mast, and R. C. Waag, Simulation of ultrasonic focus aberration and correction through human tissue. J. Acoust. Soc. Am. 113, No 2 (2003), 1166–1176.CrossRefGoogle Scholar
  96. [96]
    B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. J. Acoust. Soc. Am. 127 (2010), 2741–2748.CrossRefGoogle Scholar
  97. [97]
    B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. J. Acoust. Soc. Am. 127 (2010), 2741–2748, Section IIB.CrossRefGoogle Scholar
  98. [98]
    B. E. Treeby, J. Jaros, A. P. Rendell, and 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, No 6 (2012), 4324–4336.CrossRefGoogle Scholar
  99. [99]
    B. E. Treeby, E. Z. Zhang, and B. T. Cox, Photoacoustic tomography in absorbing acoustic media using time reversal. Inverse Probl. 26, No 11 (2010), 115003.MathSciNetCrossRefGoogle Scholar
  100. [100]
    G. Vilensky, G. ter Haar, and N. Saffari, A model of acoustic absorption in fluids based on a continuous distribution of relaxation times. Wave Motion 49, No 1 (2012), 93–108.MathSciNetCrossRefGoogle Scholar
  101. [101]
    K. R. Waters, J. Mobley, and J. G. Miller, Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion. IEEE Trans. Ultrason. Ferroelectr., Freq. Control, 52, No 5 (2005), 822–833.CrossRefGoogle Scholar
  102. [102]
    R. L. Weaver and Y. H. Pao, Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media. Journ. Math. Phys. 22 (1981), 1909–1918.MathSciNetMATHCrossRefGoogle Scholar
  103. [103]
    K. Weron and A. Klauzer, Probabilistic basis for the Cole-Cole relaxation law. Ferroelectrics 236, No 1 (2000), 59–69.CrossRefGoogle Scholar
  104. [104]
    D. Widder, An Introduction to Transform Theory, Ch. 5.13. Pure and Applied Mathematics Ser., Academic Press, 1971.Google Scholar
  105. [105]
    A. Wiman, Über den Fundamentalsatz in der Theorie der Funktionen E α(x) (About the fundamental theorem in the theory of the function E α(x)). Acta Mathematica 29 (1905), 191–201.MathSciNetMATHCrossRefGoogle Scholar
  106. [106]
    M. G. Wismer, Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. J. Acoust. Soc. Am. 120 (2006), 3493–3502.CrossRefGoogle Scholar
  107. [107]
    M. G. Wismer and R. Ludwig, An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 42, No 6 (1995), 1040–1049.CrossRefGoogle Scholar
  108. [108]
    X. Yang and R. O. Cleveland, Time domain simulation of nonlinear acoustic beams generated by rectangular pistons with application to harmonic imaging. J. Acoust. Soc. Am. 117 (2005), 113–123.CrossRefGoogle Scholar
  109. [109]
    T. K. Yasar, T. J. Royston, and R. L. Magin, Wideband MR elastography for viscoelasticity model identification. Magnet. Reson. Med., 2012, Online Version of Record published before inclusion in an issue.Google Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of OsloOsloNorway

Personalised recommendations