Abstract
In this chapter we sketch the basic mathematical notions of dynamical projection used in this book, starting from general relations and illustrating them by the simplest examples, following [1,2,3] and paying particular attention to the impact of inhomogeneities and accompanying effects. As mentioned in the introduction [see (1.1)], in the waveguide propagation, after expanding all the fields in series over the transverse coordinate basis, the coefficients \(\psi _k\) of the expansions will depend on the unique longitudinal space coordinate, say x, and time. Let \(\partial ={\partial }/{\partial x}\) denote the space derivative.
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References
S.B. Leble, A.A. Zaitsev, Novye Methody v Teorii Nelinejnych Voln [New Methods in Nonlinear Wave Theory] (in Russian) (Kaliningrad University Press, 1987)
S. Leble, General remarks on dynamic projection method. TASK Q. 20(2), 113–130 (2016)
S. Leble, A. Perelomova, The Dynamical Projector Method: Hydro and Electrodynamics (Taylor and Francis, New York, 2018)
A. Perelomova, Development of linear projecting in studies of non-linear flow. Acoustic heating induced by non-periodic sound. Phys. Lett. A 357, 42–47 (2006)
A. Perelomova, Driving force of acoustic streaming caused by aperiodic sound beam in unbounded volumes. Ultrasonics 49, 583–587 (2009)
A. Perelomova, Interaction of acoustic and thermal modes in the gas with nonequilibrium chemical reactions. Possibilities Acoust. Cool. Acta Acust. United Acust. 96, 43–48 (2010)
A. Perelomova, P. Wojda, Generation of the vorticity motion by sound in a chemically reacting gas. Invers. Acoust. Streaming Non-Equilib. Regime, Cent. Eur. J. Phys. 9(3), 740–750 (2011)
A. Perelomova, P. Wojda, Generation of the vorticity mode by sound in a vibrationally relaxing gas. Cent. Eur. J. Phys. 10(5), 1116–1124 (2012)
A. Perelomova, W. Pelc-Garska, Efficiency of acoustic heating produced in the thermoviscous flow of a fluid with relaxation. Cent. Eur. J. Phys. 8(6), 855–863 (2010)
A. Perelomova, Interaction of acoustic and thermal modes in the vibrationally relaxing gases. Acoust. Cool. Acta Phys. Pol. A 123(4), 681–687 (2013)
S. Leble, S. Vereshchagin, A wave diagnostics in geophysics: algorithmic extraction of atmosphere disturbance modes. Pure Appl. Geophys. 175(8), 3023–3035 (2018)
S. Leble, A. Perelomova, Problem of proper decomposition and initialization of acoustic and entropy modes in a gas affected by the mass force. Appl. Math. Model. 37(3), 629–635 (2013)
A. Perelomova, Nonlinear dynamics of vertically propagating acoustic waves in a stratified atmosphere. Acta Acust. 84(6), 1002–1006 (1998)
A. Perelomova, Nonlinear dynamics of directed accoustic waves in stratified and inhomogeneous liquids and gases with arbitrary equation of state. Arch. Acoust. 25, 451–463 (2000)
S. Leble, I. Vereshchagina, The method of dynamic projection operators in the theory of hyperbolic systems of partial differential equations with variable coefficients. arXiv:1403.7751
S.B. Leble, Waveguide Propagation of Nonlinear Waves in Stratified Media (in Russian), Leningrad University Press, 1988. Extended edn. in (Springer, Berlin 1990)
V.V. Belov, S.Y. Dobrohotov, T.Y. Tudorovskiy, Operator separation of variables for adiabatic problems in quantum and wave mechanics. J. Eng. Math. 55(1–4), 183–237 (2006)
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Leble, S. (2019). Evolution Operator and Projectors to Its Eigenspaces. In: Waveguide Propagation of Nonlinear Waves. Springer Series on Atomic, Optical, and Plasma Physics, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-030-22652-7_2
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