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Introduction

  • Sverre HolmEmail author
Chapter

Abstract

The interest in this book is in media that attenuate the wave with power laws of order other than two over all of the frequency range or a part of it, and the various mechanisms that can cause such attenuation. Then attenuation will follow:
$$\begin{aligned} \alpha _k = a_0 |\omega |^y, \end{aligned}$$
where \(\alpha _k\) is the attenuation (the negative imaginary part of the wave number k, hence, the index), \(\omega \) is angular frequency, and \(a_0\) and \(0 \le y \le 2\) are constants. The absolute value is used to ensure that attenuation never becomes negative. Such attenuation cannot be described with the equations above and the question that I asked myself some 15–20 years ago was whether it is possible to describe this in a better way. This book is the result of my quest for an answer.

References

  1. B. Angelsen, Ultrasonic Imaging: Waves, Signals, and Signal Processing, vol. 1–2 (Emantec AS, Trondheim, 2000)Google Scholar
  2. J.D. Barrow, Godel and physics, in Kurt Gödel and the Foundations of Mathematics, ed. by M. Baaz, C.H. Papadimitriou, H.W. Putnam, D.S. Scott, C.L. Harper Jr. (Cambridge University Press, Cambridge, 2011), pp. 255–276Google Scholar
  3. M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid I. Low-frequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956)ADSMathSciNetCrossRefGoogle Scholar
  4. G.S. Blair, M. Reiner, The rheological law underlying the Nutting equation. Appl. Sci. Res. 2(1), 225–234 (1951)CrossRefGoogle Scholar
  5. L. Boltzmann, Zur theorie der elastischen nachwirkung (On the theory of hereditary elastic effects). Ann. Phys. Chem. Bd. 7, 624–654 (1876)Google Scholar
  6. M.J. Buckingham, Wave propagation, stress relaxation, and grain-to-grain shearing in saturated, unconsolidated marine sediments. J. Acoust. Soc. Am. 108(6), 2796–2815 (2000)ADSCrossRefGoogle Scholar
  7. M.J. Buckingham, On pore-fluid viscosity and the wave properties of saturated granular materials including marine sediments. J. Acoust. Soc. Am. 122(3), 1486–1501 (2007)Google Scholar
  8. D. Cafagna, Past and present-fractional calculus: a mathematical tool from the past for present engineers. IEEE Ind. Electr. Mag. 2(1), 35–40 (2007)Google Scholar
  9. M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(1), 134–147 (1971)ADSCrossRefGoogle Scholar
  10. A. Chapman, England’s Leonardo: Robert Hooke and the Seventeenth-Century Scientific Revolution (CRC Press, New York, 2004)Google Scholar
  11. 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
  12. N.P. Chotiros, Acoustics of the Seabed as a Poroelastic Medium (Springer, ASA Press, Berlin, Switzerland, 2017)Google Scholar
  13. J. d’Alembert, Recherches sur la courbe que forme une corde tendue mise en vibrations (Research on the curve that a tense cord forms when set into vibration). Histoire de l’Académie Royale des Sciences et Belles Lettres (Année 1747) 3, 214–249 (1747)Google Scholar
  14. F.A. Duck, Physical Properties of Tissues: A Comprehensive Reference Book (Academic Press, Cambridge, 2012)Google Scholar
  15. P.M.M. Duhem, The Aim and Structure of Physical Theory (La théorie physique. Son objet, sa structure, 1906) (Princeton University Press, Princeton, 1991)Google Scholar
  16. F. Dyson, The scientist as rebel. Am. Math. Monthly 103(9), 800–805 (1996)MathSciNetCrossRefGoogle Scholar
  17. R.P. Feynman, The Character of Physical Law (MIT Press, Cambridge, 1967)Google Scholar
  18. J. Garnier, K. Sølna, Pulse propagation in random media with long-range correlation. Multiscale Model Simul. 7(3), 1302–1324 (2009)MathSciNetCrossRefGoogle Scholar
  19. D.J. Gross, The role of symmetry in fundamental physics. Proc. Natl. Acad. Sci. USA 93(25), 14256–14259 (1996)ADSMathSciNetCrossRefGoogle Scholar
  20. S. Hawking, Gödel and the End of Physics (Dirac Centennial Celebration, Cambridge, UK, 2002)Google Scholar
  21. E. Hecht, Einstein on mass and energy. Am. J. Phys. 77(9), 799–806 (2009)ADSCrossRefGoogle Scholar
  22. S. Holm, This year Easter falls on the correct date according to Newton, http://www.science20.com/view_from_the_north/this_year_easter_falls_on_the_correct_date_according_to_newton-154289. Accessed 16 June 2018
  23. R. Hooke, Lectures de potentia restitutiva, or of spring explaining the power of springing bodies, Printed for John Martyn printer to the Royal Society, at the Bell in St. Paul’s church-yard (1678)Google Scholar
  24. S.L. Jaki, The Relevance of Physics (University of Chicago Press, USA, 1966)Google Scholar
  25. L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edn. Course of Theoretical Physics, vol. 1 (Elsevier, Amsterdam, 1976)Google Scholar
  26. C. Lomnitz, Creep measurements in igneous rocks. J. Geol. 64(5), 473–479 (1956)ADSCrossRefGoogle Scholar
  27. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010)Google Scholar
  28. F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 15, 712–717 (2012)Google Scholar
  29. J.J. Markham, R.T. Beyer, R.B. Lindsay, Absorption of sound in fluids. Rev. Mod. Phys. 23(4), 353–411 (1951)ADSMathSciNetCrossRefGoogle Scholar
  30. H. Markovitz, Boltzmann and the beginnings of linear viscoelasticity. Trans. Soc. Rheol. (1957–1977) 21(3), 381–398 (1977)MathSciNetCrossRefGoogle Scholar
  31. M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, vol. 43 (Walter de Gruyter, Berlin, 2012)Google Scholar
  32. S.I. Meshkov, G.N. Pachevskaya, V.S. Postnikov, U.A. Rossikhin, Integral representations of \({\ni }_\gamma \)-functions and their application to problems in linear viscoelasticity. Int. J. Eng. Sci. 9(4), 387–398 (1971)Google Scholar
  33. S.I. Muslih, D. Baleanu, Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl. 304(2), 599–606 (2005)ADSMathSciNetCrossRefGoogle Scholar
  34. I. Newton, Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) (London, 1687)Google Scholar
  35. P. Nutting, A new general law of deformation. J. Franklin. Inst. 191(5), 679–685 (1921)CrossRefGoogle Scholar
  36. R.F. O’Doherty, N.A. Anstey, Reflections on amplitudes. Geophys. Prosp. 19, 430–458 (1971)ADSCrossRefGoogle Scholar
  37. V. Pandey, S. Holm, Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity. Phys. Rev. E 94, 032606-1–6 (2016)Google Scholar
  38. A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications (McGraw-Hill, New York, 1981). Reprinted in 1989Google Scholar
  39. F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890 (1996)ADSMathSciNetCrossRefGoogle Scholar
  40. Y.A. Rossikhin, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl. Mech. Rev. 63(1), 010701-1–12 (2010)ADSCrossRefGoogle Scholar
  41. R. Sinkus, S. Lambert, K.Z. Abd-Elmoniem, C. Morse, T. Heller, C. Guenthner, A.M. Ghanem, S. Holm, A.M. Gharib, Rheological determinants for simultaneous staging of hepatic fibrosis and inflammation in patients with chronic liver disease. NMR Biomed e3956, 1–10 (2018)Google Scholar
  42. G.G. Stokes, On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Philos. Soc. 8(part III), 287–319 (1845)Google Scholar
  43. T.L. Szabo, Diagnostic Ultrasound Imaging: Inside Out, 2nd edn. (Academic Press, Cambridge, 2014)Google Scholar
  44. 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
  45. N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction (Springer, Berlin, 1989). Reprinted in 2012CrossRefGoogle Scholar
  46. D. Valério, J.T. Machado, V. Kiryakova, Some pioneers of the applications of fractional calculus. FCAA 17(2), 552–578 (2014)Google Scholar
  47. C. Zener, Elasticity and Anelasticity of Metals (University of Chicago Press, Chicago, 1948)Google Scholar

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Authors and Affiliations

  1. 1.Department of InformaticsUniversity of OsloOsloNorway

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