Waves with Power-Law Attenuation, 1st ed. 2019

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.

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

2.2 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

3.1.6 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

4.1 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

5.1.1 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

Appendices

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.4.8 Bernstein functions

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.

Couples fractional derivatives and power laws and gives their multiple relaxation process interpretation

Investigates causes of power law attenuation and dispersion such as interaction with hierarchical models of polymer chains and non-Newtonian viscosity

Shows how fractional and multiple relaxation models are inherent in the grain shearing and extended Biot descriptions of sediment acoustics

Contains historical vignettes and side notes about the formulation of some of the concepts discussed