Waves with Power-Law Attenuation (eBook, PDF) - Holm, Sverre
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1 Introduction

1.1 Conservation laws vs constitutive equations

1.2 Conservation principles

1.3 Hookean and Newtonian medium models

1.4 Constitutive equations

1.4.1 Spring damper models

1.4.2 Exponential time responses

1.4.3 Power laws in frequency and time

1.5 Wave equations with power law solutions

1.5.1 Fractional wave equations

1.5.2 Fractal media and power law attenuation

1.5.3 Porous media

1.6 Layout

2 Classical wave equations

2.1 The lossless wave equation

2.1.1 Monochromatic plane wave

2.1.2 The wave equation in spherical coordinates

2.2 Lossless wave equations in practice

2.2.1 Acoustics

2.2.2 Elastic waves

2.2.3 Electromagnetics

2.3 Characterization of attenuation

2.3.1 Dispersion relation

2.3.2 Q, loss tangent, log decrement, and penetration depth

2.4 Viscous losses: The Kelvin-Voigt model

2.4.1 Viscous wave equation and the dispersion equation

2.4.2 Low frequency wave equation

2.5 The Zener constitutive equation

2.5.1 Wave equation

2.5.2 Dispersion relation and compressibility/compliance

2.5.3 Asymptotes

2.6 Relaxation and multiple relaxation

2.6.1 The relaxation model

2.6.2 Multiple relaxation

2.6.3 Multiple relaxation: Seawater and air

2.6.4 Higher order constitutive equations

2.6.5 Arbitrary attenuation from multiple relaxation

2.7 The Maxwell mechanical model

2.8 Losses in electromagnetics

2.8.1 A conducting medium

2.8.2 Debye dielectrics

2.8.3 Multiple Debye terms

3 Models of Linear Viscoelasticity

3.1 Constitutive equations

3.1.1 Relaxation modulus and creep compliance

3.1.2 Linear differential equation model

3.1.3 The causal fading memory model

3.1.4 Complete monotonicity

3.1.5 Relationship between descriptions

3.1.6 Spring damper model

3.2 Standard spring damper models

3.2.1 Spring and dashpot elements

3.2.2 Kelvin-Voigt model

3.2.3 Maxwell model

3.2.4 The standard linear solid

3.2.5 Higher order models

3.3 Four categories of models

3.4 Completely monotone models

3.4.1 Global vs. local passivity

3.4.2 Special role of completely monotone models

3.5 Fractional models

3.5.1 Fractional Kelvin-Voigt model

3.5.2 Fractional Zener model

3.5.3 Fractional Maxwell model

3.5.4 Fractional Newton (Scott-Blair) model

4 Wave equations with power law solutions

4.1 Generalization of the low-frequency wave equation

4.2 Causality

4.2.1 Impulse response and transfer function

4.2.2 Kramers-Kronig relations

4.3 Generalization of the viscous wave equation

4.3.1 Fractional temporal derivative

4.3.2 Fractional Laplacian loss term

4.3.3 Fractional biharmonic operator

4.4 Fractional diffusion-wave equation

4.5 Four term fractional wave equations

4.5.1 Fractional Zener wave equation

4.5.2 Constant power law for all frequencies

4.6 Power law solutions

5 Physically valid viscoelastic wave equations

5.1 Wave equations for completely monotone media

5.1.1 Wavenumber as a function of relaxation modulus

5.1.2 Bernstein property

5.1.3 Consequences of the Bernstein property

5.1.4 Asymptotic properties

5.2 Viability of two viscous wave equations

5.3 Does the viscous model represent realistic media?

5.3.1 The Navier-Stokes equation

6 Wave equations from fractional constitutive equations

6.1 The fractional Kelvin-Voigt equation

6.1.1 Dispersion relation

6.1.2 Asymptotes of attenuation and phase velocity

6.2 The fractional diffusion-wave equation

6.3 The fractional Zener wave equation

6.3.1 Dispersion relation and compressibility

6.3.2 Asymptotes of attenuation and phase velocity

6.3.3 Fractional relaxation model

6.4 The fractionalMaxwell wave equation

6.5 Hybrid viscous and fractionalmodels

6.6 Fractional conservation of mass and momentum

6.6.1 Fractional mass conservation

6.6.2 Fractional momentum conservation

6.7 The Cole-Cole model of electromagnetics

6.7.1 Circuit equivalent of the Cole-Cole model

6.7.2 Cole impedance model

7 Justification for fractional constitutive equations and power laws

8 Fractal media

9 Poroelastic and poroviscoelastic media


Appendix A List of symbols

Appendix B Acoustic, elastic, and electromagnetic wave equations

B.1 Derivation of the acoustic wave equation

B.1.1 The Navier-Stokes equation and viscosity

B.1.2 Typical media

B.2 Derivation of the elastic wave equations

B.2.1 Viscoelasticity

B.2.2 Special case for fluids and tissue

B.2.3 Typical media

B.3 The electromagnetic wave equation

Appendix C Mathematical background

C.1 Approximations

C.1.1 Power series approximation

C.1.2 McLaurin series for trigonometric functions

C.2 Mathematical operators

C.3 Fourier transform

C.3.1 Differentiation property

C.3.2 Convolution and differentiation

C.3.3 Fourier transformof an exponential decay

C.3.4 Fourier transformof a power law

C.3.5 Fourier transformof theMittag-Leffler function

C.3.6 Sign convention in Fourier transform

C.4 Fractional calculus

C.4.1 Power law function interpretation

C.4.2 Fourier interpretation

C.4.3 Convolution interpretation

C.4.4 Convolution interpretation: Two flavors

C.4.5 Fractional integral

C.4.6 The first physical problem: Abel's integral equation

C.4.7 The fractional Laplacian

C.4.8 Bernstein functions

"The book is interesting if you want to enter the field of fractional rheology. This is an actual topic of research that may be quite promising and this book is a helpful tool to take a first step in the subject. In my opinion it is well written and it is highly readable with technical details under control. I have highly appreciated the idea of integrating several ideas from physical acoustics with those from linear viscoelasticity." (Giuseppe Saccomandi, Mathematical Reviews, November, 2019)