Sverre Holm

Most cited papers

• J.-F. Synnevåg, A. Austeng, and S. Holm, "Adaptive beamforming applied to medical ultrasound imaging," IEEE Trans. Ultrason., Ferroelect., Freq. Contr. (Special issue on high resolution ultrasonic imaging), vol. 54(8), Aug. 2007, pp. 1606-1613.
• W. Chen and S. Holm, "Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency dependency," J. Acoust. Soc. Amer., pp. 1424-1430, Apr. 2004.
• J.-F. Synnevåg, A. Austeng, and S. Holm, ”Benefits of Minimum-Variance Beamforming in Medical Ultrasound Imaging”, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., Sept. 2009.
• A. Austeng and S. Holm, "Sparse 2D Arrays for 3D Phased Array Imaging - Design Methods," IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control, Aug. 2002, pp. 1073-1086.
• S. Holm, B. Elgetun, G. Dahl, "Properties of the beampattern of weight- and layout-optimized sparse arrays," IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control, vol. 44, no. 5, pp. 983-991, Sept. 1997.
• W.Chen and S. Holm, "Modified Szabo’s wave equation models for lossy media obeying frequency power law," J. Acoust. Soc. Amer., pp. 2570-2574, Nov. 2003.

• Esmonde, Harry & Holm, Sverre (2020). Fractional derivative modelling of adhesive cure. Applied Mathematical Modelling.  ISSN 0307-904X.  77(2), s 1041- 1053 . doi: 10.1016/j.apm.2019.08.021
• Esmonde, Harry & Holm, Sverre (2020). Fractional derivative modelling of adhesive cure. Applied Mathematical Modelling.  ISSN 0307-904X.  77, s 1041- 1053 . doi: https://doi.org/10.1016/j.apm.2019.08.021 Show summary

  This paper considers the modelling of curing adhesive properties using fractional derivatives. A systematic approach is adopted where results can be related to a physical interpretation of the system rather than relying on a purely data-driven approach. The method relies on selecting standard integer order models based on the pre-cure and post-cure behaviour, from which fractional order derivative models are derived. Results from dynamic mechanical testing of two chemistries, a cyanoacrylate adhesive and a methacrylate resin are used to identify the parameter values for their respective fractional models. These results are then used to interpret behaviour of the adhesives during cure such as the onset of solidification.

• Yang, Xu; Cai, Wei; Liang, Yingjie & Holm, Sverre (2020). A novel representation of time-varying viscosity with power-law and comparative study. International Journal of Non-Linear Mechanics.  ISSN 0020-7462.  119, s 1- 10 . doi: 10.1016/j.ijnonlinmec.2019.103372
• Yang, Xu; Cai, Wei; Liang, Yingjie & Holm, Sverre (2020). A novel representation of time-varying viscosity with power-law and comparative study. International Journal of Non-Linear Mechanics.  ISSN 0020-7462.  119, s 103372 . doi: https://doi.org/10.1016/j.ijnonlinmec.2019.103372
• Chandrasekaran, Sri Nivas & Holm, Sverre (2019). A multiple relaxation interpretation of the extended Biot model. Journal of the Acoustical Society of America.  ISSN 0001-4966.  146(1), s 330- 339 . doi: 10.1121/1.5116139
• Holm, Sverre (2019). Dispersion analysis for wave equations with fractional Laplacian loss operators. Fractional Calculus and Applied Analysis.  ISSN 1311-0454.  22(6), s 1596- 1606 . doi: 10.1515/fca-2019-0082
• Kvam, Johannes; Holm, Sverre & Angelsen, Bjørn Atle Johan (2019). Exploiting Ballou's rule for better tissue classification. Journal of the Acoustical Society of America.  ISSN 0001-4966.  145(4), s 2103- 2112 . doi: 10.1121/1.5096533
• Parker, Kevin; Szabo, Thomas & Holm, Sverre (2019). Towards a consensus on rheological models for elastography in soft tissues. Physics in Medicine and Biology.  ISSN 0031-9155.  64(21) . doi: 10.1088/1361-6560/ab453d Show summary

  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.

• Patz, Samuel; Fovargue, Daniel; Schregel, Katharina; Nazari, Navid; Palotai, Miklos; Barbone, Paul E.; Fabry, Ben; Hammers, Alexander; Holm, Sverre; Kozerke, Sebastian; Nordsletten, David & Sinkus, Ralph (2019). Imaging localized neuronal activity at fast time scales through biomechanics. Science Advances.  ISSN 2375-2548.  5(4) . doi: 10.1126/sciadv.aav3816 Show summary

  Mapping neuronal activity noninvasively is a key requirement for in vivo human neuroscience. Traditional functional magnetic resonance (MR) imaging, with a temporal response of seconds, cannot measure high-level cognitive processes evolving in tens of milliseconds. To advance neuroscience, imaging of fast neuronal processes is required. Here, we show in vivo imaging of fast neuronal processes at 100-ms time scales by quantifying brain biomechanics noninvasively with MR elastography. We show brain stiffness changes of ~10% in response to repetitive electric stimulation of a mouse hind paw over two orders of frequency from 0.1 to 10 Hz. We demonstrate in mice that regional patterns of stiffness modulation are synchronous with stimulus switching and evolve with frequency. For very fast stimuli (100 ms), mechanical changes are mainly located in the thalamus, the relay location for afferent cortical input. Our results demonstrate a new methodology for noninvasively tracking brain functional activity at high speed.

• Evensen, Karen Brastad; O'Rourke, Michael; Prieur, Fabrice Jean Gabriel; Holm, Sverre & Eide, Per Kristian (2018). Non-invasive Estimation of the Intracranial Pressure Waveform from the Central Arterial Blood Pressure Waveform in Idiopathic Normal Pressure Hydrocephalus Patients. Scientific Reports.  ISSN 2045-2322.  8:4714, s 1- 11 . doi: 10.1038/s41598-018-23142-7
• Evensen, Karen Brastad; Paulat, Klaus; Prieur, Fabrice Jean Gabriel; Holm, Sverre & Eide, Per Kristian (2018). Utility of the Tympanic Membrane Pressure Waveform for Non-invasive Estimation of The Intracranial Pressure Waveform. Scientific Reports.  ISSN 2045-2322.  8, s 1- 11 . doi: 10.1038/s41598-018-34083-6 Full text in Research Archive.
• Holm, Sverre (2018). Spring–damper equivalents of the fractional, poroelastic, and poroviscoelastic models for elastography. NMR in Biomedicine.  ISSN 0952-3480.  31(10), s 1- 12 . doi: 10.1002/nbm.3854 Full text in Research Archive. Show summary

  In MR elastography, it is common to use an elastic model for the tissue's response in order to interpret the results properly. 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 may all 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 of time constants or relaxation frequencies. This may open up a more physical interpretation of 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. While 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.

• Sinkus, Ralph; Lambert, Simon; Abd-Elmoniem, Khaled Z.; Morse, Caryn; Heller, Theo; Guenthner, Christian; Ghanem, Ahmed M.; Holm, Sverre & Gharib, Ahmed (2018). Rheological determinants for simultaneous staging of hepatic fibrosis and inflammation in patients with chronic liver disease. NMR in Biomedicine.  ISSN 0952-3480.  31(10) . doi: 10.1002/nbm.3956 Show summary

  The purpose of this study is to investigate the use of fundamental rheological parameters as quantified by MR elastography (MRE) to measure liver fibrosis and inflammation simultaneously in humans. MRE was performed on 45 patients at 3 T using a vibration frequency of 56 Hz. Fibrosis and inflammation scores were obtained from liver biopsies. Biomechanical properties were quantified in terms of complex shear modulus G* as well as shear wave phase velocity c and shear wave attenuation α. A rheological fractional derivative order model was used to investigate the linear dependence of the free model parameters (dispersion slope y, intrinsic speed c0, and intrinsic relaxation time τ) on histopathology. Leave‐one‐out cross‐validation was then utilized to demonstrate the effectiveness of the model. The intrinsic speed c0 increases with hepatic fibrosis, while an increased relaxation time τ is reflective of more inflammation of the liver parenchyma. The dispersion slope y does not depend either on fibrosis or on inflammation. The proposed rheological model, given this specific parameterization, establishes the functional dependences of biomechanical parameters on histological fibrosis and inflammation. The leave‐one‐out cross‐validation demonstrates that the model allows identification, from the MRE measurements, of the histology scores when grouped into low−/high‐grade fibrosis and low−/high‐grade inflammation with significance levels of P = 0.0004 (fibrosis) and P = 0.035 (inflammation). The functional dependences of intrinsic speed and relaxation time on fibrosis and inflammation, respectively, shed new light onto the impact hepatic pathological changes on liver tissue biomechanics in humans. The dispersion slope y appears to represent a structural parameter of liver parenchyma not impacted by the severity of fibrosis/inflammation present in this patient cohort. This specific parametrization of the well‐established rheological fractional order model is valuable for the clinical assessment of both fibrosis and inflammation scores, going beyond the capability of the plain shear modulus measurement commonly used for MRE.

• Wei, Cai; Chen, Wen; Fang, Jun & Holm, Sverre (2018). A Survey on Fractional Derivative Modeling of Power-Law Frequency-Dependent Viscous Dissipative and Scattering Attenuation in Acoustic Wave Propagation. Applied Mechanics Review.  ISSN 0003-6900.  70(3) . doi: 10.1115/1.4040402 Show summary

  This paper aims at presenting a survey of the fractional derivative acoustic wave equations, which have been developed in recent decades to describe the observed frequency-dependent attenuation and scattering of acoustic wave propagating through complex media. The derivation of these models and their underlying elastoviscous constitutive relationships are reviewed, and the successful applications and numerical simulations are also highlighted. The different fractional derivative acoustic wave equations characterizing viscous dissipation are analyzed and compared with each other, along with the connections and differences between these models. These model equations are mainly classified into two categories: temporal and spatial fractional derivative models. The statistical interpretation for the range of power-law indices is presented with the help of Lévy stable distribution. In addition, the fractional derivative biharmonic wave equations governing scattering attenuation are introduced and can be viewed as a generalization of viscous dissipative attenuation models.

• Bergh, Trond; Hafizovic, Ines & Holm, Sverre (2017). Acoustic imaging of sparse Sources with Orthogonal Matching Pursuit and clustering of basis vectors. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing.  ISSN 1520-6149.  s 6030- 6034 . doi: 10.1109/ICASSP.2017.7953314
• Chandrasekaran, Sri Nivas; Holm, Sverre & Prieur, Fabrice Jean Gabriel (2017). Focused ultrasound setup for the study of acoustic radiation force induced biological effects in cells. Proceedings - IEEE Ultrasonics Symposium.  ISSN 1948-5719. . doi: 10.1109/ULTSYM.2017.8091996
• Chen, Wen; Fang, Jun; Pang, Guofei & Holm, Sverre (2017). Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation. Journal of the Acoustical Society of America.  ISSN 0001-4966.  141(1), s 244- 253 . doi: 10.1121/1.4973865 Full text in Research Archive.
• Gregersen, Ida; Sandanger, Øystein; Askevold, Erik Tandberg; Sagen, Ellen Lund; Yang, Kuan; Holm, Sverre; Pedersen, Turid Margrethe; Skjelland, Mona; Krohg-Sørensen, Kirsten; Hansen, Trond Vidar; Dahl, Tuva Børresdatter; Otterdal, Kari; Espevik, Terje; Aukrust, Pål; Yndestad, Arne & Halvorsen, Bente (2017). Interleukin 27 is increased in carotid atherosclerosis and promotes NLRP3 inflammasome activation.. PLOS ONE.  ISSN 1932-6203.  12(11) . doi: 10.1371/journal.pone.0188387 Full text in Research Archive.
• Holm, Sverre & Holm, Martin Blomhoff (2017). Restrictions on wave equations for passive media. Journal of the Acoustical Society of America.  ISSN 0001-4966.  142(4), s 1888- 1896 . doi: 10.1121/1.5006059
• Prieur, Fabrice Jean Gabriel; Rindal, Ole Marius Hoel; Holm, Sverre & Austeng, Andreas (2017). Influence of the Delay-Multiply-And-Sum beamformer on the ultrasound image amplitude. Proceedings - IEEE Ultrasonics Symposium.  ISSN 1948-5719. . doi: 10.1109/ULTSYM.2017.8092637
• Rindal, Ole Marius Hoel; Aakhus, Svend; Holm, Sverre & Austeng, Andreas (2017). Hypothesis of Improved Visualization of Microstructures in the Interventricular Septum with Ultrasound and Adaptive Beamforming. Ultrasound in Medicine and Biology.  ISSN 0301-5629.  43(10), s 2494- 2499 . doi: 10.1016/j.ultrasmedbio.2017.05.023
• Bergh, Trond; Hafizovic, Ines & Holm, Sverre (2016). Multi-speaker voice activity detection using a camera-assisted microphone array. International Conference on Systems, Signals, and Image Processing.  ISSN 2157-8672.  2016-June . doi: 10.1109/IWSSIP.2016.7502768

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• Holm, Sverre (2019). Waves with Power-Law Attenuation. Springer Nature.  ISBN 978-3-030-14927-7.  312 s. Show summary

  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.

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• Holm, Sverre (2020). Abel’s 1823 Paper on Fractional Derivatives and Waves with Power-Law Attenuation. Show summary

  In 1823 Niels Henrik Abel published a paper with title "Oplösning af et par opgaver ved hjelp af bestemte integraler," (Solution of a couple of problems by means of definite integrals), Magazin for Naturviden-skaberne, Aargang I, Bind 2, Christiania, which did not appear in French translation until 1881. Here he presented a complete framework for fractional-order calculus with appropriate notation for non-integer-order integration and differentiation. This seems to have been unknown until discussed in Podlubny, Magin, Trymorush, "Niels Henrik Abel and the birth of fractional calculus," Fractional Calculus and Applied Analysis, pp. 1068-1075, 2017. I will give an introduction to non-integer order calculus and Abel's derivation of it. This will be done by going through his 1823 paper showing how he assumes a generalization of both Cauchy's formula for repeated integration and the fundamental theorem of calculus. In addition I will show how fractional calculus can be used to derive fractional-order partial differential wave equations. They have solutions with power-law attenuation which matches measurements in the complex media of medical ultrasound and elastography as well as in sediment acoustics and seismics.

• Holm, Sverre (2020). Waves with Power-Law Attenuation. Show summary

  My book with title «Waves with Power-Law Attenuation» came out in June 2019 (https://www.springer.com/gp/book/9783030149260). It 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 in frequency 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. This couples fractional derivatives and power laws and it is possible to give their multiple relaxation process interpretation. Other causes of power law attenuation and dispersion such as interaction with hierarchical models of polymer chains and non-Newtonian viscosity are also studied. Similar fractional and multiple relaxation models are as well inherent in the grain shearing and extended Biot descriptions of sediment acoustics. Finally power-law attenuation due to multiple scattering in fractal structures is covered.

• Holm, Sverre (2020). Waves with Power-Law Attenuation. Show summary

  My book with title «Waves with Power-Law Attenuation» came out in June 2019 (https://www.springer.com/gp/book/9783030149260). It 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 in frequency 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. This couples fractional derivatives and power laws and it is possible to give their multiple relaxation process interpretation. Other causes of power law attenuation and dispersion such as interaction with hierarchical models of polymer chains and non-Newtonian viscosity are also studied. Similar fractional and multiple relaxation models are as well inherent in the grain shearing and extended Biot descriptions of sediment acoustics. Finally power-law attenuation due to multiple scattering in fractal structures is covered.

• Aparicio, Joaquin; Alvarez, Fernando; Hernandez, Alvaro & Holm, Sverre (2019). A review of techniques for ultrasonic indoor localization systems.
• Holm, Sverre (2019). Er naturvitenskap Teorien om Alt?.
• Holm, Sverre (2019). Evolusjon og klassisk kristen tro.
• Holm, Sverre (2019). Evolusjon og klassisk kristen tro. Teistisk evolusjon ‐ Evolusjonær kreasjonisme.
• Holm, Sverre (2019). Troende naturforskere gjennom 400 år ved seks av dem.
• Holm, Sverre (2019). Troende naturforskere gjennom 400 år ved seks av dem.
• Holm, Sverre (2019). Troende naturforskere gjennom 400 år ved åtte av dem.
• Holm, Sverre (2019). Troende naturforskere gjennom 400 år ved åtte av dem.
• Chandrasekaran, Sri Nivas; Evensen, Karen Brastad; Ramsdal, Elisabeth Grønn & Holm, Sverre (2018). Integrating the Biot-Stoll model with contact squirt flow and shear drag (BICSQS) in the Biot viscosity extended framework. Journal of the Acoustical Society of America.  ISSN 0001-4966. . doi: https://asa.scitation.org/doi/10.1121/1.5036507
• Chandrasekaran, Sri Nivas & Holm, Sverre (2018). Integrating the Biot-Stoll model with contact squirt flow and shear drag (BICSQS) in the Biot viscosity extended framework.
• Holm, Sverre (2018). Kristen tro og naturvitenskap: Troende naturforskere og fire modeller.
• Holm, Sverre (2018). Kristne vitenskapsfolk gjennom historien.
• Holm, Sverre (2018). Stephen Hawking som filosof. Avisa Vårt land. Show summary

  La oss anerkjenne Hawking for hans enorme bidrag til teoretisk fysikk, og så kan vi la hans filosofiske utsagn langsomt gå i glemmeboken

• Holm, Sverre (2018). Vitenskap og skapertro: 8 forskere og 4 perspektiver.
• Chandrasekaran, Sri Nivas; Holm, Sverre & Prieur, Fabrice Jean Gabriel (2017). Focused ultrasound setup for the study of acoustic radiation force induced biological effects in cells. Show summary

  Low intensity pulsed ultrasound (LIPUS) is a therapy used clinically for bone fracture and soft tissue healing LIPUS induces functional alterations in cell behavior and has shown potential for cancer therapy. In order to understand the underlying biophysical mechanisms, the ability to control acoustic exposure parameters and elimination of unwanted effects such as the formation of standing wave patterns, heating, and cavitation, is vital. In this study, we designed an in-vitro focused ultrasound setup (FUS) with repeatable exposure conditions where the cells or spheroids are surrounded by an agar substrate with sufficient absorption to eliminate unwanted reflections and cell culture medium which acts as an effective coolant. The measured maximum temperature rise with the presence of coolant was 1°C and the estimated mechanical index was 1.28. The shear wave propagation induced by acoustic radiation force was monitored and no reflection was observed. We propose an in-vitro FUS cell stimulation setup which can induce non-thermal and non-cavitation biological response with no unwanted artifacts.

• Holm, Sverre (2017). Det går bra med månen. Dagen : kristelig dagblad.  ISSN 0805-4738. Show summary

  Det går bra med månen Sverre Holm, Universitetet i Oslo I spørsmålet om jordas alder og månens forflytning vekk fra jorda tar Saxe fram enda et argument fra Sarfati (Creation Ministries International, 2014) 18 august. På den ene side godtar Sarfati at tidevannsavleiringer for 900 millionar år siden viser at månen flytter seg langsomt. På den annne side mener han at en annen modell må til for å forklare at den må ha flyttet seg mye raskere før dette. Han er nok ganske alene om ikke å synes at det blir selvmotsigende. Men denne diskusjonen er antageligvis nå for detaljert til å ha allmenn interesse så jeg har skrevet mer om det på bloggen min, https://titan.uio.no/node/2448.

• Holm, Sverre (2017). Lemfeldig omgang med naturvitenskap. Dagen : kristelig dagblad.  ISSN 0805-4738.
• Holm, Sverre (2017). New insights from applying linear viscoelasticity to acoustics.
• Holm, Sverre (2017). Sannhet er det viktigste vi kan søke. Dagen : kristelig dagblad.  ISSN 0805-4738.
• Holm, Sverre (2017). Thermodynamically viable fractional wave equations for power law attenuation in viscoelastic media. Show summary

  Many complex media of great practical interest, such as in medical ultrasound and sediment acoustics, display an attenuation that increases with a power law as a function of frequency. Usually measurements can only be taken over a limited frequency range, while fractional wave equations model attenuation over all frequencies. There is therefore some freedom in how the models behave outside of this limited interval, and many different fractional wave equations have been proposed. In addition, it is desirable that a wave equation models physically viable media and for that two conditions have to be satisfied. The first is causality, and the second is a criterion that comes from thermodynamic considerations and implies that the relaxation modulus is a completely monotonic function. The latter implies that attenuation asymptotically cannot rise faster than frequency raised to the first power. These criteria will be explained and used to evaluate several of the fractional wave equations that exist.

• Holm, Sverre (2017). Thermodynamically viable wave equations for power law attenuation in viscoelastic media. Show summary

  Many complex media of great practical interest, such as in medical ultrasound, display an attenuation that increases with a power law as a function of frequency. Usually measurements can only be taken over a limited frequency range, while wave equations often model attenuation over all frequencies. There is therefore some freedom in how the models behave outside of this limited interval, and many different wave equations have been proposed, in particular fractional ones. In addition, it is desirable that a wave equation models physically viable media and for that two conditions have to be satisfied. The first is causality, and the second is a criterion that comes from thermodynamic considerations and implies that the relaxation modulus is a completely monotonic function. The latter implies that attenuation asymptotically cannot rise faster than frequency raised to the first power. These criteria will be explained and used to evaluate several of the ordinary and fractional wave equations that exist.

• Prieur, Fabrice Jean Gabriel; Rindal, Ole Marius Hoel; Holm, Sverre & Austeng, Andreas (2017). Influence of the Delay-Multiply-And-Sum beamformer on the ultrasound image amplitude.
• Holm, Sverre (2016). – Kristne bør ikke bli forskere, og særlig ikke innen biologi. Dagen : kristelig dagblad.  ISSN 0805-4738.  s 3- 3
• Holm, Sverre (2016). Bedre alternativ enn Intelligent Design. Dagen : kristelig dagblad.  ISSN 0805-4738.  s 19- 19

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