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Keywords = WKBJ

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25 pages, 736 KiB  
Article
Linear Stability Analysis of Relativistic Magnetized Jets: Methodology
by Nektarios Vlahakis
Universe 2023, 9(9), 386; https://doi.org/10.3390/universe9090386 - 26 Aug 2023
Cited by 4 | Viewed by 1340
Abstract
The stability of astrophysical jets in the linear regime is investigated by presenting a methodology to find the growth rates of the various instabilities. We perturb a cylindrical axisymmetric steady jet, linearize the relativistic ideal magnetohydrodynamic (MHD) equations, and analyze the evolution of [...] Read more.
The stability of astrophysical jets in the linear regime is investigated by presenting a methodology to find the growth rates of the various instabilities. We perturb a cylindrical axisymmetric steady jet, linearize the relativistic ideal magnetohydrodynamic (MHD) equations, and analyze the evolution of the eigenmodes of the perturbation by deriving the differential equations that need to be integrated, subject to the appropriate boundary conditions, in order to find the dispersion relation. We also apply the WKBJ approximation and, additionally, give analytical solutions in some subcases corresponding to unperturbed jets with constant bulk velocity along the symmetry axis. Full article
(This article belongs to the Section Compact Objects)
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14 pages, 363 KiB  
Article
Formal Derivations of Mode Coupling Equations in Underwater Acoustics: How the Method of Multiple Scales Results in an Expansion over Eigenfunctions and the Vectorized WKBJ Solution for the Amplitudes
by Mikhail Trofimov, Sergey Kozitskiy, Alena Zakharenko and Pavel Petrov
J. Mar. Sci. Eng. 2023, 11(4), 797; https://doi.org/10.3390/jmse11040797 - 7 Apr 2023
Cited by 6 | Viewed by 1842
Abstract
In this study formal derivation of mode coupling equations in underwater acoustics is revisited. This derivation is based on the method of multiple scales from which modal expansion of the field emerges, and the vectorized WKBJ equation for the coefficients in this expansion [...] Read more.
In this study formal derivation of mode coupling equations in underwater acoustics is revisited. This derivation is based on the method of multiple scales from which modal expansion of the field emerges, and the vectorized WKBJ equation for the coefficients in this expansion are obtained in an automatic way. Asymptotic analysis accomplished in this work also establishes a connection between coupled mode parabolic equations in three-dimensional case and the generalized WKBJ solution that emerges as its two-dimensional counterpart. Despite the fact that similar mode coupling equations can be found in literature, in our study a new systematic and formalized approach to their derivation is proposed. A theorem that guarantees asymptotic conservation of the energy flux in the considered two-dimensional waveguide is also proven. Full article
(This article belongs to the Special Issue Sound Scattering in the Ocean)
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