Waves with Power-Law Sverre Holm

Waves with Power-Law Attenuation

123 Sverre Holm Department of Informatics University of Oslo Oslo, Norway

ISBN 978-3-030-14926-0 ISBN 978-3-030-14927-7 (eBook) https://doi.org/10.1007/978-3-030-14927-7

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Editorial Board

Mark F. Hamilton (Chair), University of Texas at Austin James Cottingham, Coe College Diana Deutsch, University of California, San Diego Timothy F. Duda, Woods Hole Oceanographic Institution Robin Glosemeyer Petrone, Threshold Acoustics William M. Hartmann (Ex Officio), Michigan State University Darlene R. Ketten, Boston University James F. Lynch (Ex Officio), Woods Hole Oceanographic Institution Philip L. Marston, Washington State University Arthur N. Popper (Ex Officio), University of Maryland G. Christopher Stecker, Vanderbilt University School of Medicine Steven Thompson, Pennsylvannia State University Ning Xiang, Rensselaer Polytechnic Institute The Acoustical Society of America

On December 27, 1928 a group of scientists and engineers met at Bell Telephone Laboratories in New York City to discuss organizing a society dedicated to the field of acoustics. Plans developed rapidly and the Acoustical Society of America (ASA) held its first meeting on 10–11 May 1929 with a charter membership of about 450. Today ASA has a worldwide membership of 7000. The scope of this new society incorporated a broad range of technical areas that continues to be reflected in ASA’s present-day endeavors. Today, ASA serves the interests of its members and the acoustics community in all branches of acoustics, both theoretical and applied. To achieve this goal, ASA has established technical committees charged with keeping abreast of the developments and needs of membership in specialized fields as well as identifying new ones as they develop. The Technical Committees include acoustical oceanography, animal bioacous- tics, architectural acoustics, biomedical acoustics, engineering acoustics, musical acoustics, noise, physical acoustics, psychological and physiological acoustics, signal processing in acoustics, speech communication, structural acoustics and vibration, and underwater acoustics. This diversity is one of the Society’s unique and strongest assets since it so strongly fosters and encourages cross-disciplinary learning, collaboration, and interactions. ASA publications and meetings incorporate the diversity of these Technical Committees. In particular, publications play a major role in the Society. The Journal of the Acoustical Society of America (JASA) includes contributed papers and patent reviews. JASA Express Letters (JASA-EL) and Proceedings of Meetings on Acoustics (POMA) are online, open-access publications, offering rapid publication. Acoustics Today, published quarterly, is a popular open-access magazine. Other key features of ASA’s publishing program include books, reprints of classic acoustics texts, and videos. ASA’s biannual meetings offer opportunities for attendees to share information, with strong support throughout the career continuum, from students to retirees. Meetings incorporate many opportunities for professional and social interactions and attendees find the personal contacts a rewarding experience. These experiences result in building a robust network of fellow scientists and engineers, many of whom became lifelong friends and colleagues. From the Society’s inception, members recognized the importance of developing acoustical standards with a focus on terminology, measurement procedures, and cri- teria for determining the effects of noise and vibration. The ASA Standards Program serves as the Secretariat for four American National Standards Institute Committees and provides administrative support for several international standards committees. Throughout its history to present day, ASA’s strength resides in attracting the interest andcommitment ofscholars devotedtopromotingthe knowledge andpractical applicationsofacoustics.Theunselfishactivityofthese individualsinthe development of the Society is largely responsible for ASA’s growth and present stature. Beautiful is what we see, more beautiful is what we know, most beautiful by far is what we don't know Nicolas Steno (1638–1686) Danish anatomist, geologist, and bishop Preface

Common for the fields of imaging, medical elastography, and sediment acoustics is that waves propagate through media which may cause attenuation and dispersion—frequency-dependent reduction of amplitude and frequency-dependent propagation velocity. Compressional waves and shear waves in these media often follow much more complex attenuation laws than those of the classical viscous and relaxation models, and frequency power laws with powers between zero and two are often encountered. In the time domain, the exponential responses of the classical models will then have to be exchanged with temporal power laws. Much of the motivation for this book comes from discovering that many results that impact on acoustic and elastic wave propagation already exist in the field of linear viscoelasticity. The writing of the book has therefore been a humbling experience as it has made me aware of my initial lack of understanding of this vast field—an understanding that is still growing—and has greatly increased my appreciation for all the existing work. But more than anything else, it has been a delightful undertaking to discover how various fields fit together. The book starts off with a touch of philosophy of science in order to make a clear distinction between conservation principles and constitutive laws. In Part I, the classical models of acoustics are then reformulated in terms of standard constitutive models from linear elasticity, or actually it is rediscovered that this is where they come from. Then Part II continues with an in-depth coverage of non-classical loss models that follow power laws and which are expressed via convolution models and fractional derivatives. In addition parallels are drawn to electromagnetic waves in complex dielectric media. Other mechanisms for power-law attenuation such as multiple scattering in fractal media and those inherent in the standard models for poroviscoelasticity are also discussed, and some of them are related to the fractional models. The goal is twofold. First it is to integrate concepts from physical acoustics (Pierce 1981; Kinsler et al. 1999; Blackstock 2000) with those from linear vis- coelasticity (Tschoegl 1989), and fractional linear viscoelasticity (Mainardi 2010), in order to make the book profitable for readers in both fields.

xi xii Preface

But fractional modeling stands the risk of being considered to be a purely mathematical field (Podlubny 1999). A second goal is therefore to address how these models apply to fields such as sediment and underwater acoustics (Hovem 2012; Chotiros 2017), and medical ultrasound (Angelsen 2000; Szabo 2014). Mathematicians will therefore have to bear over with a level of rigor in the derivations which has been targeted primarily at physicists and engineers rather than mathematicians. The text is at a graduate level and requires a basic understanding of wave equations, propagating waves, and the Fourier transform.

Oslo, Norway Sverre Holm September 2018

References

B. Angelsen, Ultrasonic Imaging: Waves, Signals, and Signal Processing, vol. 1–2 (Emantec AS, Trondheim, 2000) D.T. Blackstock, Fundamentals of Physical Acoustics (Wiley, New York, 2000) N.P. Chotiros, Acoustics of the Seabed as a Poroelastic Medium (Springer and ASA Press, Switzerland, 2017) J.M. Hovem, Marine Acoustics: The Physics of Sound in Underwater Environments (Peninsula publishing, Los Altos, 2012) L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics, 4th edn. (Wiley-VCH, New York, 1999) F. Mainardi, and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010) A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications (McGraw-Hill, New York, 1981). Reprinted in 1989 I. Podlubny, Fractional Differential Equations (Academic, New York, 1999) T.L. Szabo, Diagnostic Ultrasound Imaging: Inside Out, 2nd edn. (Academic Press, Cambridge, 2014) N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction (Springer, Berlin, 1989). Reprinted in 2012 Acknowledgements

My work with wave propagation in complex media was first inspired by my Ph.D. supervisor in the Electrical Engineering Department of the Norwegian University of Science and Technology (NTNU), Jens Hovem, in the 1980’s, when he was working on wave propagation in sub-bottom sediments. But it got its present direction as I started working in the thriving medical ultrasound community in Norway in 1990 in collaboration with Kjell Kristoffersen and Kjell Arne Ingebrigtsen at GE Vingmed Ultrasound, as well as Hans Torp and Bjørn Angelsen at NTNU. Of particular importance was the study of the papers of Tom Szabo, Boston University (Szabo 1994, 1995). He had set out to tackle the challenging problem of modeling the power-law attenuation encountered in medical ultrasound imaging. In 2002 I started working on this topic with Wen Chen at Simula Research Laboratory in Oslo (now at Hohai University, Nanjing, China) in a collaboration that has continued to this day (Cai et al. 2018). He introduced me to the powerful tool of non-integer, fractional, derivatives. We were able to reformulate some of Szabo’s work in that framework and then further develop it (Chen and Holm 2003, 2004). The next milestone was a stay at the Institut Langevin in Paris 2008–2009, thanks to the hospitality of Matthias Fink and Mickaël Tanter. There I was able to delve more into the same topic as well as being introduced to elastography, medical imaging with shear waves. The collaboration with Ralph Sinkus, later at King’s College London, was especially fruitful (Holm and Sinkus 2010) and has continued since then (Sinkus et al. 2018). The models have been further developed with the help of coworkers and stu- dents. Therefore, I also want to thank in particular Sven Peter Näsholm, Fabrice Prieur, Wei Zhang, Vikash Pandey, and Sri Nivas Chandrasekaran for many stimulating discussions and joint work. Erlend Viggen, Sven Peter Näsholm, Sri Nivas Chandrasekaran, Fabrice Prieur, Vikash Pandey, Knut Sølna, Anders Kvellestad, and Kent-Andre Mardal have read parts of the book at various stages of its development and all given very constructive criticism. I also want to thank Fritz Albregtsen, Andreas Austeng, David Nordsletten, Robert McGough, Dumitru Baleanu, and Peder Tyvand for many helpful discus- sions, and Dag Langmyhr and Knut Hegna for help with LaTeX and bibliographies.

xiii xiv Acknowledgements

Thanks to anonymous reviewers of several papers through the years who have given the ideas of this book some resistance. This kind of feedback was what in the end convinced me in early 2016 that this topic warrants an entire book and not just short research papers. I am also grateful to the Department of Informatics for allowing an activity which is closer to physics than computer science to take place, and the Department as well as the University of Oslo for encouraging excellence in research. The writing of this book has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 668039.3 It has also been supported by the Centre for Innovative Ultrasound Solutions, The Research Council of Norway (237887), and MEDIMA (Multimodal medical imaging and image analysis), a strategic research initiative funded by the Faculty of Mathematics and Natural Sciences at the University of Oslo. Finally, I am indebted to my late father, Christian Holm, who instilled in me a passion for technology and electronics, to my son Martin Blomhoff Holm for collaboration (Holm and Holm 2017) and my son Thomas Holm for many helpful discussions. Above all, I want to thank my wife Lise for her friendship, love, and support over four decades.

References

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 propa- gation. Appl. Mech. Rev. (2018) W. Chen, S. Holm, Modified Szabo’s wave equation models for lossy media obeying frequency . J. Acoust. Soc. Am. 114(5), 2570–2574 (2003) 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) S. Holm, M.B. Holm, Restrictions on wave equations for passive media. J. Acoust. Soc. Am. 142 (4), (2017) S. Holm, R. Sinkus, A unifying fractional wave equation for compressional and shear waves. J. Acoust. Soc. Am. 127, 542–548 (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) T.L. Szabo, Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96, 491–500 (1994) T.L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power law. J. Acoust. Soc. Am. 97,14–24 (1995)

3This book reflects only the author’s view. The European Commission is not responsible for any use that may be made of the information it contains. Contents

1 Introduction ...... 1 1.1 A Wave Equation for Arbitrary Power-Law Attenuation? ...... 1 1.2 Conservation Laws and Constitutive Equations...... 3 1.2.1 Conservation Principles ...... 3 1.2.2 Hookean and Newtonian Medium Models ...... 6 1.2.3 Constitutive Equations ...... 7 1.3 Exponential Constitutive Laws ...... 10 1.3.1 Spring–Damper Models ...... 10 1.3.2 Exponential Time Responses ...... 11 1.4 Power-Law Constitutive Laws ...... 13 1.4.1 Fourier Definition of Fractional Derivative ...... 15 1.4.2 Fractional Derivative in Time ...... 15 1.4.3 Fractional Constitutive Laws ...... 17 1.4.4 Brief History of Fractional Viscoelasticity ...... 18 1.5 Wave Equations with Power-Law Solutions ...... 18 1.5.1 Fractional Wave Equations ...... 18 1.5.2 Power-Law Attenuation in Porous Media ...... 19 1.5.3 Power-Law Attenuation in Fractal Media ...... 19 References ...... 20

Part I Acoustics and Linear Viscoelasticity 2 Classical Wave Equations ...... 25 2.1 The Lossless Wave Equation ...... 26 2.1.1 Monochromatic Plane Wave ...... 27 2.1.2 The Wave Equation in Spherical Coordinates ...... 28 2.2 Lossless Wave Equations in Practice ...... 28 2.2.1 Acoustics ...... 29 2.2.2 Elastic Waves ...... 30 2.2.3 Electromagnetics ...... 31

xv xvi Contents

2.3 Characterization of Attenuation ...... 32 2.3.1 Dispersion Relation ...... 32 2.3.2 Attenuation per Wavelength, QÀ1, and Related Measures ...... 34 2.4 Viscous Losses: The Kelvin–Voigt Model ...... 38 2.4.1 Viscous Wave Equation and the Dispersion Equation .... 39 2.4.2 The Blackstock Equation ...... 43 2.5 The Zener Constitutive Equation ...... 43 2.5.1 Wave Equation ...... 45 2.5.2 Dispersion Relation and Compressibility/Compliance ..... 45 2.5.3 Asymptotes ...... 46 2.6 Relaxation and Multiple Relaxation ...... 49 2.6.1 The Relaxation Model ...... 49 2.6.2 Multiple Relaxation ...... 52 2.6.3 Seawater and Air ...... 54 2.7 The Maxwell Mechanical Model ...... 58 2.8 Losses in Electromagnetics ...... 59 2.8.1 A Conducting Medium ...... 59 2.8.2 Debye Dielectrics ...... 63 2.8.3 Multiple Debye Terms ...... 64 References ...... 65 3 Models of Linear Viscoelasticity ...... 67 3.1 Constitutive Equations ...... 67 3.1.1 Relaxation Modulus and Creep Compliance ...... 67 3.1.2 Linear Differential Equation Model ...... 69 3.1.3 The Causal Fading Memory Model ...... 70 3.1.4 Complete Monotonicity ...... 72 3.1.5 Relationship Between Descriptions ...... 72 3.1.6 Spring–Damper Model ...... 72 3.2 Standard Spring–Damper Models ...... 74 3.2.1 Spring and Dashpot Elements ...... 74 3.2.2 Kelvin–Voigt Model ...... 75 3.2.3 Maxwell Model ...... 78 3.2.4 The Standard Linear Solid: Zener Model...... 80 3.2.5 Higher Order Models ...... 84 3.3 Four Types of Linear Viscoelastic Models ...... 87 3.4 Completely Monotone Models ...... 88 3.4.1 Global Versus Local Passivity ...... 88 3.4.2 Special Role of Completely Monotone Models ...... 89 3.5 Wavenumber from Complex Modulus ...... 90 References ...... 92 Contents xvii

4 Absorption Mechanisms and Physical Constraints ...... 95 4.1 Viscous and Relaxation Processes in Acoustics ...... 95 4.1.1 and Heat Conduction: Monatomic Fluids ...... 96 4.1.2 Heat Relaxation: Polyatomic Gases and Normal Liquids ...... 100 4.1.3 Structural Relaxation in Water ...... 102 4.1.4 Chemical Reaction in Electrolytes ...... 104 4.1.5 Summary of Absorption Processes in Air and Seawater ... 104 4.2 Causality and Passivity ...... 105 4.2.1 Impulse Response and Transfer Function ...... 106 4.2.2 Kramers–Kronig Relations ...... 106 4.2.3 Passive Media ...... 107 4.3 Wave Equations for Passive Media ...... 108 4.3.1 Bernstein Property ...... 109 4.3.2 Consequences of the Bernstein Property ...... 109 4.3.3 Asymptotic Properties ...... 110 4.3.4 Viability of Two Viscous Wave Equations ...... 110 4.4 Does the Viscous Model Represent Realistic Media? ...... 113 4.4.1 The Validity of the Navier–Stokes Equation ...... 113 4.4.2 The Validity of the Fourier Heat Law ...... 114 References ...... 115

Part II Modeling of Power-Law Media 5 Power-Law Wave Equations from Constitutive Equations ...... 119 5.1 Empirical Power Laws in Frequency and Time ...... 119 5.2 Fractional Linear Viscoelasticity Models ...... 122 5.2.1 Fractional Kelvin–Voigt Model ...... 123 5.2.2 Fractional Zener Model ...... 124 5.2.3 Fractional Maxwell Model ...... 126 5.2.4 Fractional Newton (Spring-Pot) Model ...... 127 5.2.5 Two Spring-Pots in Parallel ...... 128 5.2.6 Classification of Fractional Models ...... 129 5.3 The Fractional Kelvin–Voigt Wave Equation ...... 130 5.3.1 Low xs Approximation ...... 131 5.3.2 High xs Approximation ...... 133 5.3.3 Asymptotes of Attenuation and Phase Velocity ...... 135 5.4 The Fractional Zener Wave Equation ...... 135 5.4.1 Frequency-Domain Properties ...... 136 5.4.2 Asymptotes of Attenuation and Phase Velocity ...... 140 5.4.3 Fractional Relaxation Model ...... 140 5.4.4 Linearly Increasing Attenuation, y ¼ 1 ...... 142 xviii Contents

5.5 The Fractional Maxwell Wave Equation ...... 142 5.5.1 Asymptotes of Attenuation and Phase Velocity ...... 146 5.6 The Fractional Diffusion-Wave Equation ...... 146 5.6.1 The Diffusion Equation ...... 147 5.6.2 Dispersion of the Fractional Diffusion-Wave Equation .... 148 5.7 Two Spring-Pots in Parallel ...... 150 5.8 Fractional Model of Electromagnetics ...... 152 5.8.1 Circuit Equivalent of the Cole–Cole Model ...... 153 5.8.2 Cole Impedance Model ...... 154 5.9 Fractional Constitutive Laws and Wave Equations ...... 155 References ...... 156 6 Phenomenological Power-Law Wave Equations ...... 161 6.1 Modified Ordinary Wave Equations ...... 161 6.1.1 Generalization of the Blackstock Equation ...... 162 6.1.2 Generalization of the Viscous Wave Equation ...... 163 6.2 Single Power-Law ...... 167 6.3 Fractional Conservation Laws ...... 168 6.3.1 Fractional Mass Conservation ...... 169 6.3.2 Fractional Momentum Conservation ...... 170 6.4 Is There a Best Model for Power-Law Attenuation? ...... 171 References ...... 172 7 Justification for Power Laws and Fractional Models ...... 173 7.1 Fractional Heat Models ...... 174 7.1.1 Fractional Heat Conduction ...... 174 7.1.2 Fractional Heat Relaxation or Diffusion ...... 176 7.2 Multiple Relaxation ...... 177 7.2.1 Capacitor Dielectric Absorption ...... 178 7.2.2 Pink Noise from Lowpass Processes ...... 179 7.2.3 Attenuation by Summing Relaxation Processes ...... 181 7.2.4 Long-Tailed Distribution of Relaxation Processes ...... 182 7.3 Multiple Relaxation in Linear Viscoelasticity ...... 188 7.3.1 Relaxation Spectral Functions ...... 188 7.3.2 Hierarchical Structures ...... 192 7.3.3 Soft Glassy Materials ...... 193 7.3.4 Example of Fitting of Maxwell–Wiechert Model ...... 194 7.4 Hierarchical Models ...... 195 7.4.1 Lumped Circuit Cable Model ...... 195 7.4.2 Self-similar Ladder ...... 197 7.4.3 The Rouse Polymer Model ...... 199 7.4.4 Polymer Models of Arbitrary Order ...... 200 7.4.5 Ladder Model for Arbitrary Fractional Order ...... 201 7.4.6 Self-similar Tree ...... 202 Contents xix

7.5 Non-Newtonian Rheology ...... 203 7.5.1 Time-Varying Non-Newtonian Media ...... 205 7.5.2 Rheopecty Modeled as a Spring-Pot ...... 206 7.6 Viscous Boundary Layer Problems ...... 208 7.6.1 Wave in Boundary Layer ...... 209 7.6.2 Stokes’ Second Problem: Oscillating Plate ...... 209 7.6.3 Elastic Interface Waves ...... 210 7.6.4 The Viscodynamic Operator and Dynamic Tortuosity .... 210 7.6.5 Flow in a Tube ...... 214 7.6.6 Gap Stiffness Model and Squirt Flow ...... 215 7.7 The Prevalence of Half-Order Models ...... 218 7.7.1 Abel’s Mechanical Problem ...... 218 7.7.2 Ladder and Tree Structures ...... 219 7.7.3 Viscous Boundary Layer Problems ...... 219 7.8 Mechanisms for Power-Law Attenuation ...... 219 References ...... 220 8 Power Laws and Porous Media ...... 225 8.1 Grain Shearing and Poroviscoelastic Models ...... 225 8.1.1 Grain Shearing Models ...... 225 8.1.2 Poroviscoelastic Models ...... 226 8.1.3 Limitations of the Models ...... 226 8.2 The Grain Shearing Family of Models ...... 227 8.2.1 Grain Shearing Model ...... 227 8.2.2 Grain Shearing with Fractional Derivatives ...... 229 8.2.3 The Viscous Grain Shearing Model ...... 232 8.3 The Poroelastic Biot Model ...... 239 8.3.1 Biot–Stoll Formulation ...... 240 8.3.2 Wave Equations ...... 241 8.3.3 Solution of the Wave Equations ...... 242 8.3.4 Low-Frequency Shear Wave Solution ...... 243 8.3.5 Low-Frequency Compressional Wave Solutions ...... 244 8.4 High-Frequency Turbulent Flow in Pores ...... 245 8.4.1 Shear Wave Solution ...... 245 8.4.2 Compressional Wave Solutions ...... 247 8.5 Viscosity in the Solid ...... 250 8.5.1 Biot–Stoll Hysteresis Modification ...... 250 8.5.2 Biot–Stoll with Contact Squirt Flow and Shear Drag (BICSQS) ...... 251 8.5.3 Biot Model with Modified Gap Stiffness (BIMGS) ...... 254 8.5.4 Extended Biot Theory ...... 254 8.5.5 Viscosity in the Fluid...... 255 xx Contents

8.6 Models for Porous Media ...... 255 References ...... 257 9 Power Laws and Fractal Scattering Media ...... 259 9.1 O’Doherty-Anstey Model ...... 259 9.1.1 Multiple Scattering Derivation ...... 259 9.1.2 Physical Interpretation ...... 262 9.2 Wave Equation in an Inhomogeneous Medium ...... 263 9.2.1 1-Dimensional Equations ...... 264 9.2.2 Travel Time Coordinate ...... 265 9.2.3 Change of Variables to ln Z ...... 265 9.3 Long-Range Correlation ...... 266 9.3.1 Mean Field Theory for Compressibility Fluctuations ..... 266 9.3.2 Asymptotic Theory and Long-Range Correlation ...... 267 9.4 Verification by Measurement in Fractal Media ...... 269 9.4.1 Characterization of a Medium’s Fractality ...... 269 9.4.2 Frequency Variation in Phase Velocity ...... 270 9.5 Effect of Fractal Scattering Process ...... 271 References ...... 272 Appendix A: Mathematical Background ...... 273 Appendix B: Wave and Heat Equations ...... 289 Index ...... 309 About the Author

Sverre Holm was born in Oslo, Norway, in 1954. He received M.S. and Ph.D. degrees in electrical engineering from the Norwegian Institute of Technology (NTNU), Trondheim in 1978 and 1982, respectively. He has academic experience from NTNU and Yarmouk University in Jordan (1984–86). Since 1995 he has been a professor of signal processing and acoustic imaging at the University of Oslo. In 2002 he was elected a member of the Norwegian Academy of Technological Sciences. His industry experience includes GE Vingmed Ultrasound (1990–94), working on digital ultrasound imaging, and Sonitor Technologies (2000–05), where he developed ultrasonic indoor positioning. He is currently involved with several startups in the Oslo area working in the areas of acoustics and ultrasonics. Dr. Holm has authored or co-authored around 220 publications and holds 12 patents. He has spent sabbaticals at GE Global Research, NY (1998), Institut Langevin, ESPCI, Paris (2008–09), and King’s College London (2014). His research interests include medical ultrasound imaging, elastography, modeling of waves in complex media, and ultrasonic positioning.

xxi Symbols

B Magnetic field in Tesla, T, or newton per meter per ampere. Also measured in gauss where 1 T = 10,000 G c Speed of propagation. Except for the lossless wave equation, it will vary with frequency and then c0 ¼ cðx ¼ 0Þ cph Phase velocity which unlike c always is a real value. In the lossless case, cph ¼ c ¼ c0, and in the general case it is given by (Eq. 2.23) cp; cv Specific heat capacities under constant pressure and constant volume conditions respectively D Electric displacement field, a vector field that accounts for the effects of free and bound charge in materials. It is measured in coulomb per squared meter E In this text it is mostly used denote any modulus of elasticity [Pa]. It can be due to compression alone or shear alone or a combination of the two. In sections dealing with electromagnetics it is electric vector field, a vector field measured in volts per meter or newtons per coulomb E~ðÞx The dynamic modulus, the frequency-dependent complex elas- ticity modulus [Pa] EY Young’s modulus [Pa] f ðnÞðtÞ Derivative of order n where n can be any integer F ðf ðtÞÞ Fourier transform of f ðtÞ GðtÞ The relaxation modulus: the stress response to a unit step in strain. The relaxation modulus describes a material at the macroscopic scale as given in Sect. 3.1.1. It must be distinguished from the relaxation processes of Chap. 4 which usually take place at the molecular scale Ge ¼ lim GðtÞ Equilibrium value of the relaxation modulus t!1 þ Gg ¼ Gð0 Þ The glass modulus or instantaneous value of the relaxation modulus

xxiii xxiv Symbols

H Magnetic field measured in amperes per meter JðtÞ The creep compliance: the strain response to a unit step in stress Je ¼ lim JðtÞ Equilibrium value of the creep compliance t!1 þ Jg ¼ Jð0 Þ The glass modulus or instantaneous value of the creep compliance k ¼ k0 þ ik00 The complex wavenumber. This is patterned after usage in electromagnetics, e ¼ e0 þ ie00 and in elastography G ¼ G0 þ iG00 K Bulk modulus [Pa] 0 0 p; p0; p Acoustic pressure [Pa], total pressure is p ¼ p0 þ p where p0 is the equilibrium or static pressure (usually 1 atmosphere) and p is the acoustic perturbation s Slowness vector given by (Eq. 2.4) 0 0 T; T0; T Temperature, absolute temperature in K is T ¼ T0 þ T, where T0 is an equilibrium temperature and T is a perturbation. Special usage is in Sect. 2.6.3.2 where T0 ¼ 293:15 K for the formula for attenuation in air, and later in Sect. 2.6.3.1, where T0 is 273.15 K so that T is temperature in  C in the formula for attenuation in seawater u; u Acoustic displacement [m] or displacement vector uHðtÞ The unit step, or Heaviside, function y Power-law exponent in expression for attenuation, see ak a Model order in fractional Kelvin-Voigt and Zener constitutive laws. Often in the low-frequency-/low-loss case y ¼ a þ 1 ak Attenuation, equal to the negative imaginary part of the 00 y wavenumber, Àk . Often it follows a power law, ak ¼ a0jf j , y where a0 is the attenuation constant in for instance Np/m/Hz or dB/m/Hzy, see discussion of the difference in Sect. 2.3.2.7.In ultrasound imaging the constant is often scaled to be dB/cm/MHzy aL;p; aL;a One-dimensional thermal expansion coefficient, hence the sub- script L for length, under constant pressure or under adiabatic conditions respectively. Only used in Sect. 4.1.2 b Second model order in fractional Zener constitutive law, usually b ¼ a c ¼ cp=cv Ratio of specific heats e In sections dealing with elastic waves it is strain, unit less relative displacement or deformation. In sections dealing with electro- magnetic waves it is the permittivity or dielectric constant. Here e0 and e1 are the values at 0 and 1 frequency ; Á gL gS Longitudinal (compressional) viscosity and shear viscosity [Pa s] j Compressibility which is the inverse of the bulk modulus j ¼ 1=K j~ðxÞ Dynamic compressibility which is 1=E~ðxÞ jh Thermal conductivity Symbols xxv k Wavelength in meters k; l The first and second Lamé parameters [Pa]. The first Lamé parameter is only used in App. B.2, everywhere else k means wavelength. The second Lamé parameter is the same as the shear modulus, l ¼ G l Permeability. It can be confused with the second Lamé parameter, but the context should make it clear which is which ; ; 0 3 0 ¼ þ q q0 q Density [kg/m ], total density is q q0 q where q0 is the density corresponding to the equilibrium pressure and q is the acoustic perturbation r Stress which is force per unit area [Pa] (similar to pressure) or conductivity, but the context should make it clear which is which se; sr Retardation time of the Kelvin-Voigt and Zener models and relaxation time of the Zener model x ¼ 2pf Angular frequency where f is frequency in Hz

A challenge when several fields are combined is that a symbol may be ambiguous. In acoustics, attenuation is usually denoted by a and the wavenumber by k ¼ b À ia. Likewise the fractional model orders in linear viscoelasticity are called a and b. Also, the thermal expansion coefficient is often called a. It is also common to use a subscript such as aL, which is the longitudinal, 1-D, version, and that is used here also. Since the fractional orders appear in exponents, the choice has been made to reserve a and b without subscripts for them. The acoustic attenuation is therefore denoted by ak and the wavenumber is k ¼ x=cph À iak. This follows Holm and Näsholm (2011, 2014). In order not to confuse the phase velocity with the specific heat for constant pressure, the first is called cph and the latter cp. Fourier and Laplace transforms of a temporal function are usually denoted by capital letters, e.g., hðtÞ,HðxÞ, in acoustics. But in linear viscoelasticity it is common to use capital letters even for function of time. Therefore, in order not to make Fourier and Laplace transforms ambiguous by using the same symbol in the two domains as in eðtÞ and eðxÞ or GðtÞ and GðxÞ, a tilde will be used for the transform, e.g. eðtÞ,~eðxÞ and GðtÞ,G~ðxÞ when the original function and its transform cannot be separated by the use of lower and upper case. List of Figures

Fig. 1.1 Kelvin–Voigt, Maxwell, and Zener models from left to right. The terminals show where stress is applied and strain measured or vice versa ...... 8 Fig. 1.2 Front suspension of a classic Vespa scooter showing a shock absorber and a spring in a coilover configuration (“coil spring over shock”) to the left. The structure to the right is a swing-arm. Image: Public domain, from Wikipedia Commons ...... 9 Fig. 1.3 A wave equation is found by combining space–time conservation principles with the material’s constitutive equation ...... 9 Fig. 1.4 Relaxation moduli of Zener model with an exponential time response, (1.13) (solid line), and for the fractional Zener model for a ¼ 0:5 (dashed line) which asymptotically approaches a power-law function, (1.30). The asymptotic values are the þ glass modulus, Gg ¼ Gð0 Þ and the equilibrium modulus, Ge for infinite time...... 12 Fig. 1.5 Maxwell–Wiechert model ...... 12 Fig. 1.6 Fractional Kelvin–Voigt (left) and Zener models of order a .... 17 Fig. 2.1 A string which is pulled up from the equilibrium position will oscillate ...... 26 Fig. 2.2 A helical spring—the model for elasticity in acoustics and for elastic waves [By User: Jean-Jacques MILAN—CC BY-SA 3.0], via Wikipedia Commons ...... 29 Fig. 2.3 Finding the dispersion relation, and attenuation and phase velocity from a wave equation, adapted from (Blackstock 2000 Fig. 9.1)...... 32 Fig. 2.4 Four examples of a damped sinusoidal wave with number of periods as abscissa. The log of the ratio between the amplitudes in the two marked peaks (at times 1 and 2) is the

xxvii xxviii List of Figures

log decrement, ddec. The penetration depth is where the horizontal line at 1=e  0:37 crosses the envelope. The lower right-hand solution is for the diffusion case and shows a non-propagating solution...... 36 Fig. 2.5 Kelvin–Voigt constitutive model...... 38 Fig. 2.6 Kelvin–Voigt model: Attenuation, phase velocity, and inverse Q (loss tangent) ...... 40 Fig. 2.7 Zener or Standard Linear Solid constitutive model ...... 44 Fig. 2.8 Zener model with se=sr ¼ 100: Attenuation, phase velocity, and inverse Q (loss tangent) ...... 47 Fig. 2.9 Relaxation model, i.e., Zener model with se=sr ¼ 1:01: Attenuation and inverse Q (loss tangent) ...... 51 Fig. 2.10 The generalized Kelvin–Voigt or Kelvin model ...... 53 Fig. 2.11 Attenuation in seawater for various oceans of the world, parameters from Ainslie and McColm (1998)...... 55 Fig. 2.12 Inverse Q (loss tangent) in seawater for various oceans of the world, parameters from Ainslie and McColm (1998) .... 56 Fig. 2.13 Attenuation in air at temperature 20 C and for 0, 55, and 100% relative humidity. The lower curve is the contribution due to viscosity and heat conduction alone, the first term of (2.92) ...... 57 Fig. 2.14 Speed of sound and dispersion in air at temperature 20 C and for 0, 55, and 100% relative humidity ...... 58 Fig. 2.15 Maxwell constitutive model ...... 59 Fig. 2.16 Circuit equivalent of a dielectric ...... 60 Fig. 2.17 Exponential distribution of current into a circular conductor and illustration of the skin depth. [By Biezl-Own work, CC BY-SA 3.0], via Wikipedia Commons ...... 62 Fig. 2.18 Circuit equivalent of the Debye model ...... 65 Fig. 3.1 A rotational rheometer that controls the applied shear stress or shear strain. Wikipedia Commons (User: Olivier Cleynen)..... 68 Fig. 3.2 The set of completely monotone models is a subset of fading memory models, which itself is a subset of all solutions to linear differential equations ...... 72 Fig. 3.3 Model for a general viscoelastic material ...... 73 Fig. 3.4 Kelvin–Voigt constitutive model...... 76 Fig. 3.5 Step responses of the Kelvin–Voigt model. Upper figure: The relaxation modulus, GðtÞ, the stress response to a unit step input in strain. Lower figure: the creep compliance, JðtÞ, the strain response to a unit step input in stress ...... 78 Fig. 3.6 Maxwell constitutive model ...... 78 Fig. 3.7 Step responses of the Maxwell model. Upper figure: The relaxation modulus, GðtÞ, the stress response to a unit step List of Figures xxix

input in strain. Lower figure: the creep compliance, JðtÞ, the strain response to a unit step input in stress ...... 79 Fig. 3.8 Zener or standard linear solid constitutive model ...... 80 Fig. 3.9 Step responses of the Zener model for se=sr ¼ 2. Upper figure: The relaxation modulus, GðtÞ, the stress response to a unit step input in strain. Lower figure: the creep compliance, JðtÞ, the strain response to a unit step input in stress ...... 82 Fig. 3.10 3-parameter Kelvin–Voigt model ...... 83 Fig. 3.11 Maxwell–Wiechert model ...... 85 Fig. 3.12 The generalized Kelvin–Voigt or Kelvin model ...... 86 Fig. 3.13 A nonphysical system where one branch has negative spring and damper constants ...... 89 Fig. 4.1 External translational modes in x, y, z (left) for the diatomic oxygen molecule, O2, and internal modes: two rotational modes (left) and one vibrational mode (right)...... 100 Fig. 4.2 Average geometry of a water molecule...... 102 Fig. 4.3 2-D sketch of hydrogen bonds between water molecules, shown in dashed lines...... 102 Fig. 4.4 Molecular dynamics simulation of hydrogen-bond structure of water prior to freezing (left) and after freezing (right). Times, (208 and 580 ns), show time from start of simulation. The lines are hydrogen bonds. Bright blue lines indicate “long-lasting” bonds, in this case those with a lifetime longer than 2 ns. Reprinted by permission from Nature (Matsumoto et al. 2002), copyright 2002 ...... 103 Fig. 4.5 Mechanical model for seawater with three relaxation process. The first relaxation frequency is very high, so the spring E1 is very small and zero if the first process is considered to be viscous...... 105 Fig. 4.6 A slab of thickness z with a plane wave traveling from left to right. Reprinted from Holm and Näsholm (2014) with permission from Elsevier ...... 106 Fig. 4.7 The set of passive, linear materials is a subset of causal wave equations, which again is a subset of possible wave equations. Reprinted with permission from Holm and Holm (2017), copyright 2017, Acoustical Society of America ...... 108 Fig. 4.8 Viscous wave equation (Stokes) with mixed derivative loss term (solid line) compared to the Blackstock equation with only temporal derivatives in the loss term (dash-dot line): Upper curve: Attenuation. Lower curve: Phase velocity, both as a function of xs, the normalized frequency. Reprinted with permission from Holm and Holm (2017), copyright 2017, Acoustical Society of America ...... 112 xxx List of Figures

Fig. 5.1 Absorption in typical tissue as well as for a representative selection of tissue types compared to the highest and lowest attenuation in water from 1 to 10 MHz ...... 120 Fig. 5.2 Fractional Kelvin–Voigt model with spring characterized by elastic modulus, E, and spring-pot given by viscosity, g, and fractional order, a ...... 123 Fig. 5.3 Fractional Zener model with two springs and a spring-pot given by viscosity, g, and fractional order, a ...... 125 Fig. 5.4 Relaxation moduli of fractional Kelvin–Voigt (upper) and fractional Zener (lower) models, a ¼ 0:5 ...... 126 Fig. 5.5 Fractional Maxwell model with a spring and a spring-pot given by viscosity, g, and fractional order, a ...... 127 Fig. 5.6 Fractional Newton or spring-pot model given by viscosity, g, and fractional order, a ...... 127 Fig. 5.7 Parallel combination of two spring-pots ...... 128 Fig. 5.8 Absolute value of dynamic modulus for the spring-pot and damper of (5.3) with a as parameter and unity viscosity parameters. The solid lines are for b ¼ 1 and the dash-dotted line is for a ¼ 0:3 and b ¼ 0:75...... 129 Fig. 5.9 Attenuation for the fractional Kelvin–Voigt model with a as parameter ...... 131 Fig. 5.10 Relative phase velocity for the fractional Kelvin–Voigt model with a as parameter ...... 132 Fig. 5.11 Inverse Q (loss tangent) for the fractional Kelvin–Voigt model with a as parameter ...... 132 Fig. 5.12 Frequency-dependent absorption for the fractional Zener wave equation for se ¼ 100sr. The horizontal axis represents normalized frequency. The fractional derivative order a has values 0.1, 0.3, 0.7, and 1. For visualization, each absorption curve is normalized to unity at xse ¼ 1 ...... 137 Fig. 5.13 Normalized frequency-dependent sound speed for the fractional Zener wave equation for se ¼ 100sr. The fractional derivative order a has values 0.1, 0.3, 0.7, and 1 ...... 137 Fig. 5.14 Inverse Q (loss tangent) for the fractional Zener wave equation for se ¼ 100sr. The horizontal axis represents normalized frequency. The fractional derivative order a has values 0.1, 0.3, 0.7, and 1 ...... 138 Fig. 5.15 Frequency-dependent absorption for the relaxation model with se ¼ 1:01sr. The fractional derivative order a has values 0.1, 0.3, 0.7, and 1. For visualization, each absorption curve is normalized to unity at xse ¼ 1 ...... 141 Fig. 5.16 Inverse Q (loss tangent) for the relaxation model with se ¼ 1:01sr. The fractional derivative order a has values 0.1, 0.3, 0.7, and 1 ...... 141 List of Figures xxxi

Fig. 5.17 Attenuation for the fractional Maxwell model with a as parameter ...... 144 Fig. 5.18 Phase velocity relative to the asymptotic value for the fractional Maxwell model with a as parameter ...... 145 Fig. 5.19 Inverse Q (loss tangent) for the fractional Maxwell model with a as parameter ...... 145 Fig. 5.20 Frequency-dependent absorption for the fractional diffusion-wave equation. The horizontal axis represents normalized frequency. The fractional derivative order a has values 0.001, 0.01, 0.1, and 0.2. The parameters are j j¼j j¼ q0 g 1 ...... 149 Fig. 5.21 Frequency-dependent sound speed for the fractional diffusion-wave equation. The fractional derivative order a has values 0.001, 0.01, 0.1, and 0.2. The parameters are j j¼j j¼ q0 g 1 ...... 149 Fig. 5.22 Attenuation for spring-pot and damper (b ¼ 1) in parallel j j¼j j¼ ¼ with a as parameter and ga gb 1, q0 1000...... 151 Fig. 5.23 Phase velocity for spring-pot and damper (b ¼ 1) in parallel j j¼j j¼ ¼ with a as parameter and ga gb 1, q0 1000...... 151 Fig. 5.24 Inverse Q (loss tangent) for spring-pot and damper (b ¼ 1) in j j¼j j¼ ¼ parallel with a as parameter and ga gb 1, q0 1000 ... 152 Fig. 5.25 Circuit equivalent of the Cole–Cole model ...... 154 Fig. 5.26 Cole impedance model ...... 155 Fig. 5.27 Attenuation of fractional diffusion, fractional Kelvin–Voigt, and fractional Zener (sr ¼ 0:0001se) models of order a ¼ 0:5. Notice how they are almost indistinguishable in the intermediate frequency range from xs ¼ 101...104 ...... 156 Fig. 7.1 Model of a realistic 1 lF capacitor with dielectric absorption (component values from Pease 1991) ...... 178 Fig. 7.2 Recovery of voltage in a 1 lF capacitor. The lower curves show the recovery of an initial voltage of 1V due to a single of the RC-terms in the model of Fig. 7.1 and the upper curve is the net effect of all six contributions. The stars show the six time constants of the RC-terms. Simulated with TopSpice 8.75, © 2018 Penzar Development ...... 179 Fig. 7.3 Band-limited pink noise from relaxation processes. Solid curve: superposition of three relaxation processes with relaxation frequencies indicated by the three stars. Dot-dashed curve: 1/f. Parameters have been selected to approximately reproduce (Schroeder 2009, Fig. 2 on p. 125) with P0 ¼ 1:6, s1 ¼ 0:024, sm þ 1=sm ¼ 8...... 180 xxxii List of Figures

Fig. 7.4 Multiple relaxation approximation to power-law attenuation with unity slope. Solid curve: superposition of five relaxation processes. Dot-dashed curve: x, the five relaxation frequencies are indicated by the stars ...... 182 Fig. 7.5 Maxwell–Wiechert model (from Figs. 1.5 and 3.11)...... 183 Fig. 7.6 Frequency-spectral function for the fractional Zener model for je ¼ 1 and sr ¼ 0:99se. Solid line a ¼ 0:8, dashed line a ¼ 0:01 ...... 184 Fig. 7.7 Multiple relaxation approximation to power-law attenuation with slope y ¼ 1:3 and underlying order of fractional Zener or Kelvin–Voigt model of a ¼ 0:3. Solid curve: superposition of five relaxation processes. Dot-dashed curve: x1:3, the five relaxation frequencies are indicated by stars ...... 186 Fig. 7.8 Multiple relaxation approximation to power-law attenuation with slope y ¼ 0:75 and underlying order of fractional diffusion-wave model of a ¼ 0:5. The solid curve is the superposition of four relaxation processes. Dot-dashed curve is the model of (5.90) with unity parameters, i.e., / x0:75. The four relaxation frequencies which are related to each other by Xn þ 1=Xn ¼ 6, are indicated by stars ...... 186 Fig. 7.9 Attenuation when two relaxation processes are fitted in order to match f 1 from 1 to 5 MHz (Tabei et al. 2003) [MATLAB code for attenuation from R. Waag] ...... 187 Fig. 7.10 Dispersion when two relaxation processes are fitted in order to match f 1 from 1 to 5 MHz ...... 188 Fig. 7.11 Time-spectral function for the fractional Zener model for je ¼ 1 and sr ¼ 0:99se. Solid line a ¼ 0:8, dashed line a ¼ 0:01 ...... 191 Fig. 7.12 Potential well picture of the dynamics of the soft glassy rheology model. Note that the relative horizontal displacement of the quadratic potential wells is arbitrary; each has its own independent zero for the scale of the local strain l. The solid vertical bars indicate the energy dissipated in the “hops” (yield events) from 1 to 2 and 3 to 4, respectively. Local additional strain is Dc. Reprinted figure and caption with permission from Sollich (1998). Copyright 1998 by the American Physical Society ...... 193 : : Fig. 7.13 Fit of 8-term relaxation model to E~ðxÞ¼ðixÞ0 3 þðixÞ0 75. Relaxation frequencies from Table 7.1 are indicated by stars : : and lie on the curves x0 3 and x0 75 ...... 194 Fig. 7.14 One section of a lumped cable model...... 195 List of Figures xxxiii

Fig. 7.15 Three sections of an infinitely long simplified cable model leading to the input current being a half-order derivative of the input voltage at low frequencies ...... 196 Fig. 7.16 Characteristic impedance, Z0, of a typical loudspeaker cable of dimension AWG 12 (3.3 mm2) and parameters R ¼ 0.01 X/m, L ¼ 390 nH/m, G ¼ 10À12 S/m, and C ¼ 76 pF/m. The dotted line is the asymptote / f À1=2. The high-frequency value is the lossless cable characteristic impedance 71.6 X ...... 197 Fig. 7.17 Hierarchical ladder model with an infinite chain of springs and dashpots. It is self-similar when all springs and dashpots are ¼ ¼ ; ... ¼ ¼ ; ... the same, E E0 E1 and g g0 g1 ...... 198 Fig. 7.18 Rouse polymer model with beads in a viscous medium and springs connecting them ...... 199 Fig. 7.19 Comparison of how well the dynamic modulus of the Rouse model of (7.55)pffiffiffi and the exact model of (7.53) and (7.54) compare to f . The model is assumed to have N ¼ 30 elements and the sums are terminated after 6 elements. The relaxation frequencies are indicated by dots...... 200 Fig. 7.20 Plot of exact and approximate (dotted) relative elastic modulus, En=E0 in a ladder model, (7.58), for order a of 0.3 and 0.7 ...... 202 Fig. 7.21 Hierarchical self-similar tree model with an infinite chain of springs and dashpots ...... 203 Fig. 7.22 Characteristics of time-independent non-Newtonian materials where the viscosity, the slope of each line, may vary with shear rate ...... 204 Fig. 7.23 A non-Newtonian fluid: A thixotropic paint where viscosity changes with time. The longer the paint is agitated, the lower its viscosity...... 205 Fig. 7.24 Maxwell spring damper with time-varying viscosity...... 206 Fig. 7.25 Gap stiffness model. a Grain-to-grain contact of sand. b Axisymmetric section through the model. The model consists of a narrow gap connected to a finite annular pore. Here, h denotes the gap separation distance; a, the contact radius; and b, the radius of the annular pore. Reprinted with permission from Kimura (2006), copyright 2006, Acoustical Society of America ...... 215 ~ Fig. 7.26 Normalized dynamic modulus, EðxÞ=Ee, for pore, (7.104), the approximation of (7.105) as in Kimura (2008, Fig. 2) compared to the Cole–Davidson model of (7.107) ...... 217 Fig. 8.1 Maxwell spring–damper with time-varying viscosity ...... 228 Fig. 8.2 Attenuation for the shear wave of the viscous grain shearing model with unityp parameterffiffiffi values and order a as parameter. Asymptotes for f and f 1 are also shown ...... 234 xxxiv List of Figures

Fig. 8.3 Phase velocity for the shear wave of the viscous grain shearing model with unity parameter values and order a as parameter ...... 234 Fig. 8.4 Inverse Q for the shear wave of the viscous grain shearing model with unity parameter values and order a as parameter ...... 235 Fig. 8.5 Attenuation for the compressional wave of the viscous grain shearing model with unity parameter values and order a as parameter. Asymptotes for f 2 and f 1 are also shown ...... 237 Fig. 8.6 Phase velocity for the compressional wave of the viscous grain shearing model with unity parameter values and order a as parameter ...... 238 Fig. 8.7 Inverse Q for the compressional wave of the viscous grain shearing model with unity parameter values and order a as parameter ...... 238 Fig. 8.8 Sediment example of Table 8.1 and shear wave attenuation of the Biot model with turbulence. The equivalent Zener model (valid for low frequencies) and the equivalent half-order fractional Zener model (valid for high frequencies) are also shown ...... 246 Fig. 8.9 Sediment example of Table 8.1 and phase velocity for the shear wave of the Biot model with turbulence ...... 247 Fig. 8.10 Brain white matter example of Table 8.1 and shear wave attenuation of the Biot model with turbulence. The equivalent Zener model (valid for low frequencies) and the equivalent half-order fractional Zener model (valid for high frequencies) are also shown ...... 248 Fig. 8.11 Brain white matter example of Table 8.1 and phase velocity for the shear wave of the Biot model with turbulence ...... 248 Fig. 8.12 Sediment example of Table 8.1 and fast compressional wave attenuation of the Biot model with turbulence. The equivalent Zener model (valid for low frequencies) and the equivalent half-order fractional Zener model (valid for high frequencies) are also shown. [Matlab code for the Biot model from J.M. Hovem]...... 249 Fig. 8.13 Sediment example of Table 8.1 and slow compressional wave attenuation of the Biot model with turbulence. The equivalent Maxwell model (valid for low frequencies) and the equivalent half-order fractional Maxwell model (valid for high frequencies) are also shown. [Matlab code for the Biot model from J.M. Hovem] ...... 250 Fig. 8.14 The equivalent of the BICSQS model for shear waves, i.e., Biot plus shear drag, is the non-standard four-parameter model of Tschoegl (1989) (left) with its conjugate to the right. The List of Figures xxxv

0 dampers g2 and g represent the difference from the Biot model ...... 253 Fig. 8.15 BICSQS model result for the sediment example of Table 8.1 with an additional shear relaxation frequency of 100kHz interpreted as a sum of contributions from a low-frequency Biot model and a shear drag model ...... 253 Fig. 9.1 O’Doherty–Anstey model showing the direct wave, and the first and second multiples propagating from left to right through a medium with discrete layers. Below is the series of reflection coefficients. Figure inspired by Sakshaug (2011, Appendix A)...... 260 Fig. 9.2 Characterization of lag-time distribution. a From particle to lag-time distribution and b verification via simulation. Reprinted with permission from Lambert et al. (2015). Copyright (2015) by the American Physical Society ...... 270 Fig. A.1 Power-law memory kernel in the convolution function of (A.48). The curves illustrate values of a from 1 in the upper curve to 0.1 in the lower curve in increments of 0.1 (inspired by Treeby and Cox 2010)...... 283 Fig. B.1 Forces and deformations that define the elastic moduli. From left to right: pressure producing a change of volume defining the bulk modulus, K; shear forces producing an angle of shear defining the shear modulus, l; linear tension giving rise to extension, defining Young’s modulus, EY ...... 290 List of Tables

Table 2.1 Weight fractions in % of constituents of the reference composition seawater of Millero et al. (2008), adapted from that paper’s Table 4. The remaining 0.05047% needed for this to add up to 100% is due to other substances which individually contribute less than carbonate...... 54 Table 2.2 Skin depth in copper ...... 63 Table 3.1 Classification of viscoelastic models ...... 88 Table 5.1 Classification of fractional viscoelastic models...... 130 Table 7.1 Table of parameters for Fig. 7.13 for fit of multiple relaxation model ...... 195 Table 8.1 Material parameters of the poroelastic Biot model. The fluid-saturated unconsolidated sand example is taken from (Chotiros and Isakson 2004, Table III) with rigid frame bulk ¼ ð þ Þ=ð ð À ÞÞ modulus estimated as Kr 2lr 1 m 3 1 2m where m ¼ 0:15 is the Poisson ratio. The brain white matter example builds on Nagashima et al. (1987) with shear modulus taken from Basser (1992) ...... 240 Table 8.2 Elastic constants of the poroelastic Biot–Stoll model according to Stoll (1977) as functions of the material parameters of Table 8.1 ...... 241 Table A.1 Sign conventions for Fourier transform. Convention 1 is used in this book. A shorter version of this table was first published in Holm and Näsholm (2014) ...... 285

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