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Hedberg, Claes
Publications (10 of 70) Show all publications
Rudenko, O. & Hedberg, C. (2019). Strong nonlinearity, anisotropy, and solitons in a lattice with holonomic constraints. Wave motion, 89, 104-115
Open this publication in new window or tab >>Strong nonlinearity, anisotropy, and solitons in a lattice with holonomic constraints
2019 (English)In: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 89, p. 104-115Article in journal (Refereed) Published
Abstract [en]

The nonlinear dynamics of a crystal lattice where the atoms are positioned along parallel rods is studied. They may move only in one direction and this constraint leads to the appearance of nonlinearity even the forces between the atoms obey the linear Hooke's law. This nonlinearity turns out to be strong. The equations of motion of the individual lattice atoms are written, and. in the continuum limit when the lattice period is small in comparison with the wavelength, a new strongly nonlinear partial differential equation is derived. The waves traveling in the direction orthogonal to the rods are purely transverse slow waves, governed by an equation of the Heisenberg type. In the direction along the rods, a fast purely longitudinal wave can propagate. In general, when the wave travels at an arbitrary angle, it is neither purely longitudinal nor transverse and the periodic structure exhibits anisotropic properties. Their velocity depends strongly on the direction of propagation and the structure exhibits properties similar to a skeletal muscle with stretched fibers. Special attention is paid to the soliton solutions of this equation and their behavior is studied. For non-stationary quasi-longitudinal waves, a new evolution equation, rich in symmetries, is derived. One of the solutions with a fixed transverse structure is described by elliptic integrals and evolves in accordance with a cubic nonlinear equation of the Klein–Gordon type. © 2019 Elsevier B.V.

Place, publisher, year, edition, pages
Elsevier B.V., 2019
Keywords
Anisotropy, Atoms, Equations of motion, Nonlinear equations, Partial differential equations, Solitons, Anisotropic property, Elliptic integrals, Evolution equations, Holonomic constraints, Longitudinal waves, Soliton solutions, Strong nonlinearity, Strongly nonlinear, Control nonlinearities
National Category
Other Mechanical Engineering
Identifiers
urn:nbn:se:bth-17777 (URN)10.1016/j.wavemoti.2019.01.001 (DOI)
Available from: 2019-04-05 Created: 2019-04-05 Last updated: 2019-04-05Bibliographically approved
Hedberg, C. & Rudenko, O. (2018). Nonlinear acoustics of structurally complex materials described by non-analytic nonlinearities. In: 25th International Congress on Sound and Vibration 2018, ICSV 2018: Hiroshima Calling. Paper presented at 25th International Congress on Sound and Vibration 2018: Hiroshima Calling, ICSV 2018; Hiroshima; Japan; 8 July 2018 through 12 July 2018 (pp. 1993-1998). International Institute of Acoustics and Vibration, IIAV
Open this publication in new window or tab >>Nonlinear acoustics of structurally complex materials described by non-analytic nonlinearities
2018 (English)In: 25th International Congress on Sound and Vibration 2018, ICSV 2018: Hiroshima Calling, International Institute of Acoustics and Vibration, IIAV , 2018, p. 1993-1998Conference paper, Published paper (Refereed)
Abstract [en]

Everybody is accustomed to that nonlinear effects amplify with increasing amplitude or intensity of a wave. When the amplitude becomes small, the nonlinearity disappears and the wave enters a linear regime. Instead, we shall consider here so-called strong nonlinearity of the first type (according to a classification introduced earlier by the authors) where the effects of nonlinearity do not disappear even for infinitesimal amplitudes. Among these nonlinearities are modular (M) and quadratically-cubic (QC). When these nonlinearities are included in partial differential equations, they form new mathematical models describing new physical effects. Such equations have been proposed over the past few years and a review of these models is given here. They are interesting because of two reasons: (i) the equations admit exact analytic solutions, and (ii) the solutions describe real physical phenomena. Among them are M- and QC-equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko types. Media with non-analytic nonlinearities exist among composites, meta-materials, and inhomogeneous and multiphase systems. Some of the physical phenomena manifested in such media are described, e.g. stable shock fronts of compression and rarefaction in QC-media. The last cannot exist in usual media and the periodic wave consists of a series of trapezoidal teeth, rather than usual triangular. In an M-nonlinear medium collision, mutual losses and annihilation of pulses are studied. These pulses exhibit corpuscular properties and, in contrast to solitons (elastic particles) and shock waves (absolutely inelastic collisions), they behave like clots of chemical reagents (fuel and oxidizer). As result of an reaction, the smaller component disappears, and the larger decreases. At equal "masses", these clots disappear or annihilate. In M-media a new stable wave - a modular soliton - exists. Other interesting physical phenomena occur for focused waves in M-media and a review of these is also included in the presentation. Copyright © (2018) by International Institute of Acoustics & Vibration.All rights reserved.

Place, publisher, year, edition, pages
International Institute of Acoustics and Vibration, IIAV, 2018
Keywords
Annihilation, Bi-modular media, Modular nonlinearity, Nonlinear partial differential equations, Strongly nonlinear systems), Acoustics, Control nonlinearities, Nonlinear optics, Nonlinear systems, Partial differential equations, Shock waves, Solitons, Analytic nonlinearities, Exact analytic solutions, New mathematical model, Non-linear acoustics, Strongly nonlinear system, Nonlinear equations
National Category
Other Mechanical Engineering
Identifiers
urn:nbn:se:bth-17460 (URN)2-s2.0-85058697135 (Scopus ID)9781510868458 (ISBN)
Conference
25th International Congress on Sound and Vibration 2018: Hiroshima Calling, ICSV 2018; Hiroshima; Japan; 8 July 2018 through 12 July 2018
Available from: 2019-01-09 Created: 2019-01-09 Last updated: 2019-01-09Bibliographically approved
Rudenko, O. & Hedberg, C. (2018). SINGLE SHOCK AND PERIODIC SAWTOOTH-SHAPED WAVES IN MEDIA WITH NON-ANALYTIC NONLINEARITIES. Mathematical Modelling of Natural Phenomena, 13(2), Article ID UNSP 18.
Open this publication in new window or tab >>SINGLE SHOCK AND PERIODIC SAWTOOTH-SHAPED WAVES IN MEDIA WITH NON-ANALYTIC NONLINEARITIES
2018 (English)In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 13, no 2, article id UNSP 18Article in journal (Refereed) Published
Abstract [en]

The review of new mathematical models containing non-analytic nonlinearities is given. These equations have been proposed recently, over the past few years. The models describe strongly nonlinear waves of the first type, according to the classification introduced earlier by the authors. These models are interesting because of two reasons: (i) equations admit exact analytic solutions, and (ii) solutions describe the real physical phenomena. Among these models are modular and quadratically cubic equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko type. Media with non-analytic nonlinearities exist among composites, meta-materials, inhomogeneous and multiphase systems. Some physical phenomena manifested in the propagation of waves in such media are described on the qualitative level of severity.

Place, publisher, year, edition, pages
EDP SCIENCES S A, 2018
Keywords
Equations of Hopf, Burgers, KdV, KZ and Ostrovsky-Vakhnenko types, exact solutions, modular solutions, quadratically-cubic nonlinearity, shocks of rarefaction, triangular and trapezoidal saw
National Category
Other Mathematics Other Mechanical Engineering
Identifiers
urn:nbn:se:bth-17017 (URN)10.1051/mmnp/2018028 (DOI)000443894600003 ()
Available from: 2018-09-20 Created: 2018-09-20 Last updated: 2018-09-20Bibliographically approved
Rudenko, O. & Hedberg, C. (2018). Wave Resonance in Media with Modular, Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations. Acoustical Physics, 64(4), 422-431
Open this publication in new window or tab >>Wave Resonance in Media with Modular, Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations
2018 (English)In: Acoustical Physics, ISSN 1063-7710, E-ISSN 1562-6865, Vol. 64, no 4, p. 422-431Article in journal (Refereed) Published
Abstract [en]

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed. © 2018, Pleiades Publishing, Ltd.

Place, publisher, year, edition, pages
Pleiades Publishing, 2018
Keywords
excitation of nonlinear wave, inhomogeneous Burgers-type equation, modular, quadratic, quadratically cubic nonlinearity, wave resonance, Nonlinear equations, Resonance, Wave propagation, Cubic nonlinearities, Nonlinear waves, Wave resonances, Control nonlinearities
National Category
Other Mathematics
Identifiers
urn:nbn:se:bth-16906 (URN)10.1134/S1063771018040127 (DOI)000439751800005 ()2-s2.0-85050096276 (Scopus ID)
Available from: 2018-08-20 Created: 2018-08-20 Last updated: 2018-08-21Bibliographically approved
Rudenko, O. & Hedberg, C. (2017). A new equation and exact solutions describing focal fields in media with modular nonlinearity. Nonlinear dynamics, 89(3), 1905-1913
Open this publication in new window or tab >>A new equation and exact solutions describing focal fields in media with modular nonlinearity
2017 (English)In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 89, no 3, p. 1905-1913Article in journal (Refereed) Published
Abstract [en]

Brand-new equations which can be regarded as modifications of Khokhlov–Zabolotskaya–Kuznetsov or Ostrovsky–Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting because of two reasons: (1) the equations admit exact analytic solutions and (2) the solutions describe real physical phenomena. The equations model nonlinear focusing of wave beams. It is shown that inside the focal zone a stationary waveform exists. Steady-state profiles are constructed by the matching of functions describing the positive and negative branches of exact solutions of an equation of Klein–Gordon type. Such profiles have been observed many times during experiments and numerical studies. The non-stationary waves can contain singularities of two types: discontinuity of the wave and of its derivative. These singularities are eliminated by introducing dissipative terms into the equations—thereby increasing their order. © 2017 The Author(s)

Place, publisher, year, edition, pages
Springer Netherlands, 2017
Keywords
Bimodular media, Exact solution, Focusing, HIFU, High-intensity focused ultrasound, Modified KZ–OV, Modular nonlinearity, Nonlinear partial differential equation, Control nonlinearities, Partial differential equations, High intensity focused ultrasound, Nonlinear partial differential equations, Nonlinear equations
National Category
Other Materials Engineering
Identifiers
urn:nbn:se:bth-14471 (URN)10.1007/s11071-017-3560-8 (DOI)000405962800025 ()2-s2.0-85019632379 (Scopus ID)
Available from: 2017-06-13 Created: 2017-06-13 Last updated: 2017-08-22Bibliographically approved
Hedberg, C. & Rudenko, O. (2017). Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium. Nonlinear dynamics, 90(3), 2083-2091
Open this publication in new window or tab >>Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium
2017 (English)In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 90, no 3, p. 2083-2091Article in journal (Refereed) Published
Abstract [en]

One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like Burgers’ equation stick together as absolutely inelastic particles when colliding. The interactions of single pulses in media with modular nonlinearity considered here reveal new physical features that are still poorly understood. There is an analogy between the single pulses collision and the interaction of clots of chemical reactants, such as fuel and oxidant, where the smaller component disappears and the larger one decreases after a reaction. At equal “masses” both clots can be annihilated. In this work various interactions of two and three pulses are considered. The conditions for which a complete annihilation of the pulses occurs are indicated. © 2017 The Author(s)

Place, publisher, year, edition, pages
Springer Netherlands, 2017
Keywords
Annihilation, Bimodular media, Modular nonlinearity, Mutual loss, Nonlinear partial differential equation, Pulse interaction, Control nonlinearities, Partial differential equations, Shock waves, Nonlinear partial differential equations, Nonlinear equations
National Category
Other Mechanical Engineering Other Physics Topics
Identifiers
urn:nbn:se:bth-15214 (URN)10.1007/s11071-017-3785-6 (DOI)000413286700038 ()2-s2.0-85028966269 (Scopus ID)
Available from: 2017-09-29 Created: 2017-09-29 Last updated: 2017-11-02Bibliographically approved
Rudenko, O. & Hedberg, C. (2016). The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves. Nonlinear dynamics, 85(2), 767-776
Open this publication in new window or tab >>The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves
2016 (English)In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 85, no 2, p. 767-776Article in journal (Refereed) Published
Abstract [en]

A modified equation of Burgers type with a quadratically cubic (QC) nonlinear term was recently pointed out as a new exactly solvable model of mathematical physics. However, its derivation, analytical solution, computer modeling, as well as its physical applications and analysis of corresponding nonlinear wave phenomena have not been published up to now. The physical meaning and generality of this QC nonlinearity are illustrated here by several examples and experimental results. The QC equation can be linearized and it describes the experimentally observed phenomena. Some of its exact solutions are given. It is shown that in a QC medium not only shocks of compression can be stable, but shocks of rarefaction as well. The formation of stationary waves with finite width of shock front resulting from the competition between nonlinearity and dissipation is traced. Single-pulse propagation is studied by computer modeling. The nonlinear evolutions of N- and S-waves in a dissipative QC medium are described, and the transformation of a harmonic wave to a sawtooth-shaped wave with periodically recurring trapezoidal teeth is analyzed. © 2016 The Author(s)

Place, publisher, year, edition, pages
Springer Netherlands, 2016
Keywords
Acoustics; Control nonlinearities; Linearization; Mathematical transformations; Nonlinear systems; Partial differential equations; Shear waves; Shock waves, Cubic equations; Exact analytical solutions; Exact linearization; Non-linear acoustics; Nonlinear partial differential equations; Shock fronts; Strongly nonlinear system, Nonlinear equations
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:bth-11867 (URN)10.1007/s11071-016-2721-5 (DOI)000378410800006 ()2-s2.0-84962622283 (Scopus ID)
Available from: 2016-05-03 Created: 2016-05-02 Last updated: 2017-06-19Bibliographically approved
Rudenko, O. & Hedberg, C. (2015). Diffraction of high-intensity field in focal region as dynamics of nonlinear system with low-frequency dispersion. Acoustical Physics, 61(1), 28-36
Open this publication in new window or tab >>Diffraction of high-intensity field in focal region as dynamics of nonlinear system with low-frequency dispersion
2015 (English)In: Acoustical Physics, ISSN 1063-7710, E-ISSN 1562-6865, Vol. 61, no 1, p. 28-36Article in journal (Refereed) Published
Abstract [en]

The stationary profile in the focal region of a focused nonlinear acoustic wave is described. Three models following from the Khokhlov-Zabolotskaya (KZ) equation with three independent variables are used: (i) the simplified one-dimensional Ostrovsky-Vakhnenko equation, (ii) the system of equations for paraxial series expansion of the acoustic field in powers of transverse coordinates, and (iii) the KZ equation reduced to two independent variables. The structure of the last equation is analogous to the Westervelt equation. Linearization through the Legendre transformation and reduction to the well-studied Euler-Tricomi equation is shown. At high intensities the stationary profiles are periodic sequences of arc sections having singularities of derivative in their matching points. The occurrence of arc profiles was pointed out by Makov. These appear in different nonlinear systems with low-frequency dispersion. Profiles containing discontinuities (shock fronts) change their form while passing through the focal region and are non-stationary waves. The numerical estimations of maximum pressure and intensity in the focus agree with computer calculations and experimental measurements. © 2015, Pleiades Publishing, Ltd.

Keywords
Acoustic fields; Acoustic waves; Acoustics; Dispersion (waves); Focusing; Nonlinear systems; One dimensional; Shock waves, HIFU; High-intensity fields; Independent variables; Legendre transformations; Limiting fields; Low-frequency dispersions; nonlinearity; Transverse coordinate, Control nonlinearities
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:bth-726 (URN)10.1134/S1063771015010091 (DOI)000348299600004 ()2-s2.0-84921869594 (Scopus ID)
Available from: 2015-06-01 Created: 2015-05-28 Last updated: 2017-12-04Bibliographically approved
Khodabandeloo, B. & Hedberg, C. (2015). NONLINEAR DYNAMICS OF MARBLE AT DIFFERENT TEMPERATURES. In: Crocker, MJ Pawelczyk, M Pedrielli, F Carletti, E Luzzi, S (Ed.), PROCEEDINGS OF THE 22ND INTERNATIONAL CONGRESS ON SOUND AND VIBRATION: MAJOR CHALLENGES IN ACOUSTICS, NOISE AND VIBRATION RESEARCH, 2015. Paper presented at 22nd International Congress on Sound and Vibration (ICSV), Florence. INT INST ACOUSTICS & VIBRATION
Open this publication in new window or tab >>NONLINEAR DYNAMICS OF MARBLE AT DIFFERENT TEMPERATURES
2015 (English)In: PROCEEDINGS OF THE 22ND INTERNATIONAL CONGRESS ON SOUND AND VIBRATION: MAJOR CHALLENGES IN ACOUSTICS, NOISE AND VIBRATION RESEARCH, 2015 / [ed] Crocker, MJ Pawelczyk, M Pedrielli, F Carletti, E Luzzi, S, INT INST ACOUSTICS & VIBRATION , 2015Conference paper, Published paper (Refereed)
Abstract [en]

The temperature influence on the acoustically amplitude dependent sound speed of a marble rod was investigated. The sound speed was monitored through the resonance frequency by a series of ultrasonic frequency sweeps with successively increasing amplitudes. For the temperatures from 15 C to 60 C, the resonance frequency was measured in 5-degree increments as a function of the resonant acoustic amplitude inside the marble rod. In most of the curves the marble exhibits a softening (i.e. the sound speed decreases) with higher amplitude, but for each test run there exist one notable exception - for one temperature - where the marble gets stiffer (i.e the sound speed increases). The test also shows that the average sound velocity level first, as expected, decreases with temperature, but for the higher temperatures it increases - to well past the starting value.

Place, publisher, year, edition, pages
INT INST ACOUSTICS & VIBRATION, 2015
National Category
Other Mechanical Engineering
Identifiers
urn:nbn:se:bth-15762 (URN)000398997003045 ()
Conference
22nd International Congress on Sound and Vibration (ICSV), Florence
Available from: 2018-01-16 Created: 2018-01-16 Last updated: 2018-01-19Bibliographically approved
Rudenko, O. & Hedberg, C. (2015). Quadratically cubic Burgers equation as exactly solvable model of mathematical physics. Doklady. Mathematics, 91(2), 232-235
Open this publication in new window or tab >>Quadratically cubic Burgers equation as exactly solvable model of mathematical physics
2015 (English)In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 91, no 2, p. 232-235Article in journal (Refereed) Published
National Category
Mathematics Other Physics Topics
Identifiers
urn:nbn:se:bth-725 (URN)10.1134/S1064562415020337 (DOI)000354207500028 ()2-s2.0-84929074814 (Scopus ID)
Available from: 2015-06-01 Created: 2015-05-28 Last updated: 2017-12-04Bibliographically approved
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