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  • 1.
    Rudenko, Oleg
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.
    Gurbatov, Sergey N.
    Lobachevsky State University of Nizhni Novgorod, RUS.
    Tyurina, A. V.
    Lobachevsky State University of Nizhni Novgorod, RUS.
    Evolution of weak noise and regular waves on dissipative shock fronts described by the Burgers model2018In: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 82, p. 20-29Article in journal (Refereed)
    Abstract [en]

    The interaction of weak noise and regular signals with a shock wave having a finite width is studied in the framework of the Burgers equation model. The temporal realization of the random process located behind the front approaches it at supersonic speed. In the process of moving to the front, the intensity of noise decreases and the correlation time increases. In the central region of the shock front, noise reveals non-trivial behaviour. For large acoustic Reynolds numbers the average intensity can increase and reach a maximum value at a definite distance. The behaviour of statistical characteristics is studied using linearized Burgers equation with variable coefficients reducible to an autonomous equation. This model allows one to take into account not only the finite width of the front, but the attenuation and diverse character of initial profiles and spectra as well. Analytical solutions of this equation are derived. Interaction of regular signals of complex shape with the front is studied by numerical methods. Some illustrative examples of ongoing processes are given. Among possible applications, the controlling the spectra of signals, in particular, noise suppression by irradiating it with shocks or sawtooth waves can be mentioned. © 2018 Elsevier B.V.

  • 2.
    Rudenko, Oleg
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.
    Hedberg, Claes
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.
    Strong nonlinearity, anisotropy, and solitons in a lattice with holonomic constraints2019In: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 89, p. 104-115Article in journal (Refereed)
    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.

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