arXiv subjects

Sverre Holm

Publications and source records attributed to Sverre Holm.

17 recordsLinked to original sources

arXiv2025

Energy landscape interpretation of universal linearly increasing absorption with frequency

Absorption of elastic waves in complex media is commonly found to increase linearly with frequency, for both longitudinal and shear waves. This ubiquitous property is observed in media such as rocks, unconsolidated sediments, and human tissue. Absorption is due to relaxation processes at the level of atomic scales and up to the sub-micron scale of biological materials. The effect of these processes is usually expressed as an integral over relaxation frequencies or relaxation times. Here we argue that these processes are thermally activated. Unusual for ultrasonics and seismics, we can therefore transform the expression for absorption from the frequency or time domains to an integral over an activation energy landscape weighted by an energy distribution. The universal power-law property surprisingly corresponds to a flat activation energy landscape. This is the solution which maximizes entropy or randomness. Therefore the linearly increasing absorption corresponds to the energy landscape with the fewest possible constraints.

cond-mat.soft↗

arXiv2023

Difference between charge-voltage relations of ordinary and fractional capacitors

In an ordinary time-varying capacitor, there is debate whether a time-domain multiplication or a time-domain convolution of capacitance and voltage determines charge. A time-varying capacitor in parallel with a resistor was implemented by a motor-driven potentiometer and op-amps. The response matched a power-law function over about two decades of time, and not an exponential, for several sets of parameters. This confirms the time-domain multiplication model. This result is the opposite of that obtained for a Constant Phase Element (CPE) in its common time- and frequency-varying capacitor interpretation. This demonstrates that a CPE is fundamentally different from an ordinary time- and frequency-varying capacitor.

physics.class-ph↗

arXiv2023

Natural Occurrence of Fractional Derivatives in Physics

Power laws in time and frequency appear in fields such as linear viscoelasticity and acoustics, viscous boundary layer problems, and dielectrics. This is consistent with fractional derivatives in the fundamental descriptions, since power laws in time and frequency are related by the Fourier transform, and also associated with fractional derivatives. Examples here include power-law frequency dependent attenuation in ultrasound, elastography and sediment acoustics. In viscous boundary problems there is a viscodynamic operator in the Biot poroviscoelastic theory which may be formulated with a fractional derivative. Power law and stretched exponential temporal responses of non-ideal capacitors can also be shown to relate to the Cole-Cole power-law dielectric model.

physics.gen-ph↗

arXiv2020

Simple Circuit Equivalents for the Constant Phase Element

The constant phase element (CPE) with a frequency-independent negative phase between current and voltage is used extensively in e.g. the bioimpedance and electrochemistry fields. Its physical meaning is only partially understood. Here we show that the responses of both the common capacitive CPE as well as the inductive CPE are exactly the same as those of simple RL and RC circuits where the inductor's or capacitor's value increases linearly with time. The resulting step and impulse responses are found and verified by simulation with the Micro-Cap simulation program. The realization with time-varying components correlates with known time-varying properties in applications.

physics.app-ph↗

arXiv2019

Towards a consensus on rheological models for shear waves in soft tissues

A rising wave of technologies and instruments are enabling more labs and clinics to make a variety of measurements related to tissue viscoelastic properties. These instruments include elastography imaging scanners, rheological shear viscometers, and a variety of calibrated stress-strain analyzers. From these many sources of disparate data, a common step in analyzing results is to fit the measurements of tissue response to some viscoelastic model. In the best scenario, this places the measurements within a theoretical framework and enables meaningful comparisons of the parameters against other types of tissues. However, there is a large set of established rheological models, even within the class of linear, causal, viscoelastic solid models, so which of these should be chosen? Is it simply a matter of best fit to a minimum mean squared error of the model to several data points? We argue that the long history of biomechanics, including the concept of the extended relaxation spectrum, along with data collected from viscoelastic soft tissues over an extended range of times and frequencies, and the theoretical framework of multiple relaxation models which model the multi-scale nature of physical tissues, all lead to the conclusion that fractional derivative models represent the most succinct and meaningful models of soft tissue viscoelastic behavior. These arguments are presented with the goal of clarifying some distinctions between, and consequences of, some of the most commonly used models, and with the longer term goal of reaching a consensus among different sub-fields in acoustics, biomechanics, and elastography that have common interests in comparing tissue measurements.

q-bio.TO↗

arXiv2017

Restrictions on wave equations for passive media

Most derivations of acoustic wave equations involve ensuring that causality is satisfied. Here we explore the consequences of also requiring that the medium should be passive. This is a stricter criterion than causality for a linear system and implies that there are restrictions on the relaxation modulus and its first few derivatives. The viscous and relaxation models of acoustics satisfy passivity and have restrictions on not only a few, but all derivatives of the relaxation modulus. This is the important class of completely monotone systems. It is the only class where the medium is modeled as a system of springs and dampers with positive parameters. It is shown here that the attenuation as a function of frequency for such media has to increase slower than a linear function. Likewise the phase velocity has to increase monotonically. This gives criteria on which one may judge whether a proposed wave equation is passive or not, as illustrated by comparing two different versions of the viscous wave equation.

physics.class-ph↗

arXiv2017

Spring-damper equivalents of the fractional, poroelastic, and poroviscoelastic models for elastography

In MR elastography it is common to use an elastic model for the tissue's response in order to properly interpret the results. More complex models such as viscoelastic, fractional viscoelastic, poroelastic, or poroviscoelastic ones are also used. These models appear at first sight to be very different, but here it is shown that they all may be expressed in terms of elementary viscoelastic models. For a medium expressed with fractional models, many elementary spring-damper combinations are added, each of them weighted according to a long-tailed distribution, hinting at a fractional distribution of time constants or relaxation frequencies. This may open up for a more physical interpretation of the fractional models. The shear wave component of the poroelastic model is shown to be modeled exactly by a three-component Zener model. The extended poroviscoelastic model is found to be equivalent to what is called a non-standard four-parameter model. Accordingly, the large number of parameters in the porous models can be reduced to the same number as in their viscoelastic equivalents. As long as the individual displacements from the solid and fluid parts cannot be measured individually the main use of the poro(visco)elastic models is therefore as a physics based method for determining parameters in a viscoelastic model.

physics.bio-ph↗

arXiv2016

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

The characteristic time-dependent viscosity of the intergranular pore-fluid in Buckingham's grain-shearing (GS) model [Buckingham, J. Acoust. Soc. Am. 108, 2796-2815 (2000)] is identified as the property of rheopecty. The property corresponds to a rare type of a non-Newtonian fluid in rheology which has largely remained unexplored. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and the shear wave equation derived from the GS model are shown to take the form of the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation respectively. Therefore, an analogy is drawn between the dispersion relations obtained from the fractional framework and those from the GS model to establish the equivalence of the respective wave equations. Further, a physical interpretation of the characteristic fractional order present in the wave equations is inferred from the GS model. The overall goal is to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials. Rather, it can also be derived from real physical processes as illustrated in this work by the example of grain-shearing.

cond-mat.soft↗

arXiv2016

Joint Beamforming and Feature Detection for Enhanced Visualization of Spinal Bone Surfaces in Ultrasound Images

We propose a framework for extracting the bone surface from B-mode images employing the eigenspace minimum variance (ESMV) beamformer and a ridge detection method. We show that an ESMV beamformer with a rank-1 signal subspace can preserve the bone anatomy and enhance the edges, despite an image which is less visually appealing due to some speckle pattern distortion. The beamformed images are post-processed using the phase symmetry (PS) technique. We validate this framework by registering the ultrasound images of a vertebra (in a water bath) against the corresponding Computed Tomography (CT) dataset. The results show a bone localization error in the same order of magnitude as the standard delay-and-sum (DAS) technique, but with approximately 20% smaller standard deviation (STD) of the image intensity distribution around the bone surface. This indicates a sharper bone surface detection. Further, the noise level inside the bone shadow is reduced by 60%. In in-vivo experiments, this framework is used for imaging the spinal anatomy. We show that PS images obtained from this beamformer setup have sharper bone boundaries in comparison with the standard DAS ones, and they are reasonably well separated from the surrounding soft tissue.

physics.med-ph↗

arXiv2016

Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity

We apply the framework of tempered fractional calculus to investigate the spatial dispersion of elastic waves in a one-dimensional elastic bar characterized by range-dependent nonlocal interactions. The measure of the interaction is given by the attenuation kernel present in the constitutive stress-strain relation of the bar, which follows from the Kröner-Eringen's model of nonlocal elasticity. We employ a fractional power-law attenuation kernel and spatially temper it, to make the model physically valid and mathematically consistent. The spatial dispersion relation is derived, but it turns out to be difficult to solve, both analytically and numerically. Consequently, we use numerical techniques to extract the real and imaginary parts of the complex wavenumber for a wide range of frequency values. From the dispersion plots, it is found that the phase velocity dispersion of elastic waves in the tempered nonlocal elastic bar is similar to that from the time-fractional Zener model. Further, we also examine the unusual attenuation pattern obtained for the elastic wave propagation in the bar.

cond-mat.mtrl-sci↗

arXiv2015

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional calculus

An analogy is drawn between the diffusion-wave equations derived from the fractional Kelvin-Voigt model and those obtained from Buckingham's grain-shearing (GS) model [J. Acoust. Soc. Am. 108, 2796-2815 (2000)] of wave propagation in saturated, unconsolidated granular materials. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and shear wave equation derived from the GS model turn out to be the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation respectively. Also, a physical interpretation of the characteristic fractional-order present in the Kelvin-Voigt fractional derivative wave equation and time-fractional diffusion-wave equation is inferred from the GS model. The shear wave equation from the GS model predicts both diffusion and wave propagation in the fractional framework. The overall goal is intended to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials, but rather it can be justified from real physical process of grain-shearing as well.

physics.geo-ph↗

arXiv2015

Four ways to justify temporal memory operators in the lossy wave equation

Attenuation of ultrasound often follows near power laws which cannot be modeled with conventional viscous or relaxation wave equations. The same is often the case for shear wave propagation in tissue also. More general temporal memory operators in the wave equation can describe such behavior. They can be justified in four ways: 1) Power laws for attenuation with exponents other than two correspond to the use of convolution operators with a temporal memory kernel which is a power law in time. 2) The corresponding constitutive equation is also a convolution, often with a temporal power law function. 3) It is also equivalent to an infinite set of relaxation processes which can be formulated via the complex compressibility. 4) The constitutive equation can also be expressed as an infinite sum of higher order derivatives. An extension to longitudinal waves in a nonlinear medium is also provided.

physics.class-ph↗

arXiv2015

Prudence in estimating coherence between planetary, solar and climate oscillations

There are claims that there is correlation between the speed of center of mass of the solar system and the global temperature anomaly. This is partly grounded in data analysis and partly in a priori expectations. The magnitude squared coherence function is the proper measure for testing such claims. It is not hard to produce high coherence estimates at periods around 15--22 and 50--60 years between these data sets. This is done in two independent ways, by wavelets and by a periodogram method. But does a coherence of high value mean that there is coherence of high significance? In order to investigate that, four different measures for significance are studied. Due to the periodic nature of the data, only Monte Carlo simulation based on a non-parametric random phase method is appropriate. None of the high values of coherence then turn out to be significant. Coupled with a lack of a physical mechanism that can connect these phenomena, the planetary hypothesis is therefore dismissed.

astro-ph.EP↗

arXiv2014

On the alleged coherence between the global temperature and the sun's movement

It has recently been claimed that there is significant coherence between the spectral peaks of the global temperature series over the last 160 years and those of the speed of the solar center of mass at periods of 10-10.5, 20-21, 30 and 60-62 years. Here it is shown that these claims are based on a comparison between spectral peaks in spectral estimates that assume that the global temperature data contains time-invariant spectral lines. However, time-frequency analysis using both windowed periodograms and the maximum entropy method shows that this is not the case. An estimate of the magnitude squared coherence shows instead that under certain conditions only coherence at a period of 15-17 years can be found in the data. As this result builds on a low number of independent averages and also is unwarranted from any physical model it is doubtful whether it is significant.

astro-ph.EP↗

arXiv2014

Comparison of fractional wave equations for power law attenuation in ultrasound and elastography

A set of wave equations with fractional loss operators in time and space are analyzed. It is shown that the fractional Szabo equation, the power law wave equation, and the fractional Laplacian wave equation in the causal and non-causal forms all are low frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations while the former wave equations are ad hoc, heuristic equations. We show that this has consequences for use in modelling and simulation especially for applications that do not satisfy the low frequency approximation, such as shear wave elastography. In such applications the wave equations based on constitutive equations are the preferred ones.

physics.class-ph↗

arXiv2012

On a Fractional Zener Elastic Wave Equation

This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.

math-ph↗

arXiv2012

A Fractional Acoustic Wave Equation from Multiple Relaxation Loss and Conservation Laws

This work concerns causal acoustical wave equations which imply frequency power-law attenuation. A connection between the five-parameter fractional Zener wave equation, which is derived from a fractional stress-strain relation plus conservations of mass and momentum, and the physically well established multiple relaxation framework is developed. It is shown that for a certain continuous distribution of relaxation mechanisms, the two descriptions are equal.

math-ph↗