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:
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.
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Notes
- 1.
Also called compressional or longitudinal waves.
- 2.
Also called translational waves.
- 3.
Calling these equations “governing” may give the impression that these laws are actually the cause of the phenomena. This may be so for legislation in a country governed by the rule of law, where laws are normative. But physics is descriptive, as this quote implies (Duhem 1991, p. 19): “A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws”.
- 4.
Gödel’s incompleteness theorem states that the proof of consistency of a set of mathematical axioms can only be found outside the set. The same limitation may therefore apply to physics (Jaki 1966, pp. 128–130; Barrow 2011). This may have implications regarding how much a “theory of everything” in physics really can encompass, as later rediscovered and made more widely known by Hawking (2002). On first glance this may cause pessimism on behalf of science, but on second thought it should rather give reason for optimism as science will never come to an end, and there will always be new discoveries to explore. The open-ended nature of physics due to Gödel’s theorem is also discussed in Dyson (1996).
- 5.
References
B. Angelsen, Ultrasonic Imaging: Waves, Signals, and Signal Processing, vol. 1–2 (Emantec AS, Trondheim, 2000)
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. 255–276
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)
G.S. Blair, M. Reiner, The rheological law underlying the Nutting equation. Appl. Sci. Res. 2(1), 225–234 (1951)
L. Boltzmann, Zur theorie der elastischen nachwirkung (On the theory of hereditary elastic effects). Ann. Phys. Chem. Bd. 7, 624–654 (1876)
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)
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)
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)
M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(1), 134–147 (1971)
A. Chapman, England’s Leonardo: Robert Hooke and the Seventeenth-Century Scientific Revolution (CRC Press, New York, 2004)
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)
N.P. Chotiros, Acoustics of the Seabed as a Poroelastic Medium (Springer, ASA Press, Berlin, Switzerland, 2017)
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)
F.A. Duck, Physical Properties of Tissues: A Comprehensive Reference Book (Academic press, Cambridge, 2012)
P.M.M. Duhem, The Aim and Structure of Physical Theory (La théorie physique. Son objet, sa structure, 1906) (Princeton University Press, Princeton, 1991)
F. Dyson, The scientist as rebel. Am. Math. Monthly 103(9), 800–805 (1996)
R.P. Feynman, The Character of Physical Law (MIT Press, Cambridge, 1967)
J. Garnier, K. Sølna, Pulse propagation in random media with long-range correlation. Multiscale Model Simul. 7(3), 1302–1324 (2009)
D.J. Gross, The role of symmetry in fundamental physics. Proc. Natl. Acad. Sci. USA 93(25), 14256–14259 (1996)
S. Hawking, Gödel and the End of Physics (Dirac Centennial Celebration, Cambridge, UK, 2002)
E. Hecht, Einstein on mass and energy. Am. J. Phys. 77(9), 799–806 (2009)
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
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)
S.L. Jaki, The Relevance of Physics (University of Chicago Press, USA, 1966)
L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edn. Course of Theoretical Physics, vol. 1 (Elsevier, Amsterdam, 1976)
C. Lomnitz, Creep measurements in igneous rocks. J. Geol. 64(5), 473–479 (1956)
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010)
F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 15, 712–717 (2012)
J.J. Markham, R.T. Beyer, R.B. Lindsay, Absorption of sound in fluids. Rev. Mod. Phys. 23(4), 353–411 (1951)
H. Markovitz, Boltzmann and the beginnings of linear viscoelasticity. Trans. Soc. Rheol. (1957–1977) 21(3), 381–398 (1977)
M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, vol. 43 (Walter de Gruyter, Berlin, 2012)
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)
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)
I. Newton, Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) (London, 1687)
P. Nutting, A new general law of deformation. J. Franklin. Inst. 191(5), 679–685 (1921)
R.F. O’Doherty, N.A. Anstey, Reflections on amplitudes. Geophys. Prosp. 19, 430–458 (1971)
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)
A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications (McGraw-Hill, New York, 1981). Reprinted in 1989
F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890 (1996)
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)
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)
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)
T.L. Szabo, Diagnostic Ultrasound Imaging: Inside Out, 2nd edn. (Academic Press, Cambridge, 2014)
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)
N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction (Springer, Berlin, 1989). Reprinted in 2012
D. Valério, J.T. Machado, V. Kiryakova, Some pioneers of the applications of fractional calculus. FCAA 17(2), 552–578 (2014)
C. Zener, Elasticity and Anelasticity of Metals (University of Chicago Press, Chicago, 1948)
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Holm, S. (2019). Introduction. In: Waves with Power-Law Attenuation. Springer, Cham. https://doi.org/10.1007/978-3-030-14927-7_1
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