Waves with Power-Law Attenuation

1. Auflage 2019
ISBN: 978-3-030-14926-0
Verlag: Springer International Publishing

Buch, Englisch, 312 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 699 g

This book integrates concepts from physical acoustics with those from linear viscoelasticity and fractional linear viscoelasticity. Compressional waves and shear waves in applications such as medical ultrasound, elastography, and sediment acoustics often follow power law attenuation and dispersion laws that cannot be described with classical viscous and relaxation models. This is accompanied by temporal power laws rather than the temporal exponential responses of classical models.The book starts by reformulating the classical models of acoustics in terms of standard models from linear elasticity. Then, non-classical loss models that follow power laws and which are expressed via convolution models and fractional derivatives are covered in depth. In addition, parallels are drawn to electromagnetic waves in complex dielectric media. The book also contains historical vignettes and important side notes about the validity of central questions. While addressed primarily to physicists and engineers working in the field of acoustics, this expert monograph will also be of interest to mathematicians, mathematical physicists, and geophysicists.

Zielgruppe

Research

Autoren/Hrsg.

Holm, Sverre

Fachgebiete

• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Naturwissenschaften Physik Mechanik Akustik, Schwingungsanalyse
• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Elastizität, Plastizität, Rheologie
• Geowissenschaften Geologie Geophysik
• Naturwissenschaften Physik Angewandte Physik Geophysik
• Technische Wissenschaften Sonstige Technologien | Angewandte Technik Akustik, Tontechnik

Inhaltsverzeichnis

1 Introduction1.1 Conservation laws vs constitutive equations1.2 Conservation principles1.3 Hookean and Newtonian medium models1.4 Constitutive equations1.4.1 Spring damper models1.4.2 Exponential time responses1.4.3 Power laws in frequency and time1.5 Wave equations with power law solutions1.5.1 Fractional wave equations1.5.2 Fractal media and power law attenuation1.5.3 Porous media1.6 Layout 2 Classical wave equations2.1 The lossless wave equation2.1.1 Monochromatic plane wave 2.1.2 The wave equation in spherical coordinates2.2 Lossless wave equations in practice2.2.1 Acoustics2.2.2 Elastic waves2.2.3 Electromagnetics2.3 Characterization of attenuation2.3.1 Dispersion relation2.3.2 Q, loss tangent, log decrement, and penetration depth2.4 Viscous losses: The Kelvin-Voigt model2.4.1 Viscous wave equation and the dispersion equation2.4.2 Low frequency wave equation2.5 The Zener constitutive equation2.5.1 Wave equation2.5.2 Dispersion relation and compressibility/compliance2.5.3 Asymptotes2.6 Relaxation and multiple relaxation2.6.1 The relaxation model2.6.2 Multiple relaxation2.6.3 Multiple relaxation: Seawater and air2.6.4 Higher order constitutive equations2.6.5 Arbitrary attenuation from multiple relaxation2.7 The Maxwell mechanical model2.8 Losses in electromagnetics2.8.1 A conducting medium2.8.2 Debye dielectrics2.8.3 Multiple Debye terms 3 Models of Linear Viscoelasticity3.1 Constitutive equations3.1.1 Relaxation modulus and creep compliance3.1.2 Linear differential equation model3.1.3 The causal fading memory model3.1.4 Complete monotonicity 3.1.5 Relationship between descriptions3.1.6 Spring damper model3.2 Standard spring damper models3.2.1 Spring and dashpot elements3.2.2 Kelvin-Voigt model3.2.3 Maxwell model3.2.4 The standard linear solid3.2.5 Higher order models3.3 Four categories of models3.4 Completely monotone models3.4.1 Global vs. local passivity3.4.2 Special role of completely monotone models3.5 Fractional models3.5.1 Fractional Kelvin-Voigt model3.5.2 Fractional Zener model3.5.3 Fractional Maxwell model3.5.4 Fractional Newton (Scott-Blair) model  4 Wave equations with power law solutions4.1 Generalization of the low-frequency wave equation4.2 Causality4.2.1 Impulse response and transfer function4.2.2 Kramers-Kronig relations4.3 Generalization of the viscous wave equation4.3.1 Fractional temporal derivative4.3.2 Fractional Laplacian loss term4.3.3 Fractional biharmonic operator4.4 Fractional diffusion-wave equation4.5 Four term fractional wave equations4.5.1 Fractional Zener wave equation4.5.2 Constant power law for all frequencies4.6 Power law solutions 5 Physically valid viscoelastic wave equations 5.1 Wave equations for completely monotone media5.1.1 Wavenumber as a function of relaxation modulus5.1.2 Bernstein property5.1.3 Consequences of the Bernstein property5.1.4 Asymptotic properties5.2 Viability of two viscous wave equations5.3 Does the viscous model represent realistic media?5.3.1 The Navier-Stokes equation 6 Wave equations from fractional constitutive equations6.1 The fractional Kelvin-Voigt equation6.1.1 Dispersion relation6.1.2 Asymptotes of attenuation and phase velocity6.2 The fractional diffusion-wave equation6.3 The fractional Zener wave equation6.3.1 Dispersion relation and compressibility6.3.2 Asymptotes of attenuation and phase velocity6.3.3 Fractional relaxation model6.4 The fractionalMaxwell wave equation6.5 Hybrid viscous and fractionalmodels6.6 Fractional conservation of mass and momentum6.6.1 Fractional mass conservation6.6.2 Fractional momentum conservation6.7 The Cole-Cole model of electromagnetics6.7.1 Circuit equivalent of the Cole-Cole model6.7.2 Cole impedance model 7 Justification for fractional constitutive equations and power laws 8 Fractal media 9 Poroelastic and poroviscoelastic media Appendices Appendix A List of symbols Appendix B Acoustic, elastic, and electromagnetic wave equationsB.1 Derivation of the acoustic wave equationB.1.1 The Navier-Stokes equation and viscosityB.1.2 Typical mediaB.2 Derivation of the elastic wave equationsB.2.1 ViscoelasticityB.2.2 Special case for fluids and tissueB.2.3 Typical mediaB.3 The electromagnetic wave equation Appendix C Mathematical backgroundC.1 ApproximationsC.1.1 Power series approximationC.1.2 McLaurin series for trigonometric functionsC.2 Mathematical operatorsC.3 Fourier transformC.3.1 Differentiation propertyC.3.2 Convolution and differentiationC.3.3 Fourier transformof an exponential decayC.3.4 Fourier transformof a power lawC.3.5 Fourier transformof theMittag–Leffler functionC.3.6 Sign convention in Fourier transformC.4 Fractional calculusC.4.1 Power law function interpretationC.4.2 Fourier interpretationC.4.3 Convolution interpretationC.4.4 Convolution interpretation: Two flavorsC.4.5 Fractional integralC.4.6 The first physical problem: Abel’s integral equation C.4.7 The fractional LaplacianC.4.8 Bernstein functions  Index

Über Autor(innen)

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.